Metamath Proof Explorer

Theorem bdayfinbndlem1

Description: Lemma for bdayfinbnd . Show the first half of the inductive step. (Contributed by Scott Fenton, 26-Feb-2026)

Ref Expression
Hypotheses bdayfinbndlem.1 
|- ( ph -> N e. NN0_s )
bdayfinbndlem.2 
|- ( ph -> A. z e. No ( ( ( bday ` z ) C_ ( bday ` N ) /\ 0s <_s z ) -> ( z = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
Assertion bdayfinbndlem1 
|- ( ph -> A. w e. No ( ( ( bday ` w ) C_ ( bday ` ( N +s 1s ) ) /\ 0s <_s w ) -> ( w = ( N +s 1s ) \/ E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b

Proof

Step Hyp Ref Expression
1 bdayfinbndlem.1 
 |-  ( ph -> N e. NN0_s )
2 bdayfinbndlem.2 
 |-  ( ph -> A. z e. No ( ( ( bday ` z ) C_ ( bday ` N ) /\ 0s <_s z ) -> ( z = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
3 bdayn0p1 
 |-  ( N e. NN0_s -> ( bday ` ( N +s 1s ) ) = suc ( bday ` N ) )
4 1 3 syl 
 |-  ( ph -> ( bday ` ( N +s 1s ) ) = suc ( bday ` N ) )
5 4 adantr 
 |-  ( ( ph /\ w e. No ) -> ( bday ` ( N +s 1s ) ) = suc ( bday ` N ) )
6 5 sseq2d 
 |-  ( ( ph /\ w e. No ) -> ( ( bday ` w ) C_ ( bday ` ( N +s 1s ) ) <-> ( bday ` w ) C_ suc ( bday ` N ) ) )
7 bdayelon 
 |-  ( bday ` w ) e. On
8 bdayelon 
 |-  ( bday ` N ) e. On
9 8 onsuci 
 |-  suc ( bday ` N ) e. On
10 onsseleq 
 |-  ( ( ( bday ` w ) e. On /\ suc ( bday ` N ) e. On ) -> ( ( bday ` w ) C_ suc ( bday ` N ) <-> ( ( bday ` w ) e. suc ( bday ` N ) \/ ( bday ` w ) = suc ( bday ` N ) ) ) )
11 7 9 10 mp2an 
 |-  ( ( bday ` w ) C_ suc ( bday ` N ) <-> ( ( bday ` w ) e. suc ( bday ` N ) \/ ( bday ` w ) = suc ( bday ` N ) ) )
12 onsssuc 
 |-  ( ( ( bday ` w ) e. On /\ ( bday ` N ) e. On ) -> ( ( bday ` w ) C_ ( bday ` N ) <-> ( bday ` w ) e. suc ( bday ` N ) ) )
13 7 8 12 mp2an 
 |-  ( ( bday ` w ) C_ ( bday ` N ) <-> ( bday ` w ) e. suc ( bday ` N ) )
14 13 orbi1i 
 |-  ( ( ( bday ` w ) C_ ( bday ` N ) \/ ( bday ` w ) = suc ( bday ` N ) ) <-> ( ( bday ` w ) e. suc ( bday ` N ) \/ ( bday ` w ) = suc ( bday ` N ) ) )
15 14 bicomi 
 |-  ( ( ( bday ` w ) e. suc ( bday ` N ) \/ ( bday ` w ) = suc ( bday ` N ) ) <-> ( ( bday ` w ) C_ ( bday ` N ) \/ ( bday ` w ) = suc ( bday ` N ) ) )
16 11 15 bitri 
 |-  ( ( bday ` w ) C_ suc ( bday ` N ) <-> ( ( bday ` w ) C_ ( bday ` N ) \/ ( bday ` w ) = suc ( bday ` N ) ) )
17 fveq2 
 |-  ( z = w -> ( bday ` z ) = ( bday ` w ) )
18 17 sseq1d 
 |-  ( z = w -> ( ( bday ` z ) C_ ( bday ` N ) <-> ( bday ` w ) C_ ( bday ` N ) ) )
19 breq2 
 |-  ( z = w -> ( 0s <_s z <-> 0s <_s w ) )
20 18 19 anbi12d 
 |-  ( z = w -> ( ( ( bday ` z ) C_ ( bday ` N ) /\ 0s <_s z ) <-> ( ( bday ` w ) C_ ( bday ` N ) /\ 0s <_s w ) ) )
21 eqeq1 
 |-  ( z = w -> ( z = N <-> w = N ) )
22 eqeq1 
 |-  ( z = w -> ( z = ( x +s ( y /su ( 2s ^su p ) ) ) <-> w = ( x +s ( y /su ( 2s ^su p ) ) ) ) )
23 22 3anbi1d 
 |-  ( z = w -> ( ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
24 23 rexbidv 
 |-  ( z = w -> ( E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. p e. NN0_s ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
25 24 2rexbidv 
 |-  ( z = w -> ( E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
26 21 25 orbi12d 
 |-  ( z = w -> ( ( z = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  ( w = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
27 20 26 imbi12d 
 |-  ( z = w -> ( ( ( ( bday ` z ) C_ ( bday ` N ) /\ 0s <_s z ) -> ( z = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  ( ( ( bday ` w ) C_ ( bday ` N ) /\ 0s <_s w ) -> ( w = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
28 27 rspccva 
 |-  ( ( A. z e. No ( ( ( bday ` z ) C_ ( bday ` N ) /\ 0s <_s z ) -> ( z = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  ( ( ( bday ` w ) C_ ( bday ` N ) /\ 0s <_s w ) -> ( w = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
29 2 28 sylan 
 |-  ( ( ph /\ w e. No ) -> ( ( ( bday ` w ) C_ ( bday ` N ) /\ 0s <_s w ) -> ( w = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
30 1 adantr 
 |-  ( ( ph /\ ( w e. No /\ w = N ) ) -> N e. NN0_s )
31 0n0s 
 |-  0s e. NN0_s
32 31 a1i 
 |-  ( ( ph /\ ( w e. No /\ w = N ) ) -> 0s e. NN0_s )
33 simprr 
 |-  ( ( ph /\ ( w e. No /\ w = N ) ) -> w = N )
34 1 n0snod 
 |-  ( ph -> N e. No )
35 34 addsridd 
 |-  ( ph -> ( N +s 0s ) = N )
36 35 adantr 
 |-  ( ( ph /\ ( w e. No /\ w = N ) ) -> ( N +s 0s ) = N )
37 33 36 eqtr4d 
 |-  ( ( ph /\ ( w e. No /\ w = N ) ) -> w = ( N +s 0s ) )
38 0slt1s 
 |-  0s
39 38 a1i 
 |-  ( ( ph /\ ( w e. No /\ w = N ) ) -> 0s
40 34 sltp1d 
 |-  ( ph -> N
41 35 40 eqbrtrd 
 |-  ( ph -> ( N +s 0s )
42 41 adantr 
 |-  ( ( ph /\ ( w e. No /\ w = N ) ) -> ( N +s 0s )
43 oveq1 
 |-  ( a = N -> ( a +s ( b /su ( 2s ^su q ) ) ) = ( N +s ( b /su ( 2s ^su q ) ) ) )
44 43 eqeq2d 
 |-  ( a = N -> ( w = ( a +s ( b /su ( 2s ^su q ) ) ) <-> w = ( N +s ( b /su ( 2s ^su q ) ) ) ) )
45 oveq1 
 |-  ( a = N -> ( a +s q ) = ( N +s q ) )
46 45 breq1d 
 |-  ( a = N -> ( ( a +s q )  ( N +s q )
47 44 46 3anbi13d 
 |-  ( a = N -> ( ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b  ( w = ( N +s ( b /su ( 2s ^su q ) ) ) /\ b
48 oveq1 
 |-  ( b = 0s -> ( b /su ( 2s ^su q ) ) = ( 0s /su ( 2s ^su q ) ) )
49 48 oveq2d 
 |-  ( b = 0s -> ( N +s ( b /su ( 2s ^su q ) ) ) = ( N +s ( 0s /su ( 2s ^su q ) ) ) )
50 49 eqeq2d 
 |-  ( b = 0s -> ( w = ( N +s ( b /su ( 2s ^su q ) ) ) <-> w = ( N +s ( 0s /su ( 2s ^su q ) ) ) ) )
51 breq1 
 |-  ( b = 0s -> ( b  0s
52 50 51 3anbi12d 
 |-  ( b = 0s -> ( ( w = ( N +s ( b /su ( 2s ^su q ) ) ) /\ b  ( w = ( N +s ( 0s /su ( 2s ^su q ) ) ) /\ 0s
53 oveq2 
 |-  ( q = 0s -> ( 2s ^su q ) = ( 2s ^su 0s ) )
54 2sno 
 |-  2s e. No
55 exps0 
 |-  ( 2s e. No -> ( 2s ^su 0s ) = 1s )
56 54 55 ax-mp 
 |-  ( 2s ^su 0s ) = 1s
57 53 56 eqtrdi 
 |-  ( q = 0s -> ( 2s ^su q ) = 1s )
58 57 oveq2d 
 |-  ( q = 0s -> ( 0s /su ( 2s ^su q ) ) = ( 0s /su 1s ) )
59 0sno 
 |-  0s e. No
60 divs1 
 |-  ( 0s e. No -> ( 0s /su 1s ) = 0s )
61 59 60 ax-mp 
 |-  ( 0s /su 1s ) = 0s
62 58 61 eqtrdi 
 |-  ( q = 0s -> ( 0s /su ( 2s ^su q ) ) = 0s )
63 62 oveq2d 
 |-  ( q = 0s -> ( N +s ( 0s /su ( 2s ^su q ) ) ) = ( N +s 0s ) )
64 63 eqeq2d 
 |-  ( q = 0s -> ( w = ( N +s ( 0s /su ( 2s ^su q ) ) ) <-> w = ( N +s 0s ) ) )
65 57 breq2d 
 |-  ( q = 0s -> ( 0s  0s
66 oveq2 
 |-  ( q = 0s -> ( N +s q ) = ( N +s 0s ) )
67 66 breq1d 
 |-  ( q = 0s -> ( ( N +s q )  ( N +s 0s )
68 64 65 67 3anbi123d 
 |-  ( q = 0s -> ( ( w = ( N +s ( 0s /su ( 2s ^su q ) ) ) /\ 0s  ( w = ( N +s 0s ) /\ 0s
69 47 52 68 rspc3ev 
 |-  ( ( ( N e. NN0_s /\ 0s e. NN0_s /\ 0s e. NN0_s ) /\ ( w = ( N +s 0s ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
70 30 32 32 37 39 42 69 syl33anc 
 |-  ( ( ph /\ ( w e. No /\ w = N ) ) -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
71 70 expr 
 |-  ( ( ph /\ w e. No ) -> ( w = N -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
72 idd 
 |-  ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) -> ( w = ( x +s ( y /su ( 2s ^su p ) ) ) -> w = ( x +s ( y /su ( 2s ^su p ) ) ) ) )
73 idd 
 |-  ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) -> ( y  y
74 n0addscl 
 |-  ( ( x e. NN0_s /\ p e. NN0_s ) -> ( x +s p ) e. NN0_s )
75 74 n0snod 
 |-  ( ( x e. NN0_s /\ p e. NN0_s ) -> ( x +s p ) e. No )
76 75 3adant2 
 |-  ( ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) -> ( x +s p ) e. No )
77 76 adantl 
 |-  ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) -> ( x +s p ) e. No )
78 77 adantr 
 |-  ( ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) /\ ( x +s p )  ( x +s p ) e. No )
79 34 adantr 
 |-  ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) -> N e. No )
80 79 adantr 
 |-  ( ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) /\ ( x +s p )  N e. No )
81 peano2no 
 |-  ( N e. No -> ( N +s 1s ) e. No )
82 80 81 syl 
 |-  ( ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) /\ ( x +s p )  ( N +s 1s ) e. No )
83 simpr 
 |-  ( ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) /\ ( x +s p )  ( x +s p )
84 79 sltp1d 
 |-  ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) -> N
85 84 adantr 
 |-  ( ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) /\ ( x +s p )  N
86 78 80 82 83 85 slttrd 
 |-  ( ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) /\ ( x +s p )  ( x +s p )
87 86 ex 
 |-  ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) -> ( ( x +s p )  ( x +s p )
88 72 73 87 3anim123d 
 |-  ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) -> ( ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
89 oveq1 
 |-  ( a = x -> ( a +s ( b /su ( 2s ^su q ) ) ) = ( x +s ( b /su ( 2s ^su q ) ) ) )
90 89 eqeq2d 
 |-  ( a = x -> ( w = ( a +s ( b /su ( 2s ^su q ) ) ) <-> w = ( x +s ( b /su ( 2s ^su q ) ) ) ) )
91 oveq1 
 |-  ( a = x -> ( a +s q ) = ( x +s q ) )
92 91 breq1d 
 |-  ( a = x -> ( ( a +s q )  ( x +s q )
93 90 92 3anbi13d 
 |-  ( a = x -> ( ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b  ( w = ( x +s ( b /su ( 2s ^su q ) ) ) /\ b
94 oveq1 
 |-  ( b = y -> ( b /su ( 2s ^su q ) ) = ( y /su ( 2s ^su q ) ) )
95 94 oveq2d 
 |-  ( b = y -> ( x +s ( b /su ( 2s ^su q ) ) ) = ( x +s ( y /su ( 2s ^su q ) ) ) )
96 95 eqeq2d 
 |-  ( b = y -> ( w = ( x +s ( b /su ( 2s ^su q ) ) ) <-> w = ( x +s ( y /su ( 2s ^su q ) ) ) ) )
97 breq1 
 |-  ( b = y -> ( b  y
98 96 97 3anbi12d 
 |-  ( b = y -> ( ( w = ( x +s ( b /su ( 2s ^su q ) ) ) /\ b  ( w = ( x +s ( y /su ( 2s ^su q ) ) ) /\ y
99 oveq2 
 |-  ( q = p -> ( 2s ^su q ) = ( 2s ^su p ) )
100 99 oveq2d 
 |-  ( q = p -> ( y /su ( 2s ^su q ) ) = ( y /su ( 2s ^su p ) ) )
101 100 oveq2d 
 |-  ( q = p -> ( x +s ( y /su ( 2s ^su q ) ) ) = ( x +s ( y /su ( 2s ^su p ) ) ) )
102 101 eqeq2d 
 |-  ( q = p -> ( w = ( x +s ( y /su ( 2s ^su q ) ) ) <-> w = ( x +s ( y /su ( 2s ^su p ) ) ) ) )
103 99 breq2d 
 |-  ( q = p -> ( y  y
104 oveq2 
 |-  ( q = p -> ( x +s q ) = ( x +s p ) )
105 104 breq1d 
 |-  ( q = p -> ( ( x +s q )  ( x +s p )
106 102 103 105 3anbi123d 
 |-  ( q = p -> ( ( w = ( x +s ( y /su ( 2s ^su q ) ) ) /\ y  ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
107 93 98 106 rspc3ev 
 |-  ( ( ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) /\ ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
108 107 ex 
 |-  ( ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) -> ( ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
109 108 adantl 
 |-  ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) -> ( ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
110 88 109 syld 
 |-  ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) -> ( ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
111 110 rexlimdvvva 
 |-  ( ph -> ( E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
112 111 adantr 
 |-  ( ( ph /\ w e. No ) -> ( E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
113 71 112 jaod 
 |-  ( ( ph /\ w e. No ) -> ( ( w = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
114 29 113 syld 
 |-  ( ( ph /\ w e. No ) -> ( ( ( bday ` w ) C_ ( bday ` N ) /\ 0s <_s w ) -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
115 114 impr 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) C_ ( bday ` N ) /\ 0s <_s w ) ) ) -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
116 115 olcd 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) C_ ( bday ` N ) /\ 0s <_s w ) ) ) -> ( w = ( N +s 1s ) \/ E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
117 116 expr 
 |-  ( ( ph /\ w e. No ) -> ( ( ( bday ` w ) C_ ( bday ` N ) /\ 0s <_s w ) -> ( w = ( N +s 1s ) \/ E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
118 117 expd 
 |-  ( ( ph /\ w e. No ) -> ( ( bday ` w ) C_ ( bday ` N ) -> ( 0s <_s w -> ( w = ( N +s 1s ) \/ E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
119 5 eqeq2d 
 |-  ( ( ph /\ w e. No ) -> ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) <-> ( bday ` w ) = suc ( bday ` N ) ) )
120 df-ne 
 |-  ( w =/= ( N +s 1s ) <-> -. w = ( N +s 1s ) )
121 simprl 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) -> w e. No )
122 sleloe 
 |-  ( ( 0s e. No /\ w e. No ) -> ( 0s <_s w <-> ( 0s
123 59 121 122 sylancr 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) -> ( 0s <_s w <-> ( 0s
124 simplrl 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w e. No )
125 simpr 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s
126 124 125 0elleft 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s e. ( _Left ` w ) )
127 126 ne0d 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( _Left ` w ) =/= (/) )
128 simprrl 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) -> ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
129 128 adantr 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
130 peano2n0s 
 |-  ( N e. NN0_s -> ( N +s 1s ) e. NN0_s )
131 1 130 syl 
 |-  ( ph -> ( N +s 1s ) e. NN0_s )
132 n0sbday 
 |-  ( ( N +s 1s ) e. NN0_s -> ( bday ` ( N +s 1s ) ) e. _om )
133 131 132 syl 
 |-  ( ph -> ( bday ` ( N +s 1s ) ) e. _om )
134 133 adantr 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) -> ( bday ` ( N +s 1s ) ) e. _om )
135 134 adantr 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` ( N +s 1s ) ) e. _om )
136 129 135 eqeltrd 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) e. _om )
137 oldfi 
 |-  ( ( bday ` w ) e. _om -> ( _Old ` ( bday ` w ) ) e. Fin )
138 136 137 syl 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( _Old ` ( bday ` w ) ) e. Fin )
139 leftssold 
 |-  ( _Left ` w ) C_ ( _Old ` ( bday ` w ) )
140 ssfi 
 |-  ( ( ( _Old ` ( bday ` w ) ) e. Fin /\ ( _Left ` w ) C_ ( _Old ` ( bday ` w ) ) ) -> ( _Left ` w ) e. Fin )
141 138 139 140 sylancl 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( _Left ` w ) e. Fin )
142 leftssno 
 |-  ( _Left ` w ) C_ No
143 sltso 
 |-
144 soss 
 |-  ( ( _Left ` w ) C_ No -> (
145 142 143 144 mp2 
 |-
146 fimax2g 
 |-  ( (  E. c e. ( _Left ` w ) A. e e. ( _Left ` w ) -. c
147 145 146 mp3an1 
 |-  ( ( ( _Left ` w ) e. Fin /\ ( _Left ` w ) =/= (/) ) -> E. c e. ( _Left ` w ) A. e e. ( _Left ` w ) -. c
148 142 sseli 
 |-  ( e e. ( _Left ` w ) -> e e. No )
149 142 sseli 
 |-  ( c e. ( _Left ` w ) -> c e. No )
150 slenlt 
 |-  ( ( e e. No /\ c e. No ) -> ( e <_s c <-> -. c
151 148 149 150 syl2an 
 |-  ( ( e e. ( _Left ` w ) /\ c e. ( _Left ` w ) ) -> ( e <_s c <-> -. c
152 151 ancoms 
 |-  ( ( c e. ( _Left ` w ) /\ e e. ( _Left ` w ) ) -> ( e <_s c <-> -. c
153 152 ralbidva 
 |-  ( c e. ( _Left ` w ) -> ( A. e e. ( _Left ` w ) e <_s c <-> A. e e. ( _Left ` w ) -. c
154 153 adantl 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( A. e e. ( _Left ` w ) e <_s c <-> A. e e. ( _Left ` w ) -. c
155 rightssold 
 |-  ( _Right ` w ) C_ ( _Old ` ( bday ` w ) )
156 ssfi 
 |-  ( ( ( _Old ` ( bday ` w ) ) e. Fin /\ ( _Right ` w ) C_ ( _Old ` ( bday ` w ) ) ) -> ( _Right ` w ) e. Fin )
157 138 155 156 sylancl 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( _Right ` w ) e. Fin )
158 157 adantr 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( _Right ` w ) e. Fin )
159 124 adantr 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w e. No )
160 simprrr 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) -> w =/= ( N +s 1s ) )
161 160 adantr 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w =/= ( N +s 1s ) )
162 161 adantr 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w =/= ( N +s 1s ) )
163 162 neneqd 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. w = ( N +s 1s ) )
164 simpr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w e. On_s )
165 131 ad4antr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) e. NN0_s )
166 n0ons 
 |-  ( ( N +s 1s ) e. NN0_s -> ( N +s 1s ) e. On_s )
167 165 166 syl 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) e. On_s )
168 129 adantr 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
169 168 adantr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
170 bday11on 
 |-  ( ( w e. On_s /\ ( N +s 1s ) e. On_s /\ ( bday ` w ) = ( bday ` ( N +s 1s ) ) ) -> w = ( N +s 1s ) )
171 164 167 169 170 syl3anc 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( N +s 1s ) )
172 163 171 mtand 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. w e. On_s )
173 elons 
 |-  ( w e. On_s <-> ( w e. No /\ ( _Right ` w ) = (/) ) )
174 173 notbii 
 |-  ( -. w e. On_s <-> -. ( w e. No /\ ( _Right ` w ) = (/) ) )
175 imnan 
 |-  ( ( w e. No -> -. ( _Right ` w ) = (/) ) <-> -. ( w e. No /\ ( _Right ` w ) = (/) ) )
176 174 175 bitr4i 
 |-  ( -. w e. On_s <-> ( w e. No -> -. ( _Right ` w ) = (/) ) )
177 172 176 sylib 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( w e. No -> -. ( _Right ` w ) = (/) ) )
178 159 177 mpd 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. ( _Right ` w ) = (/) )
179 178 neqned 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( _Right ` w ) =/= (/) )
180 rightssno 
 |-  ( _Right ` w ) C_ No
181 soss 
 |-  ( ( _Right ` w ) C_ No -> (
182 180 143 181 mp2 
 |-
183 fimin2g 
 |-  ( (  E. d e. ( _Right ` w ) A. f e. ( _Right ` w ) -. f
184 182 183 mp3an1 
 |-  ( ( ( _Right ` w ) e. Fin /\ ( _Right ` w ) =/= (/) ) -> E. d e. ( _Right ` w ) A. f e. ( _Right ` w ) -. f
185 180 sseli 
 |-  ( d e. ( _Right ` w ) -> d e. No )
186 180 sseli 
 |-  ( f e. ( _Right ` w ) -> f e. No )
187 slenlt 
 |-  ( ( d e. No /\ f e. No ) -> ( d <_s f <-> -. f
188 185 186 187 syl2an 
 |-  ( ( d e. ( _Right ` w ) /\ f e. ( _Right ` w ) ) -> ( d <_s f <-> -. f
189 188 ralbidva 
 |-  ( d e. ( _Right ` w ) -> ( A. f e. ( _Right ` w ) d <_s f <-> A. f e. ( _Right ` w ) -. f
190 189 adantl 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( A. f e. ( _Right ` w ) d <_s f <-> A. f e. ( _Right ` w ) -. f
191 simp2l 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. ( _Left ` w ) )
192 simp2r 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  A. e e. ( _Left ` w ) e <_s c )
193 simp3l 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. ( _Right ` w ) )
194 simp3r 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  A. f e. ( _Right ` w ) d <_s f )
195 191 192 193 194 cutminmax 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
196 simpl2l 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. ( _Left ` w ) )
197 139 196 sselid 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. ( _Old ` ( bday ` w ) ) )
198 142 196 sselid 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
199 oldbday 
 |-  ( ( ( bday ` w ) e. On /\ c e. No ) -> ( c e. ( _Old ` ( bday ` w ) ) <-> ( bday ` c ) e. ( bday ` w ) ) )
200 7 198 199 sylancr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( c e. ( _Old ` ( bday ` w ) ) <-> ( bday ` c ) e. ( bday ` w ) ) )
201 197 200 mpbid 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` c ) e. ( bday ` w ) )
202 129 3ad2ant1 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
203 202 adantr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
204 1 adantr 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) -> N e. NN0_s )
205 204 adantr 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. NN0_s )
206 205 3ad2ant1 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. NN0_s )
207 206 adantr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. NN0_s )
208 207 3 syl 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` ( N +s 1s ) ) = suc ( bday ` N ) )
209 203 208 eqtrd 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = suc ( bday ` N ) )
210 201 209 eleqtrd 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` c ) e. suc ( bday ` N ) )
211 bdayelon 
 |-  ( bday ` c ) e. On
212 onsssuc 
 |-  ( ( ( bday ` c ) e. On /\ ( bday ` N ) e. On ) -> ( ( bday ` c ) C_ ( bday ` N ) <-> ( bday ` c ) e. suc ( bday ` N ) ) )
213 211 8 212 mp2an 
 |-  ( ( bday ` c ) C_ ( bday ` N ) <-> ( bday ` c ) e. suc ( bday ` N ) )
214 210 213 sylibr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` c ) C_ ( bday ` N ) )
215 breq1 
 |-  ( e = 0s -> ( e <_s c <-> 0s <_s c ) )
216 simpl2r 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  A. e e. ( _Left ` w ) e <_s c )
217 simpl1 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s
218 217 126 syl 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s e. ( _Left ` w ) )
219 215 216 218 rspcdva 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s <_s c )
220 fveq2 
 |-  ( z = c -> ( bday ` z ) = ( bday ` c ) )
221 220 sseq1d 
 |-  ( z = c -> ( ( bday ` z ) C_ ( bday ` N ) <-> ( bday ` c ) C_ ( bday ` N ) ) )
222 breq2 
 |-  ( z = c -> ( 0s <_s z <-> 0s <_s c ) )
223 221 222 anbi12d 
 |-  ( z = c -> ( ( ( bday ` z ) C_ ( bday ` N ) /\ 0s <_s z ) <-> ( ( bday ` c ) C_ ( bday ` N ) /\ 0s <_s c ) ) )
224 eqeq1 
 |-  ( z = c -> ( z = N <-> c = N ) )
225 eqeq1 
 |-  ( z = c -> ( z = ( x +s ( y /su ( 2s ^su p ) ) ) <-> c = ( x +s ( y /su ( 2s ^su p ) ) ) ) )
226 225 3anbi1d 
 |-  ( z = c -> ( ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  ( c = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
227 226 rexbidv 
 |-  ( z = c -> ( E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. p e. NN0_s ( c = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
228 227 2rexbidv 
 |-  ( z = c -> ( E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( c = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
229 oveq1 
 |-  ( x = g -> ( x +s ( y /su ( 2s ^su p ) ) ) = ( g +s ( y /su ( 2s ^su p ) ) ) )
230 229 eqeq2d 
 |-  ( x = g -> ( c = ( x +s ( y /su ( 2s ^su p ) ) ) <-> c = ( g +s ( y /su ( 2s ^su p ) ) ) ) )
231 oveq1 
 |-  ( x = g -> ( x +s p ) = ( g +s p ) )
232 231 breq1d 
 |-  ( x = g -> ( ( x +s p )  ( g +s p )
233 230 232 3anbi13d 
 |-  ( x = g -> ( ( c = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  ( c = ( g +s ( y /su ( 2s ^su p ) ) ) /\ y
234 233 rexbidv 
 |-  ( x = g -> ( E. p e. NN0_s ( c = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. p e. NN0_s ( c = ( g +s ( y /su ( 2s ^su p ) ) ) /\ y
235 oveq1 
 |-  ( y = h -> ( y /su ( 2s ^su p ) ) = ( h /su ( 2s ^su p ) ) )
236 235 oveq2d 
 |-  ( y = h -> ( g +s ( y /su ( 2s ^su p ) ) ) = ( g +s ( h /su ( 2s ^su p ) ) ) )
237 236 eqeq2d 
 |-  ( y = h -> ( c = ( g +s ( y /su ( 2s ^su p ) ) ) <-> c = ( g +s ( h /su ( 2s ^su p ) ) ) ) )
238 breq1 
 |-  ( y = h -> ( y  h
239 237 238 3anbi12d 
 |-  ( y = h -> ( ( c = ( g +s ( y /su ( 2s ^su p ) ) ) /\ y  ( c = ( g +s ( h /su ( 2s ^su p ) ) ) /\ h
240 239 rexbidv 
 |-  ( y = h -> ( E. p e. NN0_s ( c = ( g +s ( y /su ( 2s ^su p ) ) ) /\ y  E. p e. NN0_s ( c = ( g +s ( h /su ( 2s ^su p ) ) ) /\ h
241 oveq2 
 |-  ( p = i -> ( 2s ^su p ) = ( 2s ^su i ) )
242 241 oveq2d 
 |-  ( p = i -> ( h /su ( 2s ^su p ) ) = ( h /su ( 2s ^su i ) ) )
243 242 oveq2d 
 |-  ( p = i -> ( g +s ( h /su ( 2s ^su p ) ) ) = ( g +s ( h /su ( 2s ^su i ) ) ) )
244 243 eqeq2d 
 |-  ( p = i -> ( c = ( g +s ( h /su ( 2s ^su p ) ) ) <-> c = ( g +s ( h /su ( 2s ^su i ) ) ) ) )
245 241 breq2d 
 |-  ( p = i -> ( h  h
246 oveq2 
 |-  ( p = i -> ( g +s p ) = ( g +s i ) )
247 246 breq1d 
 |-  ( p = i -> ( ( g +s p )  ( g +s i )
248 244 245 247 3anbi123d 
 |-  ( p = i -> ( ( c = ( g +s ( h /su ( 2s ^su p ) ) ) /\ h  ( c = ( g +s ( h /su ( 2s ^su i ) ) ) /\ h
249 248 cbvrexvw 
 |-  ( E. p e. NN0_s ( c = ( g +s ( h /su ( 2s ^su p ) ) ) /\ h  E. i e. NN0_s ( c = ( g +s ( h /su ( 2s ^su i ) ) ) /\ h
250 240 249 bitrdi 
 |-  ( y = h -> ( E. p e. NN0_s ( c = ( g +s ( y /su ( 2s ^su p ) ) ) /\ y  E. i e. NN0_s ( c = ( g +s ( h /su ( 2s ^su i ) ) ) /\ h
251 234 250 cbvrex2vw 
 |-  ( E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( c = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. g e. NN0_s E. h e. NN0_s E. i e. NN0_s ( c = ( g +s ( h /su ( 2s ^su i ) ) ) /\ h
252 228 251 bitrdi 
 |-  ( z = c -> ( E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. g e. NN0_s E. h e. NN0_s E. i e. NN0_s ( c = ( g +s ( h /su ( 2s ^su i ) ) ) /\ h
253 224 252 orbi12d 
 |-  ( z = c -> ( ( z = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  ( c = N \/ E. g e. NN0_s E. h e. NN0_s E. i e. NN0_s ( c = ( g +s ( h /su ( 2s ^su i ) ) ) /\ h
254 223 253 imbi12d 
 |-  ( z = c -> ( ( ( ( bday ` z ) C_ ( bday ` N ) /\ 0s <_s z ) -> ( z = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  ( ( ( bday ` c ) C_ ( bday ` N ) /\ 0s <_s c ) -> ( c = N \/ E. g e. NN0_s E. h e. NN0_s E. i e. NN0_s ( c = ( g +s ( h /su ( 2s ^su i ) ) ) /\ h
255 2 adantr 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) -> A. z e. No ( ( ( bday ` z ) C_ ( bday ` N ) /\ 0s <_s z ) -> ( z = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
256 255 adantr 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  A. z e. No ( ( ( bday ` z ) C_ ( bday ` N ) /\ 0s <_s z ) -> ( z = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
257 256 3ad2ant1 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  A. z e. No ( ( ( bday ` z ) C_ ( bday ` N ) /\ 0s <_s z ) -> ( z = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
258 257 adantr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  A. z e. No ( ( ( bday ` z ) C_ ( bday ` N ) /\ 0s <_s z ) -> ( z = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
259 254 258 198 rspcdva 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( bday ` c ) C_ ( bday ` N ) /\ 0s <_s c ) -> ( c = N \/ E. g e. NN0_s E. h e. NN0_s E. i e. NN0_s ( c = ( g +s ( h /su ( 2s ^su i ) ) ) /\ h
260 simp1ll 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
261 260 adantr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
262 n0ons 
 |-  ( N e. NN0_s -> N e. On_s )
263 1 262 syl 
 |-  ( ph -> N e. On_s )
264 261 263 syl 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. On_s )
265 simpl3l 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. ( _Right ` w ) )
266 155 265 sselid 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. ( _Old ` ( bday ` w ) ) )
267 oldbdayim 
 |-  ( d e. ( _Old ` ( bday ` w ) ) -> ( bday ` d ) e. ( bday ` w ) )
268 266 267 syl 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` d ) e. ( bday ` w ) )
269 268 209 eleqtrd 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` d ) e. suc ( bday ` N ) )
270 bdayelon 
 |-  ( bday ` d ) e. On
271 onsssuc 
 |-  ( ( ( bday ` d ) e. On /\ ( bday ` N ) e. On ) -> ( ( bday ` d ) C_ ( bday ` N ) <-> ( bday ` d ) e. suc ( bday ` N ) ) )
272 270 8 271 mp2an 
 |-  ( ( bday ` d ) C_ ( bday ` N ) <-> ( bday ` d ) e. suc ( bday ` N ) )
273 269 272 sylibr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` d ) C_ ( bday ` N ) )
274 180 265 sselid 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
275 madebday 
 |-  ( ( ( bday ` N ) e. On /\ d e. No ) -> ( d e. ( _Made ` ( bday ` N ) ) <-> ( bday ` d ) C_ ( bday ` N ) ) )
276 8 274 275 sylancr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( d e. ( _Made ` ( bday ` N ) ) <-> ( bday ` d ) C_ ( bday ` N ) ) )
277 273 276 mpbird 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. ( _Made ` ( bday ` N ) ) )
278 onsbnd 
 |-  ( ( N e. On_s /\ d e. ( _Made ` ( bday ` N ) ) ) -> d <_s N )
279 264 277 278 syl2anc 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d <_s N )
280 207 n0snod 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. No )
281 274 280 slenltd 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( d <_s N <-> -. N
282 279 281 mpbid 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. N
283 lltropt 
 |-  ( _Left ` w ) <
284 283 a1i 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( _Left ` w ) <
285 284 191 193 ssltsepcd 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
286 285 adantr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
287 breq1 
 |-  ( c = N -> ( c  N
288 287 bicomd 
 |-  ( c = N -> ( N  c
289 286 288 syl5ibrcom 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( c = N -> N
290 282 289 mtod 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. c = N )
291 orel1 
 |-  ( -. c = N -> ( ( c = N \/ E. g e. NN0_s E. h e. NN0_s E. i e. NN0_s ( c = ( g +s ( h /su ( 2s ^su i ) ) ) /\ h  E. g e. NN0_s E. h e. NN0_s E. i e. NN0_s ( c = ( g +s ( h /su ( 2s ^su i ) ) ) /\ h
292 290 291 syl 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( c = N \/ E. g e. NN0_s E. h e. NN0_s E. i e. NN0_s ( c = ( g +s ( h /su ( 2s ^su i ) ) ) /\ h  E. g e. NN0_s E. h e. NN0_s E. i e. NN0_s ( c = ( g +s ( h /su ( 2s ^su i ) ) ) /\ h
293 259 292 syld 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( bday ` c ) C_ ( bday ` N ) /\ 0s <_s c ) -> E. g e. NN0_s E. h e. NN0_s E. i e. NN0_s ( c = ( g +s ( h /su ( 2s ^su i ) ) ) /\ h
294 simp3l1 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
295 294 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
296 295 n0snod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
297 simp3l3 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
298 297 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
299 n0addscl 
 |-  ( ( g e. NN0_s /\ i e. NN0_s ) -> ( g +s i ) e. NN0_s )
300 295 298 299 syl2anc 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i ) e. NN0_s )
301 300 n0snod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i ) e. No )
302 260 3ad2ant1 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
303 302 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
304 303 34 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. No )
305 n0sge0 
 |-  ( i e. NN0_s -> 0s <_s i )
306 298 305 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s <_s i )
307 298 n0snod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. No )
308 296 307 addsge01d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 0s <_s i <-> g <_s ( g +s i ) ) )
309 306 308 mpbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g <_s ( g +s i ) )
310 simp3r3 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i )
311 310 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i )
312 296 301 304 309 311 slelttrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g
313 303 1 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. NN0_s )
314 n0sltp1le 
 |-  ( ( g e. NN0_s /\ N e. NN0_s ) -> ( g  ( g +s 1s ) <_s N ) )
315 295 313 314 syl2anc 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g  ( g +s 1s ) <_s N ) )
316 312 315 mpbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) <_s N )
317 sltirr 
 |-  ( N e. No -> -. N
318 304 317 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. N
319 1n0s 
 |-  1s e. NN0_s
320 n0addscl 
 |-  ( ( g e. NN0_s /\ 1s e. NN0_s ) -> ( g +s 1s ) e. NN0_s )
321 295 319 320 sylancl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. NN0_s )
322 321 n0snod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. No )
323 322 304 sltnled 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s )  -. N <_s ( g +s 1s ) ) )
324 302 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
325 131 n0snod 
 |-  ( ph -> ( N +s 1s ) e. No )
326 324 325 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) e. No )
327 324 34 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. No )
328 54 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  2s e. No )
329 327 328 subscld 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N -s 2s ) e. No )
330 1sno 
 |-  1s e. No
331 330 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. No )
332 329 331 331 addsassd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( N -s 2s ) +s 1s ) +s 1s ) = ( ( N -s 2s ) +s ( 1s +s 1s ) ) )
333 1p1e2s 
 |-  ( 1s +s 1s ) = 2s
334 333 oveq2i 
 |-  ( ( N -s 2s ) +s ( 1s +s 1s ) ) = ( ( N -s 2s ) +s 2s )
335 npcans 
 |-  ( ( N e. No /\ 2s e. No ) -> ( ( N -s 2s ) +s 2s ) = N )
336 327 54 335 sylancl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 2s ) +s 2s ) = N )
337 334 336 eqtrid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 2s ) +s ( 1s +s 1s ) ) = N )
338 332 337 eqtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( N -s 2s ) +s 1s ) +s 1s ) = N )
339 338 327 eqeltrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( N -s 2s ) +s 1s ) +s 1s ) e. No )
340 329 331 addscld 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 2s ) +s 1s ) e. No )
341 202 3ad2ant1 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
342 341 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
343 simpl2 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
344 142 191 sselid 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
345 344 3ad2ant1 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
346 345 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
347 294 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
348 peano2n0s 
 |-  ( g e. NN0_s -> ( g +s 1s ) e. NN0_s )
349 347 348 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. NN0_s )
350 349 n0snod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. No )
351 subscl 
 |-  ( ( N e. No /\ 1s e. No ) -> ( N -s 1s ) e. No )
352 34 330 351 sylancl 
 |-  ( ph -> ( N -s 1s ) e. No )
353 324 352 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N -s 1s ) e. No )
354 simp3r1 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
355 354 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
356 simp3r2 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
357 356 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
358 297 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
359 expscl 
 |-  ( ( 2s e. No /\ i e. NN0_s ) -> ( 2s ^su i ) e. No )
360 54 358 359 sylancr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) e. No )
361 360 mulslidd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 1s x.s ( 2s ^su i ) ) = ( 2s ^su i ) )
362 357 361 breqtrrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
363 simp3l2 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
364 363 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
365 364 n0snod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. No )
366 365 331 358 pw2sltdivmul2d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h /su ( 2s ^su i ) )  h
367 362 366 mpbird 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) )
368 365 358 pw2divscld 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) ) e. No )
369 347 n0snod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
370 368 331 369 sltadd2d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h /su ( 2s ^su i ) )  ( g +s ( h /su ( 2s ^su i ) ) )
371 367 370 mpbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( h /su ( 2s ^su i ) ) )
372 355 371 eqbrtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
373 n0sltp1le 
 |-  ( ( ( g +s 1s ) e. NN0_s /\ N e. NN0_s ) -> ( ( g +s 1s )  ( ( g +s 1s ) +s 1s ) <_s N ) )
374 321 313 373 syl2anc 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s )  ( ( g +s 1s ) +s 1s ) <_s N ) )
375 374 biimpd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s )  ( ( g +s 1s ) +s 1s ) <_s N ) )
376 375 impr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s ) +s 1s ) <_s N )
377 npcans 
 |-  ( ( N e. No /\ 1s e. No ) -> ( ( N -s 1s ) +s 1s ) = N )
378 327 330 377 sylancl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 1s ) +s 1s ) = N )
379 376 378 breqtrrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s ) +s 1s ) <_s ( ( N -s 1s ) +s 1s ) )
380 350 353 331 sleadd1d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s ) <_s ( N -s 1s ) <-> ( ( g +s 1s ) +s 1s ) <_s ( ( N -s 1s ) +s 1s ) ) )
381 379 380 mpbird 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) <_s ( N -s 1s ) )
382 346 350 353 372 381 sltletrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
383 333 oveq2i 
 |-  ( N -s ( 1s +s 1s ) ) = ( N -s 2s )
384 383 oveq1i 
 |-  ( ( N -s ( 1s +s 1s ) ) +s 1s ) = ( ( N -s 2s ) +s 1s )
385 327 331 331 subsubs4d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 1s ) -s 1s ) = ( N -s ( 1s +s 1s ) ) )
386 385 oveq1d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( N -s 1s ) -s 1s ) +s 1s ) = ( ( N -s ( 1s +s 1s ) ) +s 1s ) )
387 npcans 
 |-  ( ( ( N -s 1s ) e. No /\ 1s e. No ) -> ( ( ( N -s 1s ) -s 1s ) +s 1s ) = ( N -s 1s ) )
388 353 330 387 sylancl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( N -s 1s ) -s 1s ) +s 1s ) = ( N -s 1s ) )
389 386 388 eqtr3d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s ( 1s +s 1s ) ) +s 1s ) = ( N -s 1s ) )
390 384 389 eqtr3id 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 2s ) +s 1s ) = ( N -s 1s ) )
391 382 390 breqtrrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
392 346 340 391 ssltsn 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { c } <
393 180 193 sselid 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
394 393 3ad2ant1 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
395 394 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
396 simprl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = N )
397 396 oveq1d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( d -s 1s ) = ( N -s 1s ) )
398 397 eqcomd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N -s 1s ) = ( d -s 1s ) )
399 395 sltm1d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( d -s 1s )
400 398 399 eqbrtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N -s 1s )
401 390 400 eqbrtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 2s ) +s 1s )
402 340 395 401 ssltsn 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { ( ( N -s 2s ) +s 1s ) } <
403 343 340 392 402 ssltbday 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) C_ ( bday ` ( ( N -s 2s ) +s 1s ) ) )
404 342 403 eqsstrrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` ( N +s 1s ) ) C_ ( bday ` ( ( N -s 2s ) +s 1s ) ) )
405 131 166 syl 
 |-  ( ph -> ( N +s 1s ) e. On_s )
406 324 405 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) e. On_s )
407 327 331 328 addsubsd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N +s 1s ) -s 2s ) = ( ( N -s 2s ) +s 1s ) )
408 n0sge0 
 |-  ( g e. NN0_s -> 0s <_s g )
409 347 408 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s <_s g )
410 331 369 addsge01d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 0s <_s g <-> 1s <_s ( 1s +s g ) ) )
411 409 410 mpbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s <_s ( 1s +s g ) )
412 369 331 addscomd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) = ( 1s +s g ) )
413 411 412 breqtrrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s <_s ( g +s 1s ) )
414 simprr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s )
415 331 350 327 413 414 slelttrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s
416 331 327 415 sltled 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s <_s N )
417 331 327 331 sleadd1d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 1s <_s N <-> ( 1s +s 1s ) <_s ( N +s 1s ) ) )
418 416 417 mpbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 1s +s 1s ) <_s ( N +s 1s ) )
419 333 418 eqbrtrrid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  2s <_s ( N +s 1s ) )
420 2nns 
 |-  2s e. NN_s
421 nnn0s 
 |-  ( 2s e. NN_s -> 2s e. NN0_s )
422 420 421 ax-mp 
 |-  2s e. NN0_s
423 422 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  2s e. NN0_s )
424 302 131 syl 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) e. NN0_s )
425 424 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) e. NN0_s )
426 n0subs 
 |-  ( ( 2s e. NN0_s /\ ( N +s 1s ) e. NN0_s ) -> ( 2s <_s ( N +s 1s ) <-> ( ( N +s 1s ) -s 2s ) e. NN0_s ) )
427 423 425 426 syl2anc 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s <_s ( N +s 1s ) <-> ( ( N +s 1s ) -s 2s ) e. NN0_s ) )
428 419 427 mpbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N +s 1s ) -s 2s ) e. NN0_s )
429 407 428 eqeltrrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 2s ) +s 1s ) e. NN0_s )
430 n0ons 
 |-  ( ( ( N -s 2s ) +s 1s ) e. NN0_s -> ( ( N -s 2s ) +s 1s ) e. On_s )
431 429 430 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 2s ) +s 1s ) e. On_s )
432 406 431 onsled 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N +s 1s ) <_s ( ( N -s 2s ) +s 1s ) <-> ( bday ` ( N +s 1s ) ) C_ ( bday ` ( ( N -s 2s ) +s 1s ) ) ) )
433 404 432 mpbird 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) <_s ( ( N -s 2s ) +s 1s ) )
434 340 sltp1d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 2s ) +s 1s )
435 326 340 339 433 434 slelttrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s )
436 326 339 435 sltled 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) <_s ( ( ( N -s 2s ) +s 1s ) +s 1s ) )
437 436 338 breqtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) <_s N )
438 324 1 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. NN0_s )
439 n0sltp1le 
 |-  ( ( N e. NN0_s /\ N e. NN0_s ) -> ( N  ( N +s 1s ) <_s N ) )
440 438 438 439 syl2anc 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N  ( N +s 1s ) <_s N ) )
441 437 440 mpbird 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N
442 441 expr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s )  N
443 323 442 sylbird 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( -. N <_s ( g +s 1s ) -> N
444 318 443 mt3d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N <_s ( g +s 1s ) )
445 444 a1d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s ) <_s N -> N <_s ( g +s 1s ) ) )
446 316 445 jcai 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s ) <_s N /\ N <_s ( g +s 1s ) ) )
447 322 304 sletri3d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s ) = N <-> ( ( g +s 1s ) <_s N /\ N <_s ( g +s 1s ) ) ) )
448 446 447 mpbird 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) = N )
449 310 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i )
450 simprr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) = N )
451 449 450 breqtrrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i )
452 297 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
453 452 n0snod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. No )
454 330 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. No )
455 294 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
456 455 n0snod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
457 453 454 456 sltadd2d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( i  ( g +s i )
458 451 457 mpbird 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i
459 n0slt1e0 
 |-  ( i e. NN0_s -> ( i  i = 0s ) )
460 452 459 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( i  i = 0s ) )
461 458 460 mpbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i = 0s )
462 354 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
463 356 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
464 oveq2 
 |-  ( i = 0s -> ( 2s ^su i ) = ( 2s ^su 0s ) )
465 464 adantl 
 |-  ( ( ( d = N /\ ( g +s 1s ) = N ) /\ i = 0s ) -> ( 2s ^su i ) = ( 2s ^su 0s ) )
466 465 adantl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) = ( 2s ^su 0s ) )
467 466 56 eqtrdi 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) = 1s )
468 463 467 breqtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
469 363 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
470 n0slt1e0 
 |-  ( h e. NN0_s -> ( h  h = 0s ) )
471 469 470 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h  h = 0s ) )
472 468 471 mpbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h = 0s )
473 472 467 oveq12d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) ) = ( 0s /su 1s ) )
474 473 61 eqtrdi 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) ) = 0s )
475 474 oveq2d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( h /su ( 2s ^su i ) ) ) = ( g +s 0s ) )
476 294 n0snod 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
477 476 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
478 477 addsridd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 0s ) = g )
479 462 475 478 3eqtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = g )
480 56 oveq2i 
 |-  ( g /su ( 2s ^su 0s ) ) = ( g /su 1s )
481 476 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
482 divs1 
 |-  ( g e. No -> ( g /su 1s ) = g )
483 481 482 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g /su 1s ) = g )
484 simprr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = g )
485 483 484 eqtr4d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g /su 1s ) = c )
486 480 485 eqtrid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g /su ( 2s ^su 0s ) ) = c )
487 486 sneqd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { ( g /su ( 2s ^su 0s ) ) } = { c } )
488 56 oveq2i 
 |-  ( ( g +s 1s ) /su ( 2s ^su 0s ) ) = ( ( g +s 1s ) /su 1s )
489 294 348 syl 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. NN0_s )
490 489 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. NN0_s )
491 490 n0snod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. No )
492 divs1 
 |-  ( ( g +s 1s ) e. No -> ( ( g +s 1s ) /su 1s ) = ( g +s 1s ) )
493 491 492 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s ) /su 1s ) = ( g +s 1s ) )
494 simpllr 
 |-  ( ( ( ( d = N /\ ( g +s 1s ) = N ) /\ i = 0s ) /\ c = g ) -> ( g +s 1s ) = N )
495 494 adantl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) = N )
496 simplll 
 |-  ( ( ( ( d = N /\ ( g +s 1s ) = N ) /\ i = 0s ) /\ c = g ) -> d = N )
497 496 adantl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = N )
498 495 497 eqtr4d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) = d )
499 493 498 eqtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s ) /su 1s ) = d )
500 488 499 eqtrid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s ) /su ( 2s ^su 0s ) ) = d )
501 500 sneqd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { ( ( g +s 1s ) /su ( 2s ^su 0s ) ) } = { d } )
502 487 501 oveq12d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( { ( g /su ( 2s ^su 0s ) ) } |s { ( ( g +s 1s ) /su ( 2s ^su 0s ) ) } ) = ( { c } |s { d } ) )
503 294 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
504 503 n0zsd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. ZZ_s )
505 31 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s e. NN0_s )
506 504 505 pw2cutp1 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( { ( g /su ( 2s ^su 0s ) ) } |s { ( ( g +s 1s ) /su ( 2s ^su 0s ) ) } ) = ( ( ( 2s x.s g ) +s 1s ) /su ( 2s ^su ( 0s +s 1s ) ) ) )
507 506 eqcomd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s x.s g ) +s 1s ) /su ( 2s ^su ( 0s +s 1s ) ) ) = ( { ( g /su ( 2s ^su 0s ) ) } |s { ( ( g +s 1s ) /su ( 2s ^su 0s ) ) } ) )
508 simpl2 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
509 502 507 508 3eqtr4rd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( ( ( 2s x.s g ) +s 1s ) /su ( 2s ^su ( 0s +s 1s ) ) ) )
510 mulscl 
 |-  ( ( 2s e. No /\ g e. No ) -> ( 2s x.s g ) e. No )
511 54 481 510 sylancr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s g ) e. No )
512 330 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. No )
513 addslid 
 |-  ( 1s e. No -> ( 0s +s 1s ) = 1s )
514 330 513 ax-mp 
 |-  ( 0s +s 1s ) = 1s
515 514 319 eqeltri 
 |-  ( 0s +s 1s ) e. NN0_s
516 515 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 0s +s 1s ) e. NN0_s )
517 511 512 516 pw2divsdird 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s x.s g ) +s 1s ) /su ( 2s ^su ( 0s +s 1s ) ) ) = ( ( ( 2s x.s g ) /su ( 2s ^su ( 0s +s 1s ) ) ) +s ( 1s /su ( 2s ^su ( 0s +s 1s ) ) ) ) )
518 exps1 
 |-  ( 2s e. No -> ( 2s ^su 1s ) = 2s )
519 54 518 ax-mp 
 |-  ( 2s ^su 1s ) = 2s
520 519 oveq1i 
 |-  ( ( 2s ^su 1s ) x.s g ) = ( 2s x.s g )
521 520 oveq1i 
 |-  ( ( ( 2s ^su 1s ) x.s g ) /su ( 2s ^su ( 0s +s 1s ) ) ) = ( ( 2s x.s g ) /su ( 2s ^su ( 0s +s 1s ) ) )
522 319 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. NN0_s )
523 481 505 522 pw2divscan4d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g /su ( 2s ^su 0s ) ) = ( ( ( 2s ^su 1s ) x.s g ) /su ( 2s ^su ( 0s +s 1s ) ) ) )
524 480 483 eqtrid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g /su ( 2s ^su 0s ) ) = g )
525 523 524 eqtr3d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su 1s ) x.s g ) /su ( 2s ^su ( 0s +s 1s ) ) ) = g )
526 521 525 eqtr3id 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s g ) /su ( 2s ^su ( 0s +s 1s ) ) ) = g )
527 514 oveq2i 
 |-  ( 2s ^su ( 0s +s 1s ) ) = ( 2s ^su 1s )
528 527 519 eqtri 
 |-  ( 2s ^su ( 0s +s 1s ) ) = 2s
529 528 oveq2i 
 |-  ( 1s /su ( 2s ^su ( 0s +s 1s ) ) ) = ( 1s /su 2s )
530 529 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 1s /su ( 2s ^su ( 0s +s 1s ) ) ) = ( 1s /su 2s ) )
531 526 530 oveq12d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s x.s g ) /su ( 2s ^su ( 0s +s 1s ) ) ) +s ( 1s /su ( 2s ^su ( 0s +s 1s ) ) ) ) = ( g +s ( 1s /su 2s ) ) )
532 517 531 eqtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s x.s g ) +s 1s ) /su ( 2s ^su ( 0s +s 1s ) ) ) = ( g +s ( 1s /su 2s ) ) )
533 532 eqeq2d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( w = ( ( ( 2s x.s g ) +s 1s ) /su ( 2s ^su ( 0s +s 1s ) ) ) <-> w = ( g +s ( 1s /su 2s ) ) ) )
534 294 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
535 319 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. NN0_s )
536 simprr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( g +s ( 1s /su 2s ) ) )
537 sltadd1 
 |-  ( ( 0s e. No /\ 1s e. No /\ 1s e. No ) -> ( 0s  ( 0s +s 1s )
538 59 330 330 537 mp3an 
 |-  ( 0s  ( 0s +s 1s )
539 38 538 mpbi 
 |-  ( 0s +s 1s )
540 539 514 333 3brtr3i 
 |-  1s
541 540 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s
542 simp-4r 
 |-  ( ( ( ( ( d = N /\ ( g +s 1s ) = N ) /\ i = 0s ) /\ c = g ) /\ w = ( g +s ( 1s /su 2s ) ) ) -> ( g +s 1s ) = N )
543 542 adantl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) = N )
544 302 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
545 544 34 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. No )
546 545 sltp1d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N
547 543 546 eqbrtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s )
548 oveq1 
 |-  ( a = g -> ( a +s ( b /su ( 2s ^su q ) ) ) = ( g +s ( b /su ( 2s ^su q ) ) ) )
549 548 eqeq2d 
 |-  ( a = g -> ( w = ( a +s ( b /su ( 2s ^su q ) ) ) <-> w = ( g +s ( b /su ( 2s ^su q ) ) ) ) )
550 oveq1 
 |-  ( a = g -> ( a +s q ) = ( g +s q ) )
551 550 breq1d 
 |-  ( a = g -> ( ( a +s q )  ( g +s q )
552 549 551 3anbi13d 
 |-  ( a = g -> ( ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b  ( w = ( g +s ( b /su ( 2s ^su q ) ) ) /\ b
553 oveq1 
 |-  ( b = 1s -> ( b /su ( 2s ^su q ) ) = ( 1s /su ( 2s ^su q ) ) )
554 553 oveq2d 
 |-  ( b = 1s -> ( g +s ( b /su ( 2s ^su q ) ) ) = ( g +s ( 1s /su ( 2s ^su q ) ) ) )
555 554 eqeq2d 
 |-  ( b = 1s -> ( w = ( g +s ( b /su ( 2s ^su q ) ) ) <-> w = ( g +s ( 1s /su ( 2s ^su q ) ) ) ) )
556 breq1 
 |-  ( b = 1s -> ( b  1s
557 555 556 3anbi12d 
 |-  ( b = 1s -> ( ( w = ( g +s ( b /su ( 2s ^su q ) ) ) /\ b  ( w = ( g +s ( 1s /su ( 2s ^su q ) ) ) /\ 1s
558 oveq2 
 |-  ( q = 1s -> ( 2s ^su q ) = ( 2s ^su 1s ) )
559 558 519 eqtrdi 
 |-  ( q = 1s -> ( 2s ^su q ) = 2s )
560 559 oveq2d 
 |-  ( q = 1s -> ( 1s /su ( 2s ^su q ) ) = ( 1s /su 2s ) )
561 560 oveq2d 
 |-  ( q = 1s -> ( g +s ( 1s /su ( 2s ^su q ) ) ) = ( g +s ( 1s /su 2s ) ) )
562 561 eqeq2d 
 |-  ( q = 1s -> ( w = ( g +s ( 1s /su ( 2s ^su q ) ) ) <-> w = ( g +s ( 1s /su 2s ) ) ) )
563 559 breq2d 
 |-  ( q = 1s -> ( 1s  1s
564 oveq2 
 |-  ( q = 1s -> ( g +s q ) = ( g +s 1s ) )
565 564 breq1d 
 |-  ( q = 1s -> ( ( g +s q )  ( g +s 1s )
566 562 563 565 3anbi123d 
 |-  ( q = 1s -> ( ( w = ( g +s ( 1s /su ( 2s ^su q ) ) ) /\ 1s  ( w = ( g +s ( 1s /su 2s ) ) /\ 1s
567 552 557 566 rspc3ev 
 |-  ( ( ( g e. NN0_s /\ 1s e. NN0_s /\ 1s e. NN0_s ) /\ ( w = ( g +s ( 1s /su 2s ) ) /\ 1s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
568 534 535 535 536 541 547 567 syl33anc 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
569 568 expr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( w = ( g +s ( 1s /su 2s ) ) -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
570 533 569 sylbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( w = ( ( ( 2s x.s g ) +s 1s ) /su ( 2s ^su ( 0s +s 1s ) ) ) -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
571 509 570 mpd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
572 571 expr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( c = g -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
573 479 572 mpd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
574 573 expr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( i = 0s -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
575 461 574 mpd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
576 575 expr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s ) = N -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
577 448 576 mpd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
578 577 ex 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( d = N -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
579 simprr1 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( j +s ( k /su ( 2s ^su l ) ) ) )
580 simprr2 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k
581 simprl3 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. NN0_s )
582 expscl 
 |-  ( ( 2s e. No /\ l e. NN0_s ) -> ( 2s ^su l ) e. No )
583 54 581 582 sylancr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su l ) e. No )
584 583 mulslidd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 1s x.s ( 2s ^su l ) ) = ( 2s ^su l ) )
585 580 584 breqtrrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k
586 simprl2 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k e. NN0_s )
587 586 n0snod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k e. No )
588 330 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. No )
589 587 588 581 pw2sltdivmul2d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( k /su ( 2s ^su l ) )  k
590 585 589 mpbird 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( k /su ( 2s ^su l ) )
591 587 581 pw2divscld 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( k /su ( 2s ^su l ) ) e. No )
592 simprl1 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j e. NN0_s )
593 592 n0snod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j e. No )
594 591 588 593 sltadd2d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( k /su ( 2s ^su l ) )  ( j +s ( k /su ( 2s ^su l ) ) )
595 590 594 mpbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s ( k /su ( 2s ^su l ) ) )
596 579 595 eqbrtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d
597 294 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
598 597 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
599 598 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
600 599 addsridd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 0s ) = g )
601 363 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
602 601 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
603 n0sge0 
 |-  ( h e. NN0_s -> 0s <_s h )
604 602 603 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s <_s h )
605 602 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. No )
606 297 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
607 606 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
608 605 607 pw2ge0divsd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 0s <_s h <-> 0s <_s ( h /su ( 2s ^su i ) ) ) )
609 59 a1i 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s e. No )
610 605 607 pw2divscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) ) e. No )
611 609 610 599 sleadd2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 0s <_s ( h /su ( 2s ^su i ) ) <-> ( g +s 0s ) <_s ( g +s ( h /su ( 2s ^su i ) ) ) ) )
612 608 611 bitrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 0s <_s h <-> ( g +s 0s ) <_s ( g +s ( h /su ( 2s ^su i ) ) ) ) )
613 604 612 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 0s ) <_s ( g +s ( h /su ( 2s ^su i ) ) ) )
614 600 613 eqbrtrrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g <_s ( g +s ( h /su ( 2s ^su i ) ) ) )
615 354 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
616 615 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
617 614 616 breqtrrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g <_s c )
618 597 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
619 618 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
620 345 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
621 620 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
622 394 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
623 622 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
624 simprr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g <_s c )
625 286 3adant3 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
626 625 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
627 626 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
628 619 621 623 624 627 slelttrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g
629 597 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
630 629 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
631 622 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
632 592 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j e. NN0_s )
633 peano2n0s 
 |-  ( j e. NN0_s -> ( j +s 1s ) e. NN0_s )
634 632 633 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s 1s ) e. NN0_s )
635 634 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s 1s ) e. No )
636 simprr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g
637 simprll 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d
638 630 631 635 636 637 slttrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g
639 n0sleltp1 
 |-  ( ( g e. NN0_s /\ j e. NN0_s ) -> ( g <_s j <-> g
640 629 632 639 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g <_s j <-> g
641 638 640 mpbird 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g <_s j )
642 641 expr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g  g <_s j ) )
643 628 642 mpd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g <_s j )
644 593 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j e. No )
645 619 644 sleloed 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g <_s j <-> ( g
646 592 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j e. NN0_s )
647 n0sltp1le 
 |-  ( ( g e. NN0_s /\ j e. NN0_s ) -> ( g  ( g +s 1s ) <_s j ) )
648 618 646 647 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g  ( g +s 1s ) <_s j ) )
649 648 biimpd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g  ( g +s 1s ) <_s j ) )
650 649 impr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) <_s j )
651 489 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. NN0_s )
652 651 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. NN0_s )
653 652 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. No )
654 592 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j e. NN0_s )
655 654 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j e. No )
656 622 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
657 simprr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) <_s j )
658 586 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k e. NN0_s )
659 n0sge0 
 |-  ( k e. NN0_s -> 0s <_s k )
660 658 659 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s <_s k )
661 658 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k e. No )
662 581 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. NN0_s )
663 661 662 pw2ge0divsd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 0s <_s k <-> 0s <_s ( k /su ( 2s ^su l ) ) ) )
664 661 662 pw2divscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( k /su ( 2s ^su l ) ) e. No )
665 655 664 addsge01d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 0s <_s ( k /su ( 2s ^su l ) ) <-> j <_s ( j +s ( k /su ( 2s ^su l ) ) ) ) )
666 663 665 bitrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 0s <_s k <-> j <_s ( j +s ( k /su ( 2s ^su l ) ) ) ) )
667 660 666 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j <_s ( j +s ( k /su ( 2s ^su l ) ) ) )
668 579 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( j +s ( k /su ( 2s ^su l ) ) ) )
669 667 668 breqtrrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j <_s d )
670 653 655 656 657 669 sletrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) <_s d )
671 592 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j e. NN0_s )
672 581 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. NN0_s )
673 n0addscl 
 |-  ( ( j e. NN0_s /\ l e. NN0_s ) -> ( j +s l ) e. NN0_s )
674 671 672 673 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s l ) e. NN0_s )
675 674 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s l ) e. No )
676 302 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
677 676 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
678 677 34 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. No )
679 677 325 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) e. No )
680 simprr3 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s l )
681 680 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s l )
682 678 sltp1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N
683 675 678 679 681 682 slttrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s l )
684 675 679 sltnled 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( j +s l )  -. ( N +s 1s ) <_s ( j +s l ) ) )
685 683 684 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. ( N +s 1s ) <_s ( j +s l ) )
686 651 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. NN0_s )
687 686 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. No )
688 622 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
689 687 688 sltnled 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s )  -. d <_s ( g +s 1s ) ) )
690 679 adantrr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) e. No )
691 593 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j e. No )
692 675 adantrr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s l ) e. No )
693 651 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. NN0_s )
694 693 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. No )
695 341 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
696 695 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
697 simpll2 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
698 620 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
699 615 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
700 356 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
701 700 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
702 606 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
703 54 702 359 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) e. No )
704 703 mulslidd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 1s x.s ( 2s ^su i ) ) = ( 2s ^su i ) )
705 701 704 breqtrrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
706 601 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
707 706 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. No )
708 330 a1i 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. No )
709 707 708 702 pw2sltdivmul2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h /su ( 2s ^su i ) )  h
710 705 709 mpbird 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) )
711 707 702 pw2divscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) ) e. No )
712 597 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
713 712 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
714 711 708 713 sltadd2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h /su ( 2s ^su i ) )  ( g +s ( h /su ( 2s ^su i ) ) )
715 710 714 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( h /su ( 2s ^su i ) ) )
716 699 715 eqbrtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
717 698 694 716 ssltsn 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { c } <
718 622 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
719 simprr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s )
720 694 718 719 ssltsn 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { ( g +s 1s ) } <
721 697 694 717 720 ssltbday 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) C_ ( bday ` ( g +s 1s ) ) )
722 696 721 eqsstrrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` ( N +s 1s ) ) C_ ( bday ` ( g +s 1s ) ) )
723 676 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
724 723 405 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) e. On_s )
725 n0ons 
 |-  ( ( g +s 1s ) e. NN0_s -> ( g +s 1s ) e. On_s )
726 693 725 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. On_s )
727 724 726 onsled 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N +s 1s ) <_s ( g +s 1s ) <-> ( bday ` ( N +s 1s ) ) C_ ( bday ` ( g +s 1s ) ) ) )
728 722 727 mpbird 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) <_s ( g +s 1s ) )
729 simpllr 
 |-  ( ( ( ( ( ( d  ( g +s 1s ) <_s j )
730 729 adantl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) <_s j )
731 690 694 691 728 730 sletrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) <_s j )
732 581 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. NN0_s )
733 n0sge0 
 |-  ( l e. NN0_s -> 0s <_s l )
734 732 733 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s <_s l )
735 732 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. No )
736 691 735 addsge01d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 0s <_s l <-> j <_s ( j +s l ) ) )
737 734 736 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j <_s ( j +s l ) )
738 690 691 692 731 737 sletrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) <_s ( j +s l ) )
739 738 expr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s )  ( N +s 1s ) <_s ( j +s l ) ) )
740 689 739 sylbird 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( -. d <_s ( g +s 1s ) -> ( N +s 1s ) <_s ( j +s l ) ) )
741 685 740 mt3d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d <_s ( g +s 1s ) )
742 simprr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) <_s d )
743 688 687 sletri3d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( d = ( g +s 1s ) <-> ( d <_s ( g +s 1s ) /\ ( g +s 1s ) <_s d ) ) )
744 741 742 743 mpbir2and 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( g +s 1s ) )
745 700 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
746 601 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
747 606 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
748 n0expscl 
 |-  ( ( 2s e. NN0_s /\ i e. NN0_s ) -> ( 2s ^su i ) e. NN0_s )
749 422 747 748 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) e. NN0_s )
750 n0sltp1le 
 |-  ( ( h e. NN0_s /\ ( 2s ^su i ) e. NN0_s ) -> ( h  ( h +s 1s ) <_s ( 2s ^su i ) ) )
751 746 749 750 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h  ( h +s 1s ) <_s ( 2s ^su i ) ) )
752 745 751 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s ) <_s ( 2s ^su i ) )
753 peano2n0s 
 |-  ( h e. NN0_s -> ( h +s 1s ) e. NN0_s )
754 363 753 syl 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s ) e. NN0_s )
755 754 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s ) e. NN0_s )
756 755 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s ) e. NN0_s )
757 756 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s ) e. No )
758 749 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) e. No )
759 757 758 sleloed 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h +s 1s ) <_s ( 2s ^su i ) <-> ( ( h +s 1s )
760 676 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
761 34 325 sltnled 
 |-  ( ph -> ( N  -. ( N +s 1s ) <_s N ) )
762 40 761 mpbid 
 |-  ( ph -> -. ( N +s 1s ) <_s N )
763 760 762 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. ( N +s 1s ) <_s N )
764 695 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
765 simpll2 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
766 597 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
767 766 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
768 755 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s ) e. NN0_s )
769 768 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s ) e. No )
770 606 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
771 769 770 pw2divscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h +s 1s ) /su ( 2s ^su i ) ) e. No )
772 767 771 addscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( ( h +s 1s ) /su ( 2s ^su i ) ) ) e. No )
773 620 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
774 615 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
775 601 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
776 775 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. No )
777 776 sltp1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
778 776 769 770 pw2sltdiv1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h  ( h /su ( 2s ^su i ) )
779 777 778 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) )
780 776 770 pw2divscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) ) e. No )
781 780 771 767 sltadd2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h /su ( 2s ^su i ) )  ( g +s ( h /su ( 2s ^su i ) ) )
782 779 781 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( h /su ( 2s ^su i ) ) )
783 774 782 eqbrtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
784 773 772 783 ssltsn 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { c } <
785 622 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
786 simprr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s )
787 54 770 359 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) e. No )
788 787 mulslidd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 1s x.s ( 2s ^su i ) ) = ( 2s ^su i ) )
789 786 788 breqtrrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s )
790 330 a1i 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. No )
791 769 790 770 pw2sltdivmul2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( h +s 1s ) /su ( 2s ^su i ) )  ( h +s 1s )
792 791 bicomd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h +s 1s )  ( ( h +s 1s ) /su ( 2s ^su i ) )
793 771 790 767 sltadd2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( h +s 1s ) /su ( 2s ^su i ) )  ( g +s ( ( h +s 1s ) /su ( 2s ^su i ) ) )
794 792 793 bitrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h +s 1s )  ( g +s ( ( h +s 1s ) /su ( 2s ^su i ) ) )
795 789 794 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( ( h +s 1s ) /su ( 2s ^su i ) ) )
796 simprlr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( g +s 1s ) )
797 795 796 breqtrrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( ( h +s 1s ) /su ( 2s ^su i ) ) )
798 772 785 797 ssltsn 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { ( g +s ( ( h +s 1s ) /su ( 2s ^su i ) ) ) } <
799 765 772 784 798 ssltbday 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) C_ ( bday ` ( g +s ( ( h +s 1s ) /su ( 2s ^su i ) ) ) ) )
800 764 799 eqsstrrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` ( N +s 1s ) ) C_ ( bday ` ( g +s ( ( h +s 1s ) /su ( 2s ^su i ) ) ) ) )
801 676 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
802 801 1 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. NN0_s )
803 310 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i )
804 803 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i )
805 802 766 768 770 786 804 bdaypw2bnd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` ( g +s ( ( h +s 1s ) /su ( 2s ^su i ) ) ) ) C_ ( bday ` N ) )
806 800 805 sstrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` ( N +s 1s ) ) C_ ( bday ` N ) )
807 onslt 
 |-  ( ( N e. On_s /\ ( N +s 1s ) e. On_s ) -> ( N  ( bday ` N ) e. ( bday ` ( N +s 1s ) ) ) )
808 263 405 807 syl2anc 
 |-  ( ph -> ( N  ( bday ` N ) e. ( bday ` ( N +s 1s ) ) ) )
809 801 808 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N  ( bday ` N ) e. ( bday ` ( N +s 1s ) ) ) )
810 809 notbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( -. N  -. ( bday ` N ) e. ( bday ` ( N +s 1s ) ) ) )
811 325 34 slenltd 
 |-  ( ph -> ( ( N +s 1s ) <_s N <-> -. N
812 801 811 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N +s 1s ) <_s N <-> -. N
813 bdayelon 
 |-  ( bday ` ( N +s 1s ) ) e. On
814 ontri1 
 |-  ( ( ( bday ` ( N +s 1s ) ) e. On /\ ( bday ` N ) e. On ) -> ( ( bday ` ( N +s 1s ) ) C_ ( bday ` N ) <-> -. ( bday ` N ) e. ( bday ` ( N +s 1s ) ) ) )
815 813 8 814 mp2an 
 |-  ( ( bday ` ( N +s 1s ) ) C_ ( bday ` N ) <-> -. ( bday ` N ) e. ( bday ` ( N +s 1s ) ) )
816 815 a1i 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( bday ` ( N +s 1s ) ) C_ ( bday ` N ) <-> -. ( bday ` N ) e. ( bday ` ( N +s 1s ) ) ) )
817 810 812 816 3bitr4d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N +s 1s ) <_s N <-> ( bday ` ( N +s 1s ) ) C_ ( bday ` N ) ) )
818 806 817 mpbird 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) <_s N )
819 818 expr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h +s 1s )  ( N +s 1s ) <_s N ) )
820 763 819 mtod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. ( h +s 1s )
821 orel1 
 |-  ( -. ( h +s 1s )  ( ( ( h +s 1s )  ( h +s 1s ) = ( 2s ^su i ) ) )
822 820 821 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( h +s 1s )  ( h +s 1s ) = ( 2s ^su i ) ) )
823 597 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
824 601 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
825 n0mulscl 
 |-  ( ( 2s e. NN0_s /\ h e. NN0_s ) -> ( 2s x.s h ) e. NN0_s )
826 422 824 825 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s h ) e. NN0_s )
827 n0addscl 
 |-  ( ( ( 2s x.s h ) e. NN0_s /\ 1s e. NN0_s ) -> ( ( 2s x.s h ) +s 1s ) e. NN0_s )
828 826 319 827 sylancl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s h ) +s 1s ) e. NN0_s )
829 peano2n0s 
 |-  ( i e. NN0_s -> ( i +s 1s ) e. NN0_s )
830 606 829 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( i +s 1s ) e. NN0_s )
831 830 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( i +s 1s ) e. NN0_s )
832 simpll2 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
833 615 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
834 54 606 359 sylancr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) e. No )
835 834 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) e. No )
836 823 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
837 835 836 mulscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su i ) x.s g ) e. No )
838 601 n0snod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. No )
839 838 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. No )
840 606 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
841 837 839 840 pw2divsdird 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su i ) x.s g ) +s h ) /su ( 2s ^su i ) ) = ( ( ( ( 2s ^su i ) x.s g ) /su ( 2s ^su i ) ) +s ( h /su ( 2s ^su i ) ) ) )
842 836 840 pw2divscan3d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su i ) x.s g ) /su ( 2s ^su i ) ) = g )
843 842 oveq1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su i ) x.s g ) /su ( 2s ^su i ) ) +s ( h /su ( 2s ^su i ) ) ) = ( g +s ( h /su ( 2s ^su i ) ) ) )
844 841 843 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su i ) x.s g ) +s h ) /su ( 2s ^su i ) ) = ( g +s ( h /su ( 2s ^su i ) ) ) )
845 833 844 eqtr4d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( ( ( ( 2s ^su i ) x.s g ) +s h ) /su ( 2s ^su i ) ) )
846 845 sneqd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { c } = { ( ( ( ( 2s ^su i ) x.s g ) +s h ) /su ( 2s ^su i ) ) } )
847 simprlr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( g +s 1s ) )
848 330 a1i 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. No )
849 837 839 848 addsassd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su i ) x.s g ) +s h ) +s 1s ) = ( ( ( 2s ^su i ) x.s g ) +s ( h +s 1s ) ) )
850 849 oveq1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( ( 2s ^su i ) x.s g ) +s h ) +s 1s ) /su ( 2s ^su i ) ) = ( ( ( ( 2s ^su i ) x.s g ) +s ( h +s 1s ) ) /su ( 2s ^su i ) ) )
851 simprr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s ) = ( 2s ^su i ) )
852 851 835 eqeltrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s ) e. No )
853 837 852 840 pw2divsdird 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su i ) x.s g ) +s ( h +s 1s ) ) /su ( 2s ^su i ) ) = ( ( ( ( 2s ^su i ) x.s g ) /su ( 2s ^su i ) ) +s ( ( h +s 1s ) /su ( 2s ^su i ) ) ) )
854 851 oveq1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h +s 1s ) /su ( 2s ^su i ) ) = ( ( 2s ^su i ) /su ( 2s ^su i ) ) )
855 840 pw2divsidd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su i ) /su ( 2s ^su i ) ) = 1s )
856 854 855 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h +s 1s ) /su ( 2s ^su i ) ) = 1s )
857 842 856 oveq12d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su i ) x.s g ) /su ( 2s ^su i ) ) +s ( ( h +s 1s ) /su ( 2s ^su i ) ) ) = ( g +s 1s ) )
858 853 857 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su i ) x.s g ) +s ( h +s 1s ) ) /su ( 2s ^su i ) ) = ( g +s 1s ) )
859 850 858 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( ( 2s ^su i ) x.s g ) +s h ) +s 1s ) /su ( 2s ^su i ) ) = ( g +s 1s ) )
860 847 859 eqtr4d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( ( ( ( ( 2s ^su i ) x.s g ) +s h ) +s 1s ) /su ( 2s ^su i ) ) )
861 860 sneqd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { d } = { ( ( ( ( ( 2s ^su i ) x.s g ) +s h ) +s 1s ) /su ( 2s ^su i ) ) } )
862 846 861 oveq12d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( { c } |s { d } ) = ( { ( ( ( ( 2s ^su i ) x.s g ) +s h ) /su ( 2s ^su i ) ) } |s { ( ( ( ( ( 2s ^su i ) x.s g ) +s h ) +s 1s ) /su ( 2s ^su i ) ) } ) )
863 422 840 748 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) e. NN0_s )
864 n0mulscl 
 |-  ( ( ( 2s ^su i ) e. NN0_s /\ g e. NN0_s ) -> ( ( 2s ^su i ) x.s g ) e. NN0_s )
865 863 823 864 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su i ) x.s g ) e. NN0_s )
866 n0addscl 
 |-  ( ( ( ( 2s ^su i ) x.s g ) e. NN0_s /\ h e. NN0_s ) -> ( ( ( 2s ^su i ) x.s g ) +s h ) e. NN0_s )
867 865 824 866 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su i ) x.s g ) +s h ) e. NN0_s )
868 867 n0zsd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su i ) x.s g ) +s h ) e. ZZ_s )
869 868 840 pw2cutp1 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( { ( ( ( ( 2s ^su i ) x.s g ) +s h ) /su ( 2s ^su i ) ) } |s { ( ( ( ( ( 2s ^su i ) x.s g ) +s h ) +s 1s ) /su ( 2s ^su i ) ) } ) = ( ( ( 2s x.s ( ( ( 2s ^su i ) x.s g ) +s h ) ) +s 1s ) /su ( 2s ^su ( i +s 1s ) ) ) )
870 54 a1i 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  2s e. No )
871 870 837 839 addsdid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s ( ( ( 2s ^su i ) x.s g ) +s h ) ) = ( ( 2s x.s ( ( 2s ^su i ) x.s g ) ) +s ( 2s x.s h ) ) )
872 expsp1 
 |-  ( ( 2s e. No /\ i e. NN0_s ) -> ( 2s ^su ( i +s 1s ) ) = ( ( 2s ^su i ) x.s 2s ) )
873 54 840 872 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( i +s 1s ) ) = ( ( 2s ^su i ) x.s 2s ) )
874 835 870 mulscomd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su i ) x.s 2s ) = ( 2s x.s ( 2s ^su i ) ) )
875 873 874 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( i +s 1s ) ) = ( 2s x.s ( 2s ^su i ) ) )
876 875 oveq1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( i +s 1s ) ) x.s g ) = ( ( 2s x.s ( 2s ^su i ) ) x.s g ) )
877 870 835 836 mulsassd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s ( 2s ^su i ) ) x.s g ) = ( 2s x.s ( ( 2s ^su i ) x.s g ) ) )
878 876 877 eqtr2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s ( ( 2s ^su i ) x.s g ) ) = ( ( 2s ^su ( i +s 1s ) ) x.s g ) )
879 878 oveq1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s ( ( 2s ^su i ) x.s g ) ) +s ( 2s x.s h ) ) = ( ( ( 2s ^su ( i +s 1s ) ) x.s g ) +s ( 2s x.s h ) ) )
880 871 879 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s ( ( ( 2s ^su i ) x.s g ) +s h ) ) = ( ( ( 2s ^su ( i +s 1s ) ) x.s g ) +s ( 2s x.s h ) ) )
881 880 oveq1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s ( ( ( 2s ^su i ) x.s g ) +s h ) ) +s 1s ) = ( ( ( ( 2s ^su ( i +s 1s ) ) x.s g ) +s ( 2s x.s h ) ) +s 1s ) )
882 881 oveq1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s x.s ( ( ( 2s ^su i ) x.s g ) +s h ) ) +s 1s ) /su ( 2s ^su ( i +s 1s ) ) ) = ( ( ( ( ( 2s ^su ( i +s 1s ) ) x.s g ) +s ( 2s x.s h ) ) +s 1s ) /su ( 2s ^su ( i +s 1s ) ) ) )
883 869 882 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( { ( ( ( ( 2s ^su i ) x.s g ) +s h ) /su ( 2s ^su i ) ) } |s { ( ( ( ( ( 2s ^su i ) x.s g ) +s h ) +s 1s ) /su ( 2s ^su i ) ) } ) = ( ( ( ( ( 2s ^su ( i +s 1s ) ) x.s g ) +s ( 2s x.s h ) ) +s 1s ) /su ( 2s ^su ( i +s 1s ) ) ) )
884 832 862 883 3eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( ( ( ( ( 2s ^su ( i +s 1s ) ) x.s g ) +s ( 2s x.s h ) ) +s 1s ) /su ( 2s ^su ( i +s 1s ) ) ) )
885 expscl 
 |-  ( ( 2s e. No /\ ( i +s 1s ) e. NN0_s ) -> ( 2s ^su ( i +s 1s ) ) e. No )
886 54 831 885 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( i +s 1s ) ) e. No )
887 886 836 mulscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( i +s 1s ) ) x.s g ) e. No )
888 826 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s h ) e. No )
889 887 888 848 addsassd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su ( i +s 1s ) ) x.s g ) +s ( 2s x.s h ) ) +s 1s ) = ( ( ( 2s ^su ( i +s 1s ) ) x.s g ) +s ( ( 2s x.s h ) +s 1s ) ) )
890 889 oveq1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( ( 2s ^su ( i +s 1s ) ) x.s g ) +s ( 2s x.s h ) ) +s 1s ) /su ( 2s ^su ( i +s 1s ) ) ) = ( ( ( ( 2s ^su ( i +s 1s ) ) x.s g ) +s ( ( 2s x.s h ) +s 1s ) ) /su ( 2s ^su ( i +s 1s ) ) ) )
891 884 890 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( ( ( ( 2s ^su ( i +s 1s ) ) x.s g ) +s ( ( 2s x.s h ) +s 1s ) ) /su ( 2s ^su ( i +s 1s ) ) ) )
892 828 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s h ) +s 1s ) e. No )
893 887 892 831 pw2divsdird 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su ( i +s 1s ) ) x.s g ) +s ( ( 2s x.s h ) +s 1s ) ) /su ( 2s ^su ( i +s 1s ) ) ) = ( ( ( ( 2s ^su ( i +s 1s ) ) x.s g ) /su ( 2s ^su ( i +s 1s ) ) ) +s ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su ( i +s 1s ) ) ) ) )
894 836 831 pw2divscan3d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su ( i +s 1s ) ) x.s g ) /su ( 2s ^su ( i +s 1s ) ) ) = g )
895 894 oveq1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su ( i +s 1s ) ) x.s g ) /su ( 2s ^su ( i +s 1s ) ) ) +s ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su ( i +s 1s ) ) ) ) = ( g +s ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su ( i +s 1s ) ) ) ) )
896 893 895 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su ( i +s 1s ) ) x.s g ) +s ( ( 2s x.s h ) +s 1s ) ) /su ( 2s ^su ( i +s 1s ) ) ) = ( g +s ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su ( i +s 1s ) ) ) ) )
897 891 896 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( g +s ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su ( i +s 1s ) ) ) ) )
898 848 870 888 sltadd2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 1s  ( ( 2s x.s h ) +s 1s )
899 540 898 mpbii 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s h ) +s 1s )
900 851 oveq2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s ( h +s 1s ) ) = ( 2s x.s ( 2s ^su i ) ) )
901 870 839 848 addsdid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s ( h +s 1s ) ) = ( ( 2s x.s h ) +s ( 2s x.s 1s ) ) )
902 mulsrid 
 |-  ( 2s e. No -> ( 2s x.s 1s ) = 2s )
903 54 902 ax-mp 
 |-  ( 2s x.s 1s ) = 2s
904 903 oveq2i 
 |-  ( ( 2s x.s h ) +s ( 2s x.s 1s ) ) = ( ( 2s x.s h ) +s 2s )
905 904 a1i 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s h ) +s ( 2s x.s 1s ) ) = ( ( 2s x.s h ) +s 2s ) )
906 901 905 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s ( h +s 1s ) ) = ( ( 2s x.s h ) +s 2s ) )
907 900 906 eqtr3d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s ( 2s ^su i ) ) = ( ( 2s x.s h ) +s 2s ) )
908 874 907 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su i ) x.s 2s ) = ( ( 2s x.s h ) +s 2s ) )
909 873 908 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( i +s 1s ) ) = ( ( 2s x.s h ) +s 2s ) )
910 899 909 breqtrrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s h ) +s 1s )
911 840 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. No )
912 836 911 848 addsassd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s i ) +s 1s ) = ( g +s ( i +s 1s ) ) )
913 803 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i )
914 836 911 addscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i ) e. No )
915 676 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
916 915 34 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. No )
917 914 916 848 sltadd1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s i )  ( ( g +s i ) +s 1s )
918 913 917 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s i ) +s 1s )
919 912 918 eqbrtrrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( i +s 1s ) )
920 oveq1 
 |-  ( b = ( ( 2s x.s h ) +s 1s ) -> ( b /su ( 2s ^su q ) ) = ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su q ) ) )
921 920 oveq2d 
 |-  ( b = ( ( 2s x.s h ) +s 1s ) -> ( g +s ( b /su ( 2s ^su q ) ) ) = ( g +s ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su q ) ) ) )
922 921 eqeq2d 
 |-  ( b = ( ( 2s x.s h ) +s 1s ) -> ( w = ( g +s ( b /su ( 2s ^su q ) ) ) <-> w = ( g +s ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su q ) ) ) ) )
923 breq1 
 |-  ( b = ( ( 2s x.s h ) +s 1s ) -> ( b  ( ( 2s x.s h ) +s 1s )
924 922 923 3anbi12d 
 |-  ( b = ( ( 2s x.s h ) +s 1s ) -> ( ( w = ( g +s ( b /su ( 2s ^su q ) ) ) /\ b  ( w = ( g +s ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su q ) ) ) /\ ( ( 2s x.s h ) +s 1s )
925 oveq2 
 |-  ( q = ( i +s 1s ) -> ( 2s ^su q ) = ( 2s ^su ( i +s 1s ) ) )
926 925 oveq2d 
 |-  ( q = ( i +s 1s ) -> ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su q ) ) = ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su ( i +s 1s ) ) ) )
927 926 oveq2d 
 |-  ( q = ( i +s 1s ) -> ( g +s ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su q ) ) ) = ( g +s ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su ( i +s 1s ) ) ) ) )
928 927 eqeq2d 
 |-  ( q = ( i +s 1s ) -> ( w = ( g +s ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su q ) ) ) <-> w = ( g +s ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su ( i +s 1s ) ) ) ) ) )
929 925 breq2d 
 |-  ( q = ( i +s 1s ) -> ( ( ( 2s x.s h ) +s 1s )  ( ( 2s x.s h ) +s 1s )
930 oveq2 
 |-  ( q = ( i +s 1s ) -> ( g +s q ) = ( g +s ( i +s 1s ) ) )
931 930 breq1d 
 |-  ( q = ( i +s 1s ) -> ( ( g +s q )  ( g +s ( i +s 1s ) )
932 928 929 931 3anbi123d 
 |-  ( q = ( i +s 1s ) -> ( ( w = ( g +s ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su q ) ) ) /\ ( ( 2s x.s h ) +s 1s )  ( w = ( g +s ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su ( i +s 1s ) ) ) ) /\ ( ( 2s x.s h ) +s 1s )
933 552 924 932 rspc3ev 
 |-  ( ( ( g e. NN0_s /\ ( ( 2s x.s h ) +s 1s ) e. NN0_s /\ ( i +s 1s ) e. NN0_s ) /\ ( w = ( g +s ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su ( i +s 1s ) ) ) ) /\ ( ( 2s x.s h ) +s 1s )  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
934 823 828 831 897 910 919 933 syl33anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
935 934 expr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h +s 1s ) = ( 2s ^su i ) -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
936 822 935 syld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( h +s 1s )  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
937 759 936 sylbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h +s 1s ) <_s ( 2s ^su i ) -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
938 752 937 mpd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
939 938 expr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( d = ( g +s 1s ) -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
940 939 adantrr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( d = ( g +s 1s ) -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
941 744 940 mpd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
942 941 expr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s ) <_s d -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
943 670 942 mpd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
944 943 expr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s ) <_s j -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
945 650 944 mpd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
946 945 expr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
947 626 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
948 615 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
949 579 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( j +s ( k /su ( 2s ^su l ) ) ) )
950 simprr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g = j )
951 950 oveq1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( k /su ( 2s ^su l ) ) ) = ( j +s ( k /su ( 2s ^su l ) ) ) )
952 949 951 eqtr4d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( g +s ( k /su ( 2s ^su l ) ) ) )
953 947 948 952 3brtr3d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( h /su ( 2s ^su i ) ) )
954 838 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. No )
955 606 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
956 954 955 pw2divscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) ) e. No )
957 587 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k e. No )
958 581 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. NN0_s )
959 957 958 pw2divscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( k /su ( 2s ^su l ) ) e. No )
960 597 n0snod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
961 960 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
962 956 959 961 sltadd2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h /su ( 2s ^su i ) )  ( g +s ( h /su ( 2s ^su i ) ) )
963 953 962 mpbird 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) )
964 601 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
965 simpl3 
 |-  ( ( ( j e. NN0_s /\ k e. NN0_s /\ l e. NN0_s ) /\ ( d = ( j +s ( k /su ( 2s ^su l ) ) ) /\ k  l e. NN0_s )
966 965 adantl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. NN0_s )
967 966 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. NN0_s )
968 606 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
969 n0subs 
 |-  ( ( l e. NN0_s /\ i e. NN0_s ) -> ( l <_s i <-> ( i -s l ) e. NN0_s ) )
970 967 968 969 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( l <_s i <-> ( i -s l ) e. NN0_s ) )
971 970 biimpd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( l <_s i -> ( i -s l ) e. NN0_s ) )
972 971 impr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( i -s l ) e. NN0_s )
973 n0expscl 
 |-  ( ( 2s e. NN0_s /\ ( i -s l ) e. NN0_s ) -> ( 2s ^su ( i -s l ) ) e. NN0_s )
974 422 972 973 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( i -s l ) ) e. NN0_s )
975 586 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k e. NN0_s )
976 n0mulscl 
 |-  ( ( ( 2s ^su ( i -s l ) ) e. NN0_s /\ k e. NN0_s ) -> ( ( 2s ^su ( i -s l ) ) x.s k ) e. NN0_s )
977 974 975 976 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( i -s l ) ) x.s k ) e. NN0_s )
978 606 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
979 simprlr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) )
980 966 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. NN0_s )
981 980 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. No )
982 972 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( i -s l ) e. No )
983 981 982 addscomd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( l +s ( i -s l ) ) = ( ( i -s l ) +s l ) )
984 978 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. No )
985 npcans 
 |-  ( ( i e. No /\ l e. No ) -> ( ( i -s l ) +s l ) = i )
986 984 981 985 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( i -s l ) +s l ) = i )
987 983 986 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( l +s ( i -s l ) ) = i )
988 987 oveq2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( l +s ( i -s l ) ) ) = ( 2s ^su i ) )
989 988 oveq2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su ( l +s ( i -s l ) ) ) ) = ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su i ) ) )
990 989 eqcomd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su i ) ) = ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su ( l +s ( i -s l ) ) ) ) )
991 975 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k e. No )
992 991 980 972 pw2divscan4d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( k /su ( 2s ^su l ) ) = ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su ( l +s ( i -s l ) ) ) ) )
993 990 992 eqtr4d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su i ) ) = ( k /su ( 2s ^su l ) ) )
994 993 eqcomd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( k /su ( 2s ^su l ) ) = ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su i ) ) )
995 979 994 breqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) )
996 964 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. No )
997 977 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( i -s l ) ) x.s k ) e. No )
998 996 997 978 pw2sltdiv1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h  ( h /su ( 2s ^su i ) )
999 995 998 mpbird 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
1000 700 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
1001 580 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k
1002 n0expscl 
 |-  ( ( 2s e. NN0_s /\ l e. NN0_s ) -> ( 2s ^su l ) e. NN0_s )
1003 422 980 1002 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su l ) e. NN0_s )
1004 1003 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su l ) e. No )
1005 974 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( i -s l ) ) e. No )
1006 nnsgt0 
 |-  ( 2s e. NN_s -> 0s
1007 420 1006 ax-mp 
 |-  0s
1008 expsgt0 
 |-  ( ( 2s e. No /\ ( i -s l ) e. NN0_s /\ 0s  0s
1009 54 1007 1008 mp3an13 
 |-  ( ( i -s l ) e. NN0_s -> 0s
1010 972 1009 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s
1011 991 1004 1005 1010 sltmul2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( k  ( ( 2s ^su ( i -s l ) ) x.s k )
1012 1001 1011 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( i -s l ) ) x.s k )
1013 expadds 
 |-  ( ( 2s e. No /\ ( i -s l ) e. NN0_s /\ l e. NN0_s ) -> ( 2s ^su ( ( i -s l ) +s l ) ) = ( ( 2s ^su ( i -s l ) ) x.s ( 2s ^su l ) ) )
1014 54 1013 mp3an1 
 |-  ( ( ( i -s l ) e. NN0_s /\ l e. NN0_s ) -> ( 2s ^su ( ( i -s l ) +s l ) ) = ( ( 2s ^su ( i -s l ) ) x.s ( 2s ^su l ) ) )
1015 972 980 1014 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( ( i -s l ) +s l ) ) = ( ( 2s ^su ( i -s l ) ) x.s ( 2s ^su l ) ) )
1016 986 oveq2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( ( i -s l ) +s l ) ) = ( 2s ^su i ) )
1017 1015 1016 eqtr3d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( i -s l ) ) x.s ( 2s ^su l ) ) = ( 2s ^su i ) )
1018 1012 1017 breqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( i -s l ) ) x.s k )
1019 999 1000 1018 3jca 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h
1020 615 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
1021 579 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( j +s ( k /su ( 2s ^su l ) ) ) )
1022 simpllr 
 |-  ( ( ( ( ( d  g = j )
1023 1022 adantl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g = j )
1024 1023 993 oveq12d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su i ) ) ) = ( j +s ( k /su ( 2s ^su l ) ) ) )
1025 1021 1024 eqtr4d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( g +s ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su i ) ) ) )
1026 803 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i )
1027 1020 1025 1026 3jca 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( c = ( g +s ( h /su ( 2s ^su i ) ) ) /\ d = ( g +s ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su i ) ) ) /\ ( g +s i )
1028 breq1 
 |-  ( m = h -> ( m  h
1029 breq1 
 |-  ( m = h -> ( m  h
1030 1028 1029 3anbi12d 
 |-  ( m = h -> ( ( m  ( h
1031 oveq1 
 |-  ( m = h -> ( m /su ( 2s ^su o ) ) = ( h /su ( 2s ^su o ) ) )
1032 1031 oveq2d 
 |-  ( m = h -> ( g +s ( m /su ( 2s ^su o ) ) ) = ( g +s ( h /su ( 2s ^su o ) ) ) )
1033 1032 eqeq2d 
 |-  ( m = h -> ( c = ( g +s ( m /su ( 2s ^su o ) ) ) <-> c = ( g +s ( h /su ( 2s ^su o ) ) ) ) )
1034 1033 3anbi1d 
 |-  ( m = h -> ( ( c = ( g +s ( m /su ( 2s ^su o ) ) ) /\ d = ( g +s ( n /su ( 2s ^su o ) ) ) /\ ( g +s o )  ( c = ( g +s ( h /su ( 2s ^su o ) ) ) /\ d = ( g +s ( n /su ( 2s ^su o ) ) ) /\ ( g +s o )
1035 1030 1034 anbi12d 
 |-  ( m = h -> ( ( ( m  ( ( h
1036 breq2 
 |-  ( n = ( ( 2s ^su ( i -s l ) ) x.s k ) -> ( h  h
1037 breq1 
 |-  ( n = ( ( 2s ^su ( i -s l ) ) x.s k ) -> ( n  ( ( 2s ^su ( i -s l ) ) x.s k )
1038 1036 1037 3anbi13d 
 |-  ( n = ( ( 2s ^su ( i -s l ) ) x.s k ) -> ( ( h  ( h
1039 oveq1 
 |-  ( n = ( ( 2s ^su ( i -s l ) ) x.s k ) -> ( n /su ( 2s ^su o ) ) = ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su o ) ) )
1040 1039 oveq2d 
 |-  ( n = ( ( 2s ^su ( i -s l ) ) x.s k ) -> ( g +s ( n /su ( 2s ^su o ) ) ) = ( g +s ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su o ) ) ) )
1041 1040 eqeq2d 
 |-  ( n = ( ( 2s ^su ( i -s l ) ) x.s k ) -> ( d = ( g +s ( n /su ( 2s ^su o ) ) ) <-> d = ( g +s ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su o ) ) ) ) )
1042 1041 3anbi2d 
 |-  ( n = ( ( 2s ^su ( i -s l ) ) x.s k ) -> ( ( c = ( g +s ( h /su ( 2s ^su o ) ) ) /\ d = ( g +s ( n /su ( 2s ^su o ) ) ) /\ ( g +s o )  ( c = ( g +s ( h /su ( 2s ^su o ) ) ) /\ d = ( g +s ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su o ) ) ) /\ ( g +s o )
1043 1038 1042 anbi12d 
 |-  ( n = ( ( 2s ^su ( i -s l ) ) x.s k ) -> ( ( ( h  ( ( h
1044 oveq2 
 |-  ( o = i -> ( 2s ^su o ) = ( 2s ^su i ) )
1045 1044 breq2d 
 |-  ( o = i -> ( h  h
1046 1044 breq2d 
 |-  ( o = i -> ( ( ( 2s ^su ( i -s l ) ) x.s k )  ( ( 2s ^su ( i -s l ) ) x.s k )
1047 1045 1046 3anbi23d 
 |-  ( o = i -> ( ( h  ( h
1048 1044 oveq2d 
 |-  ( o = i -> ( h /su ( 2s ^su o ) ) = ( h /su ( 2s ^su i ) ) )
1049 1048 oveq2d 
 |-  ( o = i -> ( g +s ( h /su ( 2s ^su o ) ) ) = ( g +s ( h /su ( 2s ^su i ) ) ) )
1050 1049 eqeq2d 
 |-  ( o = i -> ( c = ( g +s ( h /su ( 2s ^su o ) ) ) <-> c = ( g +s ( h /su ( 2s ^su i ) ) ) ) )
1051 1044 oveq2d 
 |-  ( o = i -> ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su o ) ) = ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su i ) ) )
1052 1051 oveq2d 
 |-  ( o = i -> ( g +s ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su o ) ) ) = ( g +s ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su i ) ) ) )
1053 1052 eqeq2d 
 |-  ( o = i -> ( d = ( g +s ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su o ) ) ) <-> d = ( g +s ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su i ) ) ) ) )
1054 oveq2 
 |-  ( o = i -> ( g +s o ) = ( g +s i ) )
1055 1054 breq1d 
 |-  ( o = i -> ( ( g +s o )  ( g +s i )
1056 1050 1053 1055 3anbi123d 
 |-  ( o = i -> ( ( c = ( g +s ( h /su ( 2s ^su o ) ) ) /\ d = ( g +s ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su o ) ) ) /\ ( g +s o )  ( c = ( g +s ( h /su ( 2s ^su i ) ) ) /\ d = ( g +s ( ( ( 2s ^su ( i -s l ) ) x.s k ) /su ( 2s ^su i ) ) ) /\ ( g +s i )
1057 1047 1056 anbi12d 
 |-  ( o = i -> ( ( ( h  ( ( h
1058 1035 1043 1057 rspc3ev 
 |-  ( ( ( h e. NN0_s /\ ( ( 2s ^su ( i -s l ) ) x.s k ) e. NN0_s /\ i e. NN0_s ) /\ ( ( h  E. m e. NN0_s E. n e. NN0_s E. o e. NN0_s ( ( m
1059 964 977 978 1019 1027 1058 syl32anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. m e. NN0_s E. n e. NN0_s E. o e. NN0_s ( ( m
1060 1059 expr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( l <_s i -> E. m e. NN0_s E. n e. NN0_s E. o e. NN0_s ( ( m
1061 n0subs 
 |-  ( ( i e. NN0_s /\ l e. NN0_s ) -> ( i <_s l <-> ( l -s i ) e. NN0_s ) )
1062 968 967 1061 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( i <_s l <-> ( l -s i ) e. NN0_s ) )
1063 1062 biimpd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( i <_s l -> ( l -s i ) e. NN0_s ) )
1064 1063 impr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( l -s i ) e. NN0_s )
1065 n0expscl 
 |-  ( ( 2s e. NN0_s /\ ( l -s i ) e. NN0_s ) -> ( 2s ^su ( l -s i ) ) e. NN0_s )
1066 422 1064 1065 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( l -s i ) ) e. NN0_s )
1067 601 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
1068 n0mulscl 
 |-  ( ( ( 2s ^su ( l -s i ) ) e. NN0_s /\ h e. NN0_s ) -> ( ( 2s ^su ( l -s i ) ) x.s h ) e. NN0_s )
1069 1066 1067 1068 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( l -s i ) ) x.s h ) e. NN0_s )
1070 586 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k e. NN0_s )
1071 966 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. NN0_s )
1072 1067 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. No )
1073 606 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
1074 1072 1073 1064 pw2divscan4d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) ) = ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su ( i +s ( l -s i ) ) ) ) )
1075 1073 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. No )
1076 1064 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( l -s i ) e. No )
1077 1075 1076 addscomd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( i +s ( l -s i ) ) = ( ( l -s i ) +s i ) )
1078 1077 oveq2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( i +s ( l -s i ) ) ) = ( 2s ^su ( ( l -s i ) +s i ) ) )
1079 1078 oveq2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su ( i +s ( l -s i ) ) ) ) = ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su ( ( l -s i ) +s i ) ) ) )
1080 1074 1079 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) ) = ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su ( ( l -s i ) +s i ) ) ) )
1081 1071 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. No )
1082 npcans 
 |-  ( ( l e. No /\ i e. No ) -> ( ( l -s i ) +s i ) = l )
1083 1081 1075 1082 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( l -s i ) +s i ) = l )
1084 1083 oveq2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( ( l -s i ) +s i ) ) = ( 2s ^su l ) )
1085 1084 oveq2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su ( ( l -s i ) +s i ) ) ) = ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su l ) ) )
1086 1080 1085 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) ) = ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su l ) ) )
1087 simprlr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) )
1088 1086 1087 eqbrtrrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su l ) )
1089 expscl 
 |-  ( ( 2s e. No /\ ( l -s i ) e. NN0_s ) -> ( 2s ^su ( l -s i ) ) e. No )
1090 54 1064 1089 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( l -s i ) ) e. No )
1091 1090 1072 mulscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( l -s i ) ) x.s h ) e. No )
1092 1070 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k e. No )
1093 1091 1092 1071 pw2sltdiv1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su ( l -s i ) ) x.s h )  ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su l ) )
1094 1088 1093 mpbird 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( l -s i ) ) x.s h )
1095 700 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
1096 54 1073 359 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) e. No )
1097 expsgt0 
 |-  ( ( 2s e. No /\ ( l -s i ) e. NN0_s /\ 0s  0s
1098 54 1007 1097 mp3an13 
 |-  ( ( l -s i ) e. NN0_s -> 0s
1099 1064 1098 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s
1100 1072 1096 1090 1099 sltmul2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h  ( ( 2s ^su ( l -s i ) ) x.s h )
1101 1095 1100 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( l -s i ) ) x.s h )
1102 expadds 
 |-  ( ( 2s e. No /\ ( l -s i ) e. NN0_s /\ i e. NN0_s ) -> ( 2s ^su ( ( l -s i ) +s i ) ) = ( ( 2s ^su ( l -s i ) ) x.s ( 2s ^su i ) ) )
1103 54 1102 mp3an1 
 |-  ( ( ( l -s i ) e. NN0_s /\ i e. NN0_s ) -> ( 2s ^su ( ( l -s i ) +s i ) ) = ( ( 2s ^su ( l -s i ) ) x.s ( 2s ^su i ) ) )
1104 1064 1073 1103 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( ( l -s i ) +s i ) ) = ( ( 2s ^su ( l -s i ) ) x.s ( 2s ^su i ) ) )
1105 1104 1084 eqtr3d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( l -s i ) ) x.s ( 2s ^su i ) ) = ( 2s ^su l ) )
1106 1101 1105 breqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( l -s i ) ) x.s h )
1107 580 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k
1108 1094 1106 1107 3jca 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su ( l -s i ) ) x.s h )
1109 615 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
1110 1086 oveq2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( h /su ( 2s ^su i ) ) ) = ( g +s ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su l ) ) ) )
1111 1109 1110 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su l ) ) ) )
1112 579 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( j +s ( k /su ( 2s ^su l ) ) ) )
1113 simpllr 
 |-  ( ( ( ( ( d  g = j )
1114 1113 adantl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g = j )
1115 1114 oveq1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( k /su ( 2s ^su l ) ) ) = ( j +s ( k /su ( 2s ^su l ) ) ) )
1116 1112 1115 eqtr4d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( g +s ( k /su ( 2s ^su l ) ) ) )
1117 1114 oveq1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s l ) = ( j +s l ) )
1118 680 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s l )
1119 1117 1118 eqbrtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s l )
1120 1111 1116 1119 3jca 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( c = ( g +s ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su l ) ) ) /\ d = ( g +s ( k /su ( 2s ^su l ) ) ) /\ ( g +s l )
1121 breq1 
 |-  ( m = ( ( 2s ^su ( l -s i ) ) x.s h ) -> ( m  ( ( 2s ^su ( l -s i ) ) x.s h )
1122 breq1 
 |-  ( m = ( ( 2s ^su ( l -s i ) ) x.s h ) -> ( m  ( ( 2s ^su ( l -s i ) ) x.s h )
1123 1121 1122 3anbi12d 
 |-  ( m = ( ( 2s ^su ( l -s i ) ) x.s h ) -> ( ( m  ( ( ( 2s ^su ( l -s i ) ) x.s h )
1124 oveq1 
 |-  ( m = ( ( 2s ^su ( l -s i ) ) x.s h ) -> ( m /su ( 2s ^su o ) ) = ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su o ) ) )
1125 1124 oveq2d 
 |-  ( m = ( ( 2s ^su ( l -s i ) ) x.s h ) -> ( g +s ( m /su ( 2s ^su o ) ) ) = ( g +s ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su o ) ) ) )
1126 1125 eqeq2d 
 |-  ( m = ( ( 2s ^su ( l -s i ) ) x.s h ) -> ( c = ( g +s ( m /su ( 2s ^su o ) ) ) <-> c = ( g +s ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su o ) ) ) ) )
1127 1126 3anbi1d 
 |-  ( m = ( ( 2s ^su ( l -s i ) ) x.s h ) -> ( ( c = ( g +s ( m /su ( 2s ^su o ) ) ) /\ d = ( g +s ( n /su ( 2s ^su o ) ) ) /\ ( g +s o )  ( c = ( g +s ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su o ) ) ) /\ d = ( g +s ( n /su ( 2s ^su o ) ) ) /\ ( g +s o )
1128 1123 1127 anbi12d 
 |-  ( m = ( ( 2s ^su ( l -s i ) ) x.s h ) -> ( ( ( m  ( ( ( ( 2s ^su ( l -s i ) ) x.s h )
1129 breq2 
 |-  ( n = k -> ( ( ( 2s ^su ( l -s i ) ) x.s h )  ( ( 2s ^su ( l -s i ) ) x.s h )
1130 breq1 
 |-  ( n = k -> ( n  k
1131 1129 1130 3anbi13d 
 |-  ( n = k -> ( ( ( ( 2s ^su ( l -s i ) ) x.s h )  ( ( ( 2s ^su ( l -s i ) ) x.s h )
1132 oveq1 
 |-  ( n = k -> ( n /su ( 2s ^su o ) ) = ( k /su ( 2s ^su o ) ) )
1133 1132 oveq2d 
 |-  ( n = k -> ( g +s ( n /su ( 2s ^su o ) ) ) = ( g +s ( k /su ( 2s ^su o ) ) ) )
1134 1133 eqeq2d 
 |-  ( n = k -> ( d = ( g +s ( n /su ( 2s ^su o ) ) ) <-> d = ( g +s ( k /su ( 2s ^su o ) ) ) ) )
1135 1134 3anbi2d 
 |-  ( n = k -> ( ( c = ( g +s ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su o ) ) ) /\ d = ( g +s ( n /su ( 2s ^su o ) ) ) /\ ( g +s o )  ( c = ( g +s ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su o ) ) ) /\ d = ( g +s ( k /su ( 2s ^su o ) ) ) /\ ( g +s o )
1136 1131 1135 anbi12d 
 |-  ( n = k -> ( ( ( ( ( 2s ^su ( l -s i ) ) x.s h )  ( ( ( ( 2s ^su ( l -s i ) ) x.s h )
1137 oveq2 
 |-  ( o = l -> ( 2s ^su o ) = ( 2s ^su l ) )
1138 1137 breq2d 
 |-  ( o = l -> ( ( ( 2s ^su ( l -s i ) ) x.s h )  ( ( 2s ^su ( l -s i ) ) x.s h )
1139 1137 breq2d 
 |-  ( o = l -> ( k  k
1140 1138 1139 3anbi23d 
 |-  ( o = l -> ( ( ( ( 2s ^su ( l -s i ) ) x.s h )  ( ( ( 2s ^su ( l -s i ) ) x.s h )
1141 1137 oveq2d 
 |-  ( o = l -> ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su o ) ) = ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su l ) ) )
1142 1141 oveq2d 
 |-  ( o = l -> ( g +s ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su o ) ) ) = ( g +s ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su l ) ) ) )
1143 1142 eqeq2d 
 |-  ( o = l -> ( c = ( g +s ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su o ) ) ) <-> c = ( g +s ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su l ) ) ) ) )
1144 1137 oveq2d 
 |-  ( o = l -> ( k /su ( 2s ^su o ) ) = ( k /su ( 2s ^su l ) ) )
1145 1144 oveq2d 
 |-  ( o = l -> ( g +s ( k /su ( 2s ^su o ) ) ) = ( g +s ( k /su ( 2s ^su l ) ) ) )
1146 1145 eqeq2d 
 |-  ( o = l -> ( d = ( g +s ( k /su ( 2s ^su o ) ) ) <-> d = ( g +s ( k /su ( 2s ^su l ) ) ) ) )
1147 oveq2 
 |-  ( o = l -> ( g +s o ) = ( g +s l ) )
1148 1147 breq1d 
 |-  ( o = l -> ( ( g +s o )  ( g +s l )
1149 1143 1146 1148 3anbi123d 
 |-  ( o = l -> ( ( c = ( g +s ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su o ) ) ) /\ d = ( g +s ( k /su ( 2s ^su o ) ) ) /\ ( g +s o )  ( c = ( g +s ( ( ( 2s ^su ( l -s i ) ) x.s h ) /su ( 2s ^su l ) ) ) /\ d = ( g +s ( k /su ( 2s ^su l ) ) ) /\ ( g +s l )
1150 1140 1149 anbi12d 
 |-  ( o = l -> ( ( ( ( ( 2s ^su ( l -s i ) ) x.s h )  ( ( ( ( 2s ^su ( l -s i ) ) x.s h )
1151 1128 1136 1150 rspc3ev 
 |-  ( ( ( ( ( 2s ^su ( l -s i ) ) x.s h ) e. NN0_s /\ k e. NN0_s /\ l e. NN0_s ) /\ ( ( ( ( 2s ^su ( l -s i ) ) x.s h )  E. m e. NN0_s E. n e. NN0_s E. o e. NN0_s ( ( m
1152 1069 1070 1071 1108 1120 1151 syl32anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. m e. NN0_s E. n e. NN0_s E. o e. NN0_s ( ( m
1153 1152 expr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( i <_s l -> E. m e. NN0_s E. n e. NN0_s E. o e. NN0_s ( ( m
1154 967 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. No )
1155 968 n0snod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. No )
1156 sletric 
 |-  ( ( l e. No /\ i e. No ) -> ( l <_s i \/ i <_s l ) )
1157 1154 1155 1156 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( l <_s i \/ i <_s l ) )
1158 1060 1153 1157 mpjaod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. m e. NN0_s E. n e. NN0_s E. o e. NN0_s ( ( m
1159 597 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
1160 1159 adantr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
1161 simprl1 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  m e. NN0_s )
1162 n0mulscl 
 |-  ( ( 2s e. NN0_s /\ m e. NN0_s ) -> ( 2s x.s m ) e. NN0_s )
1163 422 1161 1162 sylancr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s m ) e. NN0_s )
1164 peano2n0s 
 |-  ( ( 2s x.s m ) e. NN0_s -> ( ( 2s x.s m ) +s 1s ) e. NN0_s )
1165 1163 1164 syl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s m ) +s 1s ) e. NN0_s )
1166 simprl3 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  o e. NN0_s )
1167 peano2n0s 
 |-  ( o e. NN0_s -> ( o +s 1s ) e. NN0_s )
1168 1166 1167 syl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( o +s 1s ) e. NN0_s )
1169 simpl2 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
1170 1169 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
1171 1170 adantr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
1172 simprr1 
 |-  ( ( ( m e. NN0_s /\ n e. NN0_s /\ o e. NN0_s ) /\ ( ( m  c = ( g +s ( m /su ( 2s ^su o ) ) ) )
1173 1172 adantl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( m /su ( 2s ^su o ) ) ) )
1174 n0expscl 
 |-  ( ( 2s e. NN0_s /\ o e. NN0_s ) -> ( 2s ^su o ) e. NN0_s )
1175 422 1166 1174 sylancr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su o ) e. NN0_s )
1176 n0mulscl 
 |-  ( ( ( 2s ^su o ) e. NN0_s /\ g e. NN0_s ) -> ( ( 2s ^su o ) x.s g ) e. NN0_s )
1177 1175 1160 1176 syl2anc 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su o ) x.s g ) e. NN0_s )
1178 1177 n0snod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su o ) x.s g ) e. No )
1179 1161 n0snod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  m e. No )
1180 1178 1179 1166 pw2divsdird 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su o ) x.s g ) +s m ) /su ( 2s ^su o ) ) = ( ( ( ( 2s ^su o ) x.s g ) /su ( 2s ^su o ) ) +s ( m /su ( 2s ^su o ) ) ) )
1181 1160 n0snod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
1182 1181 1166 pw2divscan3d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su o ) x.s g ) /su ( 2s ^su o ) ) = g )
1183 1182 oveq1d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su o ) x.s g ) /su ( 2s ^su o ) ) +s ( m /su ( 2s ^su o ) ) ) = ( g +s ( m /su ( 2s ^su o ) ) ) )
1184 1180 1183 eqtrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su o ) x.s g ) +s m ) /su ( 2s ^su o ) ) = ( g +s ( m /su ( 2s ^su o ) ) ) )
1185 1173 1184 eqtr4d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( ( ( ( 2s ^su o ) x.s g ) +s m ) /su ( 2s ^su o ) ) )
1186 1185 sneqd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { c } = { ( ( ( ( 2s ^su o ) x.s g ) +s m ) /su ( 2s ^su o ) ) } )
1187 simprr2 
 |-  ( ( ( m e. NN0_s /\ n e. NN0_s /\ o e. NN0_s ) /\ ( ( m  d = ( g +s ( n /su ( 2s ^su o ) ) ) )
1188 1187 adantl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( g +s ( n /su ( 2s ^su o ) ) ) )
1189 676 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
1190 1189 adantr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
1191 1190 762 syl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. ( N +s 1s ) <_s N )
1192 330 a1i 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. No )
1193 simprl2 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  n e. NN0_s )
1194 1193 n0snod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  n e. No )
1195 1194 1179 subscld 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( n -s m ) e. No )
1196 1192 1195 sltnled 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 1s  -. ( n -s m ) <_s 1s ) )
1197 695 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
1198 1197 adantr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
1199 1170 adantr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
1200 1159 adantr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
1201 1200 n0snod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
1202 1161 adantrr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  m e. NN0_s )
1203 peano2n0s 
 |-  ( m e. NN0_s -> ( m +s 1s ) e. NN0_s )
1204 1202 1203 syl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s ) e. NN0_s )
1205 1204 n0snod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s ) e. No )
1206 1166 adantrr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  o e. NN0_s )
1207 1205 1206 pw2divscld 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( m +s 1s ) /su ( 2s ^su o ) ) e. No )
1208 1201 1207 addscld 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( ( m +s 1s ) /su ( 2s ^su o ) ) ) e. No )
1209 620 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
1210 1209 adantr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
1211 1173 adantrr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( m /su ( 2s ^su o ) ) ) )
1212 1179 adantrr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  m e. No )
1213 1212 sltp1d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  m
1214 1212 1205 1206 pw2sltdiv1d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m  ( m /su ( 2s ^su o ) )
1215 1212 1206 pw2divscld 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m /su ( 2s ^su o ) ) e. No )
1216 1215 1207 1201 sltadd2d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( m /su ( 2s ^su o ) )  ( g +s ( m /su ( 2s ^su o ) ) )
1217 1214 1216 bitrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m  ( g +s ( m /su ( 2s ^su o ) ) )
1218 1213 1217 mpbid 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( m /su ( 2s ^su o ) ) )
1219 1211 1218 eqbrtrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
1220 1210 1208 1219 ssltsn 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { c } <
1221 622 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
1222 1221 adantr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
1223 1179 1192 addscomd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s ) = ( 1s +s m ) )
1224 1223 breq1d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( m +s 1s )  ( 1s +s m )
1225 1192 1179 1194 sltaddsubd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 1s +s m )  1s
1226 1224 1225 bitrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( m +s 1s )  1s
1227 1226 biimprd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 1s  ( m +s 1s )
1228 1227 impr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s )
1229 1193 adantrr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  n e. NN0_s )
1230 1229 n0snod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  n e. No )
1231 1205 1230 1206 pw2sltdiv1d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( m +s 1s )  ( ( m +s 1s ) /su ( 2s ^su o ) )
1232 1230 1206 pw2divscld 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( n /su ( 2s ^su o ) ) e. No )
1233 1207 1232 1201 sltadd2d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( m +s 1s ) /su ( 2s ^su o ) )  ( g +s ( ( m +s 1s ) /su ( 2s ^su o ) ) )
1234 1231 1233 bitrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( m +s 1s )  ( g +s ( ( m +s 1s ) /su ( 2s ^su o ) ) )
1235 1228 1234 mpbid 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( ( m +s 1s ) /su ( 2s ^su o ) ) )
1236 1188 adantrr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( g +s ( n /su ( 2s ^su o ) ) ) )
1237 1235 1236 breqtrrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( ( m +s 1s ) /su ( 2s ^su o ) ) )
1238 1208 1222 1237 ssltsn 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { ( g +s ( ( m +s 1s ) /su ( 2s ^su o ) ) ) } <
1239 1199 1208 1220 1238 ssltbday 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) C_ ( bday ` ( g +s ( ( m +s 1s ) /su ( 2s ^su o ) ) ) ) )
1240 1198 1239 eqsstrrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` ( N +s 1s ) ) C_ ( bday ` ( g +s ( ( m +s 1s ) /su ( 2s ^su o ) ) ) ) )
1241 1189 adantr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
1242 1241 1 syl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. NN0_s )
1243 expscl 
 |-  ( ( 2s e. No /\ o e. NN0_s ) -> ( 2s ^su o ) e. No )
1244 54 1166 1243 sylancr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su o ) e. No )
1245 1244 adantrr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su o ) e. No )
1246 simprl1 
 |-  ( ( ( m e. NN0_s /\ n e. NN0_s /\ o e. NN0_s ) /\ ( ( m  m
1247 1246 adantl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  m
1248 1247 adantrr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  m
1249 n0sltp1le 
 |-  ( ( m e. NN0_s /\ n e. NN0_s ) -> ( m  ( m +s 1s ) <_s n ) )
1250 1202 1229 1249 syl2anc 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m  ( m +s 1s ) <_s n ) )
1251 1248 1250 mpbid 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s ) <_s n )
1252 simprl3 
 |-  ( ( ( m e. NN0_s /\ n e. NN0_s /\ o e. NN0_s ) /\ ( ( m  n
1253 1252 adantl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  n
1254 1253 adantrr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  n
1255 1205 1230 1245 1251 1254 slelttrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s )
1256 simprr3 
 |-  ( ( ( m e. NN0_s /\ n e. NN0_s /\ o e. NN0_s ) /\ ( ( m  ( g +s o )
1257 1256 adantr 
 |-  ( ( ( ( m e. NN0_s /\ n e. NN0_s /\ o e. NN0_s ) /\ ( ( m  ( g +s o )
1258 1257 adantl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s o )
1259 1242 1200 1204 1206 1255 1258 bdaypw2bnd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` ( g +s ( ( m +s 1s ) /su ( 2s ^su o ) ) ) ) C_ ( bday ` N ) )
1260 1240 1259 sstrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` ( N +s 1s ) ) C_ ( bday ` N ) )
1261 405 263 onsled 
 |-  ( ph -> ( ( N +s 1s ) <_s N <-> ( bday ` ( N +s 1s ) ) C_ ( bday ` N ) ) )
1262 1241 1261 syl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N +s 1s ) <_s N <-> ( bday ` ( N +s 1s ) ) C_ ( bday ` N ) ) )
1263 1260 1262 mpbird 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) <_s N )
1264 1263 expr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 1s  ( N +s 1s ) <_s N ) )
1265 1196 1264 sylbird 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( -. ( n -s m ) <_s 1s -> ( N +s 1s ) <_s N ) )
1266 1191 1265 mt3d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( n -s m ) <_s 1s )
1267 1161 1193 1249 syl2anc 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m  ( m +s 1s ) <_s n ) )
1268 npcans 
 |-  ( ( n e. No /\ 1s e. No ) -> ( ( n -s 1s ) +s 1s ) = n )
1269 1194 330 1268 sylancl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( n -s 1s ) +s 1s ) = n )
1270 1269 breq2d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( m +s 1s ) <_s ( ( n -s 1s ) +s 1s ) <-> ( m +s 1s ) <_s n ) )
1271 1267 1270 bitr4d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m  ( m +s 1s ) <_s ( ( n -s 1s ) +s 1s ) ) )
1272 1194 1192 subscld 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( n -s 1s ) e. No )
1273 1179 1272 1192 sleadd1d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m <_s ( n -s 1s ) <-> ( m +s 1s ) <_s ( ( n -s 1s ) +s 1s ) ) )
1274 1271 1273 bitr4d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m  m <_s ( n -s 1s ) ) )
1275 1179 1194 1192 slesubd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m <_s ( n -s 1s ) <-> 1s <_s ( n -s m ) ) )
1276 1274 1275 bitrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m  1s <_s ( n -s m ) ) )
1277 1247 1276 mpbid 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s <_s ( n -s m ) )
1278 1266 1277 jca 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( n -s m ) <_s 1s /\ 1s <_s ( n -s m ) ) )
1279 1195 1192 sletri3d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( n -s m ) = 1s <-> ( ( n -s m ) <_s 1s /\ 1s <_s ( n -s m ) ) ) )
1280 1278 1279 mpbird 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( n -s m ) = 1s )
1281 1194 1179 1192 subaddsd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( n -s m ) = 1s <-> ( m +s 1s ) = n ) )
1282 1280 1281 mpbid 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s ) = n )
1283 1282 eqcomd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  n = ( m +s 1s ) )
1284 1283 oveq1d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( n /su ( 2s ^su o ) ) = ( ( m +s 1s ) /su ( 2s ^su o ) ) )
1285 1284 oveq2d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( n /su ( 2s ^su o ) ) ) = ( g +s ( ( m +s 1s ) /su ( 2s ^su o ) ) ) )
1286 1188 1285 eqtrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( g +s ( ( m +s 1s ) /su ( 2s ^su o ) ) ) )
1287 1182 oveq1d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su o ) x.s g ) /su ( 2s ^su o ) ) +s ( ( m +s 1s ) /su ( 2s ^su o ) ) ) = ( g +s ( ( m +s 1s ) /su ( 2s ^su o ) ) ) )
1288 1287 eqcomd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( ( m +s 1s ) /su ( 2s ^su o ) ) ) = ( ( ( ( 2s ^su o ) x.s g ) /su ( 2s ^su o ) ) +s ( ( m +s 1s ) /su ( 2s ^su o ) ) ) )
1289 1161 1203 syl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s ) e. NN0_s )
1290 1289 n0snod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s ) e. No )
1291 1178 1290 1166 pw2divsdird 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su o ) x.s g ) +s ( m +s 1s ) ) /su ( 2s ^su o ) ) = ( ( ( ( 2s ^su o ) x.s g ) /su ( 2s ^su o ) ) +s ( ( m +s 1s ) /su ( 2s ^su o ) ) ) )
1292 1288 1291 eqtr4d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( ( m +s 1s ) /su ( 2s ^su o ) ) ) = ( ( ( ( 2s ^su o ) x.s g ) +s ( m +s 1s ) ) /su ( 2s ^su o ) ) )
1293 1178 1179 1192 addsassd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su o ) x.s g ) +s m ) +s 1s ) = ( ( ( 2s ^su o ) x.s g ) +s ( m +s 1s ) ) )
1294 1293 oveq1d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( ( 2s ^su o ) x.s g ) +s m ) +s 1s ) /su ( 2s ^su o ) ) = ( ( ( ( 2s ^su o ) x.s g ) +s ( m +s 1s ) ) /su ( 2s ^su o ) ) )
1295 1294 eqcomd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su o ) x.s g ) +s ( m +s 1s ) ) /su ( 2s ^su o ) ) = ( ( ( ( ( 2s ^su o ) x.s g ) +s m ) +s 1s ) /su ( 2s ^su o ) ) )
1296 1286 1292 1295 3eqtrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( ( ( ( ( 2s ^su o ) x.s g ) +s m ) +s 1s ) /su ( 2s ^su o ) ) )
1297 1296 sneqd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { d } = { ( ( ( ( ( 2s ^su o ) x.s g ) +s m ) +s 1s ) /su ( 2s ^su o ) ) } )
1298 1186 1297 oveq12d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( { c } |s { d } ) = ( { ( ( ( ( 2s ^su o ) x.s g ) +s m ) /su ( 2s ^su o ) ) } |s { ( ( ( ( ( 2s ^su o ) x.s g ) +s m ) +s 1s ) /su ( 2s ^su o ) ) } ) )
1299 n0addscl 
 |-  ( ( ( ( 2s ^su o ) x.s g ) e. NN0_s /\ m e. NN0_s ) -> ( ( ( 2s ^su o ) x.s g ) +s m ) e. NN0_s )
1300 1177 1161 1299 syl2anc 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su o ) x.s g ) +s m ) e. NN0_s )
1301 1300 n0zsd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su o ) x.s g ) +s m ) e. ZZ_s )
1302 1301 1166 pw2cutp1 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( { ( ( ( ( 2s ^su o ) x.s g ) +s m ) /su ( 2s ^su o ) ) } |s { ( ( ( ( ( 2s ^su o ) x.s g ) +s m ) +s 1s ) /su ( 2s ^su o ) ) } ) = ( ( ( 2s x.s ( ( ( 2s ^su o ) x.s g ) +s m ) ) +s 1s ) /su ( 2s ^su ( o +s 1s ) ) ) )
1303 1171 1298 1302 3eqtrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( ( ( 2s x.s ( ( ( 2s ^su o ) x.s g ) +s m ) ) +s 1s ) /su ( 2s ^su ( o +s 1s ) ) ) )
1304 54 a1i 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  2s e. No )
1305 1304 1178 1179 addsdid 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s ( ( ( 2s ^su o ) x.s g ) +s m ) ) = ( ( 2s x.s ( ( 2s ^su o ) x.s g ) ) +s ( 2s x.s m ) ) )
1306 expsp1 
 |-  ( ( 2s e. No /\ o e. NN0_s ) -> ( 2s ^su ( o +s 1s ) ) = ( ( 2s ^su o ) x.s 2s ) )
1307 54 1166 1306 sylancr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( o +s 1s ) ) = ( ( 2s ^su o ) x.s 2s ) )
1308 1244 1304 mulscomd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su o ) x.s 2s ) = ( 2s x.s ( 2s ^su o ) ) )
1309 1307 1308 eqtrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( o +s 1s ) ) = ( 2s x.s ( 2s ^su o ) ) )
1310 1309 oveq1d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( o +s 1s ) ) x.s g ) = ( ( 2s x.s ( 2s ^su o ) ) x.s g ) )
1311 1304 1244 1181 mulsassd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s ( 2s ^su o ) ) x.s g ) = ( 2s x.s ( ( 2s ^su o ) x.s g ) ) )
1312 1310 1311 eqtrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( o +s 1s ) ) x.s g ) = ( 2s x.s ( ( 2s ^su o ) x.s g ) ) )
1313 1312 oveq1d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su ( o +s 1s ) ) x.s g ) +s ( 2s x.s m ) ) = ( ( 2s x.s ( ( 2s ^su o ) x.s g ) ) +s ( 2s x.s m ) ) )
1314 1305 1313 eqtr4d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s ( ( ( 2s ^su o ) x.s g ) +s m ) ) = ( ( ( 2s ^su ( o +s 1s ) ) x.s g ) +s ( 2s x.s m ) ) )
1315 1314 oveq1d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s ( ( ( 2s ^su o ) x.s g ) +s m ) ) +s 1s ) = ( ( ( ( 2s ^su ( o +s 1s ) ) x.s g ) +s ( 2s x.s m ) ) +s 1s ) )
1316 n0expscl 
 |-  ( ( 2s e. NN0_s /\ ( o +s 1s ) e. NN0_s ) -> ( 2s ^su ( o +s 1s ) ) e. NN0_s )
1317 422 1168 1316 sylancr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( o +s 1s ) ) e. NN0_s )
1318 n0mulscl 
 |-  ( ( ( 2s ^su ( o +s 1s ) ) e. NN0_s /\ g e. NN0_s ) -> ( ( 2s ^su ( o +s 1s ) ) x.s g ) e. NN0_s )
1319 1317 1160 1318 syl2anc 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( o +s 1s ) ) x.s g ) e. NN0_s )
1320 1319 n0snod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( o +s 1s ) ) x.s g ) e. No )
1321 1163 n0snod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s m ) e. No )
1322 1320 1321 1192 addsassd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su ( o +s 1s ) ) x.s g ) +s ( 2s x.s m ) ) +s 1s ) = ( ( ( 2s ^su ( o +s 1s ) ) x.s g ) +s ( ( 2s x.s m ) +s 1s ) ) )
1323 1315 1322 eqtrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s ( ( ( 2s ^su o ) x.s g ) +s m ) ) +s 1s ) = ( ( ( 2s ^su ( o +s 1s ) ) x.s g ) +s ( ( 2s x.s m ) +s 1s ) ) )
1324 1323 oveq1d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s x.s ( ( ( 2s ^su o ) x.s g ) +s m ) ) +s 1s ) /su ( 2s ^su ( o +s 1s ) ) ) = ( ( ( ( 2s ^su ( o +s 1s ) ) x.s g ) +s ( ( 2s x.s m ) +s 1s ) ) /su ( 2s ^su ( o +s 1s ) ) ) )
1325 1303 1324 eqtrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( ( ( ( 2s ^su ( o +s 1s ) ) x.s g ) +s ( ( 2s x.s m ) +s 1s ) ) /su ( 2s ^su ( o +s 1s ) ) ) )
1326 1165 n0snod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s m ) +s 1s ) e. No )
1327 1320 1326 1168 pw2divsdird 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su ( o +s 1s ) ) x.s g ) +s ( ( 2s x.s m ) +s 1s ) ) /su ( 2s ^su ( o +s 1s ) ) ) = ( ( ( ( 2s ^su ( o +s 1s ) ) x.s g ) /su ( 2s ^su ( o +s 1s ) ) ) +s ( ( ( 2s x.s m ) +s 1s ) /su ( 2s ^su ( o +s 1s ) ) ) ) )
1328 1181 1168 pw2divscan3d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su ( o +s 1s ) ) x.s g ) /su ( 2s ^su ( o +s 1s ) ) ) = g )
1329 1328 oveq1d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( ( 2s ^su ( o +s 1s ) ) x.s g ) /su ( 2s ^su ( o +s 1s ) ) ) +s ( ( ( 2s x.s m ) +s 1s ) /su ( 2s ^su ( o +s 1s ) ) ) ) = ( g +s ( ( ( 2s x.s m ) +s 1s ) /su ( 2s ^su ( o +s 1s ) ) ) ) )
1330 1325 1327 1329 3eqtrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( g +s ( ( ( 2s x.s m ) +s 1s ) /su ( 2s ^su ( o +s 1s ) ) ) ) )
1331 n0mulscl 
 |-  ( ( 2s e. NN0_s /\ ( m +s 1s ) e. NN0_s ) -> ( 2s x.s ( m +s 1s ) ) e. NN0_s )
1332 422 1289 1331 sylancr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s ( m +s 1s ) ) e. NN0_s )
1333 1332 n0snod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s ( m +s 1s ) ) e. No )
1334 1317 n0snod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( o +s 1s ) ) e. No )
1335 1192 1304 1321 sltadd2d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 1s  ( ( 2s x.s m ) +s 1s )
1336 540 1335 mpbii 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s m ) +s 1s )
1337 1304 1179 1192 addsdid 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s ( m +s 1s ) ) = ( ( 2s x.s m ) +s ( 2s x.s 1s ) ) )
1338 903 oveq2i 
 |-  ( ( 2s x.s m ) +s ( 2s x.s 1s ) ) = ( ( 2s x.s m ) +s 2s )
1339 1337 1338 eqtrdi 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s ( m +s 1s ) ) = ( ( 2s x.s m ) +s 2s ) )
1340 1336 1339 breqtrrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s m ) +s 1s )
1341 simprl2 
 |-  ( ( ( m e. NN0_s /\ n e. NN0_s /\ o e. NN0_s ) /\ ( ( m  m
1342 1341 adantl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  m
1343 n0sltp1le 
 |-  ( ( m e. NN0_s /\ ( 2s ^su o ) e. NN0_s ) -> ( m  ( m +s 1s ) <_s ( 2s ^su o ) ) )
1344 1161 1175 1343 syl2anc 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m  ( m +s 1s ) <_s ( 2s ^su o ) ) )
1345 1342 1344 mpbid 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s ) <_s ( 2s ^su o ) )
1346 1007 a1i 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s
1347 1290 1244 1304 1346 slemul2d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( m +s 1s ) <_s ( 2s ^su o ) <-> ( 2s x.s ( m +s 1s ) ) <_s ( 2s x.s ( 2s ^su o ) ) ) )
1348 1345 1347 mpbid 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s ( m +s 1s ) ) <_s ( 2s x.s ( 2s ^su o ) ) )
1349 1348 1309 breqtrrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s ( m +s 1s ) ) <_s ( 2s ^su ( o +s 1s ) ) )
1350 1326 1333 1334 1340 1349 sltletrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s m ) +s 1s )
1351 1166 n0snod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  o e. No )
1352 1181 1351 1192 addsassd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s o ) +s 1s ) = ( g +s ( o +s 1s ) ) )
1353 1256 adantl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s o )
1354 n0addscl 
 |-  ( ( g e. NN0_s /\ o e. NN0_s ) -> ( g +s o ) e. NN0_s )
1355 1160 1166 1354 syl2anc 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s o ) e. NN0_s )
1356 1355 n0snod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s o ) e. No )
1357 1190 34 syl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. No )
1358 1356 1357 1192 sltadd1d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s o )  ( ( g +s o ) +s 1s )
1359 1353 1358 mpbid 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s o ) +s 1s )
1360 1352 1359 eqbrtrrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( o +s 1s ) )
1361 oveq1 
 |-  ( b = ( ( 2s x.s m ) +s 1s ) -> ( b /su ( 2s ^su q ) ) = ( ( ( 2s x.s m ) +s 1s ) /su ( 2s ^su q ) ) )
1362 1361 oveq2d 
 |-  ( b = ( ( 2s x.s m ) +s 1s ) -> ( g +s ( b /su ( 2s ^su q ) ) ) = ( g +s ( ( ( 2s x.s m ) +s 1s ) /su ( 2s ^su q ) ) ) )
1363 1362 eqeq2d 
 |-  ( b = ( ( 2s x.s m ) +s 1s ) -> ( w = ( g +s ( b /su ( 2s ^su q ) ) ) <-> w = ( g +s ( ( ( 2s x.s m ) +s 1s ) /su ( 2s ^su q ) ) ) ) )
1364 breq1 
 |-  ( b = ( ( 2s x.s m ) +s 1s ) -> ( b  ( ( 2s x.s m ) +s 1s )
1365 1363 1364 3anbi12d 
 |-  ( b = ( ( 2s x.s m ) +s 1s ) -> ( ( w = ( g +s ( b /su ( 2s ^su q ) ) ) /\ b  ( w = ( g +s ( ( ( 2s x.s m ) +s 1s ) /su ( 2s ^su q ) ) ) /\ ( ( 2s x.s m ) +s 1s )
1366 oveq2 
 |-  ( q = ( o +s 1s ) -> ( 2s ^su q ) = ( 2s ^su ( o +s 1s ) ) )
1367 1366 oveq2d 
 |-  ( q = ( o +s 1s ) -> ( ( ( 2s x.s m ) +s 1s ) /su ( 2s ^su q ) ) = ( ( ( 2s x.s m ) +s 1s ) /su ( 2s ^su ( o +s 1s ) ) ) )
1368 1367 oveq2d 
 |-  ( q = ( o +s 1s ) -> ( g +s ( ( ( 2s x.s m ) +s 1s ) /su ( 2s ^su q ) ) ) = ( g +s ( ( ( 2s x.s m ) +s 1s ) /su ( 2s ^su ( o +s 1s ) ) ) ) )
1369 1368 eqeq2d 
 |-  ( q = ( o +s 1s ) -> ( w = ( g +s ( ( ( 2s x.s m ) +s 1s ) /su ( 2s ^su q ) ) ) <-> w = ( g +s ( ( ( 2s x.s m ) +s 1s ) /su ( 2s ^su ( o +s 1s ) ) ) ) ) )
1370 1366 breq2d 
 |-  ( q = ( o +s 1s ) -> ( ( ( 2s x.s m ) +s 1s )  ( ( 2s x.s m ) +s 1s )
1371 oveq2 
 |-  ( q = ( o +s 1s ) -> ( g +s q ) = ( g +s ( o +s 1s ) ) )
1372 1371 breq1d 
 |-  ( q = ( o +s 1s ) -> ( ( g +s q )  ( g +s ( o +s 1s ) )
1373 1369 1370 1372 3anbi123d 
 |-  ( q = ( o +s 1s ) -> ( ( w = ( g +s ( ( ( 2s x.s m ) +s 1s ) /su ( 2s ^su q ) ) ) /\ ( ( 2s x.s m ) +s 1s )  ( w = ( g +s ( ( ( 2s x.s m ) +s 1s ) /su ( 2s ^su ( o +s 1s ) ) ) ) /\ ( ( 2s x.s m ) +s 1s )
1374 552 1365 1373 rspc3ev 
 |-  ( ( ( g e. NN0_s /\ ( ( 2s x.s m ) +s 1s ) e. NN0_s /\ ( o +s 1s ) e. NN0_s ) /\ ( w = ( g +s ( ( ( 2s x.s m ) +s 1s ) /su ( 2s ^su ( o +s 1s ) ) ) ) /\ ( ( 2s x.s m ) +s 1s )  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1375 1160 1165 1168 1330 1350 1360 1374 syl33anc 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1376 1375 expr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( m  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1377 1376 rexlimdvvva 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( E. m e. NN0_s E. n e. NN0_s E. o e. NN0_s ( ( m  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1378 1158 1377 mpd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1379 1378 expr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h /su ( 2s ^su i ) )  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1380 963 1379 mpd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1381 1380 expr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g = j -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1382 946 1381 jaod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1383 645 1382 sylbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g <_s j -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1384 643 1383 mpd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1385 1384 expr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g <_s c -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1386 617 1385 mpd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1387 1386 ex 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( d  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1388 596 1387 mpd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1389 1388 expr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( d = ( j +s ( k /su ( 2s ^su l ) ) ) /\ k  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1390 1389 rexlimdvvva 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( E. j e. NN0_s E. k e. NN0_s E. l e. NN0_s ( d = ( j +s ( k /su ( 2s ^su l ) ) ) /\ k  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1391 273 3adant3 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` d ) C_ ( bday ` N ) )
1392 59 a1i 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s e. No )
1393 124 3ad2ant1 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w e. No )
1394 simp1r 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s
1395 1393 1394 0elleft 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s e. ( _Left ` w ) )
1396 215 192 1395 rspcdva 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s <_s c )
1397 1392 344 393 1396 285 slelttrd 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s
1398 1392 393 1397 sltled 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s <_s d )
1399 1398 3ad2ant1 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s <_s d )
1400 fveq2 
 |-  ( z = d -> ( bday ` z ) = ( bday ` d ) )
1401 1400 sseq1d 
 |-  ( z = d -> ( ( bday ` z ) C_ ( bday ` N ) <-> ( bday ` d ) C_ ( bday ` N ) ) )
1402 breq2 
 |-  ( z = d -> ( 0s <_s z <-> 0s <_s d ) )
1403 1401 1402 anbi12d 
 |-  ( z = d -> ( ( ( bday ` z ) C_ ( bday ` N ) /\ 0s <_s z ) <-> ( ( bday ` d ) C_ ( bday ` N ) /\ 0s <_s d ) ) )
1404 eqeq1 
 |-  ( z = d -> ( z = N <-> d = N ) )
1405 eqeq1 
 |-  ( z = d -> ( z = ( x +s ( y /su ( 2s ^su p ) ) ) <-> d = ( x +s ( y /su ( 2s ^su p ) ) ) ) )
1406 1405 3anbi1d 
 |-  ( z = d -> ( ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  ( d = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
1407 1406 rexbidv 
 |-  ( z = d -> ( E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. p e. NN0_s ( d = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
1408 1407 2rexbidv 
 |-  ( z = d -> ( E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( d = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
1409 oveq1 
 |-  ( x = j -> ( x +s ( y /su ( 2s ^su p ) ) ) = ( j +s ( y /su ( 2s ^su p ) ) ) )
1410 1409 eqeq2d 
 |-  ( x = j -> ( d = ( x +s ( y /su ( 2s ^su p ) ) ) <-> d = ( j +s ( y /su ( 2s ^su p ) ) ) ) )
1411 oveq1 
 |-  ( x = j -> ( x +s p ) = ( j +s p ) )
1412 1411 breq1d 
 |-  ( x = j -> ( ( x +s p )  ( j +s p )
1413 1410 1412 3anbi13d 
 |-  ( x = j -> ( ( d = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  ( d = ( j +s ( y /su ( 2s ^su p ) ) ) /\ y
1414 1413 rexbidv 
 |-  ( x = j -> ( E. p e. NN0_s ( d = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. p e. NN0_s ( d = ( j +s ( y /su ( 2s ^su p ) ) ) /\ y
1415 oveq1 
 |-  ( y = k -> ( y /su ( 2s ^su p ) ) = ( k /su ( 2s ^su p ) ) )
1416 1415 oveq2d 
 |-  ( y = k -> ( j +s ( y /su ( 2s ^su p ) ) ) = ( j +s ( k /su ( 2s ^su p ) ) ) )
1417 1416 eqeq2d 
 |-  ( y = k -> ( d = ( j +s ( y /su ( 2s ^su p ) ) ) <-> d = ( j +s ( k /su ( 2s ^su p ) ) ) ) )
1418 breq1 
 |-  ( y = k -> ( y  k
1419 1417 1418 3anbi12d 
 |-  ( y = k -> ( ( d = ( j +s ( y /su ( 2s ^su p ) ) ) /\ y  ( d = ( j +s ( k /su ( 2s ^su p ) ) ) /\ k
1420 1419 rexbidv 
 |-  ( y = k -> ( E. p e. NN0_s ( d = ( j +s ( y /su ( 2s ^su p ) ) ) /\ y  E. p e. NN0_s ( d = ( j +s ( k /su ( 2s ^su p ) ) ) /\ k
1421 oveq2 
 |-  ( p = l -> ( 2s ^su p ) = ( 2s ^su l ) )
1422 1421 oveq2d 
 |-  ( p = l -> ( k /su ( 2s ^su p ) ) = ( k /su ( 2s ^su l ) ) )
1423 1422 oveq2d 
 |-  ( p = l -> ( j +s ( k /su ( 2s ^su p ) ) ) = ( j +s ( k /su ( 2s ^su l ) ) ) )
1424 1423 eqeq2d 
 |-  ( p = l -> ( d = ( j +s ( k /su ( 2s ^su p ) ) ) <-> d = ( j +s ( k /su ( 2s ^su l ) ) ) ) )
1425 1421 breq2d 
 |-  ( p = l -> ( k  k
1426 oveq2 
 |-  ( p = l -> ( j +s p ) = ( j +s l ) )
1427 1426 breq1d 
 |-  ( p = l -> ( ( j +s p )  ( j +s l )
1428 1424 1425 1427 3anbi123d 
 |-  ( p = l -> ( ( d = ( j +s ( k /su ( 2s ^su p ) ) ) /\ k  ( d = ( j +s ( k /su ( 2s ^su l ) ) ) /\ k
1429 1428 cbvrexvw 
 |-  ( E. p e. NN0_s ( d = ( j +s ( k /su ( 2s ^su p ) ) ) /\ k  E. l e. NN0_s ( d = ( j +s ( k /su ( 2s ^su l ) ) ) /\ k
1430 1420 1429 bitrdi 
 |-  ( y = k -> ( E. p e. NN0_s ( d = ( j +s ( y /su ( 2s ^su p ) ) ) /\ y  E. l e. NN0_s ( d = ( j +s ( k /su ( 2s ^su l ) ) ) /\ k
1431 1414 1430 cbvrex2vw 
 |-  ( E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( d = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. j e. NN0_s E. k e. NN0_s E. l e. NN0_s ( d = ( j +s ( k /su ( 2s ^su l ) ) ) /\ k
1432 1408 1431 bitrdi 
 |-  ( z = d -> ( E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  E. j e. NN0_s E. k e. NN0_s E. l e. NN0_s ( d = ( j +s ( k /su ( 2s ^su l ) ) ) /\ k
1433 1404 1432 orbi12d 
 |-  ( z = d -> ( ( z = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  ( d = N \/ E. j e. NN0_s E. k e. NN0_s E. l e. NN0_s ( d = ( j +s ( k /su ( 2s ^su l ) ) ) /\ k
1434 1403 1433 imbi12d 
 |-  ( z = d -> ( ( ( ( bday ` z ) C_ ( bday ` N ) /\ 0s <_s z ) -> ( z = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  ( ( ( bday ` d ) C_ ( bday ` N ) /\ 0s <_s d ) -> ( d = N \/ E. j e. NN0_s E. k e. NN0_s E. l e. NN0_s ( d = ( j +s ( k /su ( 2s ^su l ) ) ) /\ k
1435 302 2 syl 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  A. z e. No ( ( ( bday ` z ) C_ ( bday ` N ) /\ 0s <_s z ) -> ( z = N \/ E. x e. NN0_s E. y e. NN0_s E. p e. NN0_s ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
1436 1434 1435 394 rspcdva 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( bday ` d ) C_ ( bday ` N ) /\ 0s <_s d ) -> ( d = N \/ E. j e. NN0_s E. k e. NN0_s E. l e. NN0_s ( d = ( j +s ( k /su ( 2s ^su l ) ) ) /\ k
1437 1391 1399 1436 mp2and 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( d = N \/ E. j e. NN0_s E. k e. NN0_s E. l e. NN0_s ( d = ( j +s ( k /su ( 2s ^su l ) ) ) /\ k
1438 578 1390 1437 mpjaod 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1439 1438 3expa 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1440 1439 expr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( c = ( g +s ( h /su ( 2s ^su i ) ) ) /\ h  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1441 1440 rexlimdvvva 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( E. g e. NN0_s E. h e. NN0_s E. i e. NN0_s ( c = ( g +s ( h /su ( 2s ^su i ) ) ) /\ h  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1442 293 1441 syld 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( bday ` c ) C_ ( bday ` N ) /\ 0s <_s c ) -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1443 214 219 1442 mp2and 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1444 1443 ex 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( w = ( { c } |s { d } ) -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1445 195 1444 mpd 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1446 1445 3expa 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1447 1446 expr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( A. f e. ( _Right ` w ) d <_s f -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1448 190 1447 sylbird 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( A. f e. ( _Right ` w ) -. f  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1449 1448 rexlimdva 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( E. d e. ( _Right ` w ) A. f e. ( _Right ` w ) -. f  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1450 184 1449 syl5 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( _Right ` w ) e. Fin /\ ( _Right ` w ) =/= (/) ) -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1451 158 179 1450 mp2and 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1452 1451 expr 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( A. e e. ( _Left ` w ) e <_s c -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1453 154 1452 sylbird 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( A. e e. ( _Left ` w ) -. c  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1454 1453 rexlimdva 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( E. c e. ( _Left ` w ) A. e e. ( _Left ` w ) -. c  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1455 147 1454 syl5 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( _Left ` w ) e. Fin /\ ( _Left ` w ) =/= (/) ) -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1456 141 1455 mpand 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( _Left ` w ) =/= (/) -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1457 127 1456 mpd 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1458 1457 ex 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) -> ( 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1459 id 
 |-  ( ph -> ph )
1460 n0p1nns 
 |-  ( N e. NN0_s -> ( N +s 1s ) e. NN_s )
1461 1 1460 syl 
 |-  ( ph -> ( N +s 1s ) e. NN_s )
1462 nnsgt0 
 |-  ( ( N +s 1s ) e. NN_s -> 0s
1463 1461 1462 syl 
 |-  ( ph -> 0s
1464 addslid 
 |-  ( 0s e. No -> ( 0s +s 0s ) = 0s )
1465 59 1464 ax-mp 
 |-  ( 0s +s 0s ) = 0s
1466 1465 eqcomi 
 |-  0s = ( 0s +s 0s )
1467 31 31 31 3pm3.2i 
 |-  ( 0s e. NN0_s /\ 0s e. NN0_s /\ 0s e. NN0_s )
1468 oveq1 
 |-  ( a = 0s -> ( a +s ( b /su ( 2s ^su q ) ) ) = ( 0s +s ( b /su ( 2s ^su q ) ) ) )
1469 1468 eqeq2d 
 |-  ( a = 0s -> ( 0s = ( a +s ( b /su ( 2s ^su q ) ) ) <-> 0s = ( 0s +s ( b /su ( 2s ^su q ) ) ) ) )
1470 oveq1 
 |-  ( a = 0s -> ( a +s q ) = ( 0s +s q ) )
1471 1470 breq1d 
 |-  ( a = 0s -> ( ( a +s q )  ( 0s +s q )
1472 1469 1471 3anbi13d 
 |-  ( a = 0s -> ( ( 0s = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b  ( 0s = ( 0s +s ( b /su ( 2s ^su q ) ) ) /\ b
1473 48 oveq2d 
 |-  ( b = 0s -> ( 0s +s ( b /su ( 2s ^su q ) ) ) = ( 0s +s ( 0s /su ( 2s ^su q ) ) ) )
1474 1473 eqeq2d 
 |-  ( b = 0s -> ( 0s = ( 0s +s ( b /su ( 2s ^su q ) ) ) <-> 0s = ( 0s +s ( 0s /su ( 2s ^su q ) ) ) ) )
1475 1474 51 3anbi12d 
 |-  ( b = 0s -> ( ( 0s = ( 0s +s ( b /su ( 2s ^su q ) ) ) /\ b  ( 0s = ( 0s +s ( 0s /su ( 2s ^su q ) ) ) /\ 0s
1476 62 oveq2d 
 |-  ( q = 0s -> ( 0s +s ( 0s /su ( 2s ^su q ) ) ) = ( 0s +s 0s ) )
1477 1476 eqeq2d 
 |-  ( q = 0s -> ( 0s = ( 0s +s ( 0s /su ( 2s ^su q ) ) ) <-> 0s = ( 0s +s 0s ) ) )
1478 oveq2 
 |-  ( q = 0s -> ( 0s +s q ) = ( 0s +s 0s ) )
1479 1478 1465 eqtrdi 
 |-  ( q = 0s -> ( 0s +s q ) = 0s )
1480 1479 breq1d 
 |-  ( q = 0s -> ( ( 0s +s q )  0s
1481 1477 65 1480 3anbi123d 
 |-  ( q = 0s -> ( ( 0s = ( 0s +s ( 0s /su ( 2s ^su q ) ) ) /\ 0s  ( 0s = ( 0s +s 0s ) /\ 0s
1482 1472 1475 1481 rspc3ev 
 |-  ( ( ( 0s e. NN0_s /\ 0s e. NN0_s /\ 0s e. NN0_s ) /\ ( 0s = ( 0s +s 0s ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( 0s = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1483 1467 1482 mpan 
 |-  ( ( 0s = ( 0s +s 0s ) /\ 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( 0s = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1484 1466 38 1483 mp3an12 
 |-  ( 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( 0s = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1485 1463 1484 syl 
 |-  ( ph -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( 0s = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1486 1459 1485 syl 
 |-  ( ph -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( 0s = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1487 eqeq1 
 |-  ( 0s = w -> ( 0s = ( a +s ( b /su ( 2s ^su q ) ) ) <-> w = ( a +s ( b /su ( 2s ^su q ) ) ) ) )
1488 1487 3anbi1d 
 |-  ( 0s = w -> ( ( 0s = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b  ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1489 1488 rexbidv 
 |-  ( 0s = w -> ( E. q e. NN0_s ( 0s = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b  E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1490 1489 2rexbidv 
 |-  ( 0s = w -> ( E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( 0s = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1491 1486 1490 syl5ibcom 
 |-  ( ph -> ( 0s = w -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1492 1491 adantr 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) -> ( 0s = w -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1493 1458 1492 jaod 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) -> ( ( 0s  E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1494 123 1493 sylbid 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) -> ( 0s <_s w -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1495 1494 expr 
 |-  ( ( ph /\ w e. No ) -> ( ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) -> ( 0s <_s w -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1496 1495 expd 
 |-  ( ( ph /\ w e. No ) -> ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) -> ( w =/= ( N +s 1s ) -> ( 0s <_s w -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1497 1496 com34 
 |-  ( ( ph /\ w e. No ) -> ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) -> ( 0s <_s w -> ( w =/= ( N +s 1s ) -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1498 1497 impd 
 |-  ( ( ph /\ w e. No ) -> ( ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ 0s <_s w ) -> ( w =/= ( N +s 1s ) -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1499 1498 impr 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ 0s <_s w ) ) ) -> ( w =/= ( N +s 1s ) -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1500 120 1499 biimtrrid 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ 0s <_s w ) ) ) -> ( -. w = ( N +s 1s ) -> E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1501 1500 orrd 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ 0s <_s w ) ) ) -> ( w = ( N +s 1s ) \/ E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1502 1501 expr 
 |-  ( ( ph /\ w e. No ) -> ( ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ 0s <_s w ) -> ( w = ( N +s 1s ) \/ E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1503 1502 expd 
 |-  ( ( ph /\ w e. No ) -> ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) -> ( 0s <_s w -> ( w = ( N +s 1s ) \/ E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1504 119 1503 sylbird 
 |-  ( ( ph /\ w e. No ) -> ( ( bday ` w ) = suc ( bday ` N ) -> ( 0s <_s w -> ( w = ( N +s 1s ) \/ E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1505 118 1504 jaod 
 |-  ( ( ph /\ w e. No ) -> ( ( ( bday ` w ) C_ ( bday ` N ) \/ ( bday ` w ) = suc ( bday ` N ) ) -> ( 0s <_s w -> ( w = ( N +s 1s ) \/ E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1506 16 1505 biimtrid 
 |-  ( ( ph /\ w e. No ) -> ( ( bday ` w ) C_ suc ( bday ` N ) -> ( 0s <_s w -> ( w = ( N +s 1s ) \/ E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1507 6 1506 sylbid 
 |-  ( ( ph /\ w e. No ) -> ( ( bday ` w ) C_ ( bday ` ( N +s 1s ) ) -> ( 0s <_s w -> ( w = ( N +s 1s ) \/ E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1508 1507 impd 
 |-  ( ( ph /\ w e. No ) -> ( ( ( bday ` w ) C_ ( bday ` ( N +s 1s ) ) /\ 0s <_s w ) -> ( w = ( N +s 1s ) \/ E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1509 1508 ralrimiva 
 |-  ( ph -> A. w e. No ( ( ( bday ` w ) C_ ( bday ` ( N +s 1s ) ) /\ 0s <_s w ) -> ( w = ( N +s 1s ) \/ E. a e. NN0_s E. b e. NN0_s E. q e. NN0_s ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b