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 bdayon 
 |-  ( bday ` w ) e. On
8 bdayon 
 |-  ( bday ` N ) e. On
9 8 onsuci 
 |-  suc ( bday ` N ) e. On
10 7 9 onsseli 
 |-  ( ( bday ` w ) C_ suc ( bday ` N ) <-> ( ( bday ` w ) e. suc ( bday ` N ) \/ ( bday ` w ) = suc ( bday ` N ) ) )
11 onsssuc 
 |-  ( ( ( bday ` w ) e. On /\ ( bday ` N ) e. On ) -> ( ( bday ` w ) C_ ( bday ` N ) <-> ( bday ` w ) e. suc ( bday ` N ) ) )
12 7 8 11 mp2an 
 |-  ( ( bday ` w ) C_ ( bday ` N ) <-> ( bday ` w ) e. suc ( bday ` N ) )
13 12 orbi1i 
 |-  ( ( ( bday ` w ) C_ ( bday ` N ) \/ ( bday ` w ) = suc ( bday ` N ) ) <-> ( ( bday ` w ) e. suc ( bday ` N ) \/ ( bday ` w ) = suc ( bday ` N ) ) )
14 10 13 bitr4i 
 |-  ( ( bday ` w ) C_ suc ( bday ` N ) <-> ( ( bday ` w ) C_ ( bday ` N ) \/ ( bday ` w ) = suc ( bday ` N ) ) )
15 fveq2 
 |-  ( z = w -> ( bday ` z ) = ( bday ` w ) )
16 15 sseq1d 
 |-  ( z = w -> ( ( bday ` z ) C_ ( bday ` N ) <-> ( bday ` w ) C_ ( bday ` N ) ) )
17 breq2 
 |-  ( z = w -> ( 0s <_s z <-> 0s <_s w ) )
18 16 17 anbi12d 
 |-  ( z = w -> ( ( ( bday ` z ) C_ ( bday ` N ) /\ 0s <_s z ) <-> ( ( bday ` w ) C_ ( bday ` N ) /\ 0s <_s w ) ) )
19 eqeq1 
 |-  ( z = w -> ( z = N <-> w = N ) )
20 eqeq1 
 |-  ( z = w -> ( z = ( x +s ( y /su ( 2s ^su p ) ) ) <-> w = ( x +s ( y /su ( 2s ^su p ) ) ) ) )
21 20 3anbi1d 
 |-  ( z = w -> ( ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
22 21 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
23 22 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
24 19 23 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
25 18 24 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
26 25 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
27 2 26 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
28 1 adantr 
 |-  ( ( ph /\ ( w e. No /\ w = N ) ) -> N e. NN0_s )
29 0n0s 
 |-  0s e. NN0_s
30 29 a1i 
 |-  ( ( ph /\ ( w e. No /\ w = N ) ) -> 0s e. NN0_s )
31 simprr 
 |-  ( ( ph /\ ( w e. No /\ w = N ) ) -> w = N )
32 1 n0nod 
 |-  ( ph -> N e. No )
33 32 addsridd 
 |-  ( ph -> ( N +s 0s ) = N )
34 33 adantr 
 |-  ( ( ph /\ ( w e. No /\ w = N ) ) -> ( N +s 0s ) = N )
35 31 34 eqtr4d 
 |-  ( ( ph /\ ( w e. No /\ w = N ) ) -> w = ( N +s 0s ) )
36 0lt1s 
 |-  0s
37 36 a1i 
 |-  ( ( ph /\ ( w e. No /\ w = N ) ) -> 0s
38 32 ltsp1d 
 |-  ( ph -> N
39 33 38 eqbrtrd 
 |-  ( ph -> ( N +s 0s )
40 39 adantr 
 |-  ( ( ph /\ ( w e. No /\ w = N ) ) -> ( N +s 0s )
41 oveq1 
 |-  ( a = N -> ( a +s ( b /su ( 2s ^su q ) ) ) = ( N +s ( b /su ( 2s ^su q ) ) ) )
42 41 eqeq2d 
 |-  ( a = N -> ( w = ( a +s ( b /su ( 2s ^su q ) ) ) <-> w = ( N +s ( b /su ( 2s ^su q ) ) ) ) )
43 oveq1 
 |-  ( a = N -> ( a +s q ) = ( N +s q ) )
44 43 breq1d 
 |-  ( a = N -> ( ( a +s q )  ( N +s q )
45 42 44 3anbi13d 
 |-  ( a = N -> ( ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b  ( w = ( N +s ( b /su ( 2s ^su q ) ) ) /\ b
46 oveq1 
 |-  ( b = 0s -> ( b /su ( 2s ^su q ) ) = ( 0s /su ( 2s ^su q ) ) )
47 46 oveq2d 
 |-  ( b = 0s -> ( N +s ( b /su ( 2s ^su q ) ) ) = ( N +s ( 0s /su ( 2s ^su q ) ) ) )
48 47 eqeq2d 
 |-  ( b = 0s -> ( w = ( N +s ( b /su ( 2s ^su q ) ) ) <-> w = ( N +s ( 0s /su ( 2s ^su q ) ) ) ) )
49 breq1 
 |-  ( b = 0s -> ( b  0s
50 48 49 3anbi12d 
 |-  ( b = 0s -> ( ( w = ( N +s ( b /su ( 2s ^su q ) ) ) /\ b  ( w = ( N +s ( 0s /su ( 2s ^su q ) ) ) /\ 0s
51 oveq2 
 |-  ( q = 0s -> ( 2s ^su q ) = ( 2s ^su 0s ) )
52 2no 
 |-  2s e. No
53 exps0 
 |-  ( 2s e. No -> ( 2s ^su 0s ) = 1s )
54 52 53 ax-mp 
 |-  ( 2s ^su 0s ) = 1s
55 51 54 eqtrdi 
 |-  ( q = 0s -> ( 2s ^su q ) = 1s )
56 55 oveq2d 
 |-  ( q = 0s -> ( 0s /su ( 2s ^su q ) ) = ( 0s /su 1s ) )
57 0no 
 |-  0s e. No
58 divs1 
 |-  ( 0s e. No -> ( 0s /su 1s ) = 0s )
59 57 58 ax-mp 
 |-  ( 0s /su 1s ) = 0s
60 56 59 eqtrdi 
 |-  ( q = 0s -> ( 0s /su ( 2s ^su q ) ) = 0s )
61 60 oveq2d 
 |-  ( q = 0s -> ( N +s ( 0s /su ( 2s ^su q ) ) ) = ( N +s 0s ) )
62 61 eqeq2d 
 |-  ( q = 0s -> ( w = ( N +s ( 0s /su ( 2s ^su q ) ) ) <-> w = ( N +s 0s ) ) )
63 55 breq2d 
 |-  ( q = 0s -> ( 0s  0s
64 oveq2 
 |-  ( q = 0s -> ( N +s q ) = ( N +s 0s ) )
65 64 breq1d 
 |-  ( q = 0s -> ( ( N +s q )  ( N +s 0s )
66 62 63 65 3anbi123d 
 |-  ( q = 0s -> ( ( w = ( N +s ( 0s /su ( 2s ^su q ) ) ) /\ 0s  ( w = ( N +s 0s ) /\ 0s
67 45 50 66 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
68 28 30 30 35 37 40 67 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
69 68 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
70 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 ) ) ) ) )
71 idd 
 |-  ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) -> ( y  y
72 n0addscl 
 |-  ( ( x e. NN0_s /\ p e. NN0_s ) -> ( x +s p ) e. NN0_s )
73 72 n0nod 
 |-  ( ( x e. NN0_s /\ p e. NN0_s ) -> ( x +s p ) e. No )
74 73 3adant2 
 |-  ( ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) -> ( x +s p ) e. No )
75 74 adantl 
 |-  ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) -> ( x +s p ) e. No )
76 75 adantr 
 |-  ( ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) /\ ( x +s p )  ( x +s p ) e. No )
77 32 adantr 
 |-  ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) -> N e. No )
78 77 adantr 
 |-  ( ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) /\ ( x +s p )  N e. No )
79 peano2no 
 |-  ( N e. No -> ( N +s 1s ) e. No )
80 78 79 syl 
 |-  ( ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) /\ ( x +s p )  ( N +s 1s ) e. No )
81 simpr 
 |-  ( ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) /\ ( x +s p )  ( x +s p )
82 77 ltsp1d 
 |-  ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) -> N
83 82 adantr 
 |-  ( ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) /\ ( x +s p )  N
84 76 78 80 81 83 ltstrd 
 |-  ( ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) /\ ( x +s p )  ( x +s p )
85 84 ex 
 |-  ( ( ph /\ ( x e. NN0_s /\ y e. NN0_s /\ p e. NN0_s ) ) -> ( ( x +s p )  ( x +s p )
86 70 71 85 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
87 oveq1 
 |-  ( a = x -> ( a +s ( b /su ( 2s ^su q ) ) ) = ( x +s ( b /su ( 2s ^su q ) ) ) )
88 87 eqeq2d 
 |-  ( a = x -> ( w = ( a +s ( b /su ( 2s ^su q ) ) ) <-> w = ( x +s ( b /su ( 2s ^su q ) ) ) ) )
89 oveq1 
 |-  ( a = x -> ( a +s q ) = ( x +s q ) )
90 89 breq1d 
 |-  ( a = x -> ( ( a +s q )  ( x +s q )
91 88 90 3anbi13d 
 |-  ( a = x -> ( ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b  ( w = ( x +s ( b /su ( 2s ^su q ) ) ) /\ b
92 oveq1 
 |-  ( b = y -> ( b /su ( 2s ^su q ) ) = ( y /su ( 2s ^su q ) ) )
93 92 oveq2d 
 |-  ( b = y -> ( x +s ( b /su ( 2s ^su q ) ) ) = ( x +s ( y /su ( 2s ^su q ) ) ) )
94 93 eqeq2d 
 |-  ( b = y -> ( w = ( x +s ( b /su ( 2s ^su q ) ) ) <-> w = ( x +s ( y /su ( 2s ^su q ) ) ) ) )
95 breq1 
 |-  ( b = y -> ( b  y
96 94 95 3anbi12d 
 |-  ( b = y -> ( ( w = ( x +s ( b /su ( 2s ^su q ) ) ) /\ b  ( w = ( x +s ( y /su ( 2s ^su q ) ) ) /\ y
97 oveq2 
 |-  ( q = p -> ( 2s ^su q ) = ( 2s ^su p ) )
98 97 oveq2d 
 |-  ( q = p -> ( y /su ( 2s ^su q ) ) = ( y /su ( 2s ^su p ) ) )
99 98 oveq2d 
 |-  ( q = p -> ( x +s ( y /su ( 2s ^su q ) ) ) = ( x +s ( y /su ( 2s ^su p ) ) ) )
100 99 eqeq2d 
 |-  ( q = p -> ( w = ( x +s ( y /su ( 2s ^su q ) ) ) <-> w = ( x +s ( y /su ( 2s ^su p ) ) ) ) )
101 97 breq2d 
 |-  ( q = p -> ( y  y
102 oveq2 
 |-  ( q = p -> ( x +s q ) = ( x +s p ) )
103 102 breq1d 
 |-  ( q = p -> ( ( x +s q )  ( x +s p )
104 100 101 103 3anbi123d 
 |-  ( q = p -> ( ( w = ( x +s ( y /su ( 2s ^su q ) ) ) /\ y  ( w = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
105 91 96 104 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
106 105 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
107 106 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
108 86 107 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
109 108 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
110 109 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
111 69 110 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
112 27 111 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
113 112 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
114 113 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
115 114 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
116 115 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
117 5 eqeq2d 
 |-  ( ( ph /\ w e. No ) -> ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) <-> ( bday ` w ) = suc ( bday ` N ) ) )
118 df-ne 
 |-  ( w =/= ( N +s 1s ) <-> -. w = ( N +s 1s ) )
119 simprl 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) -> w e. No )
120 lesloe 
 |-  ( ( 0s e. No /\ w e. No ) -> ( 0s <_s w <-> ( 0s
121 57 119 120 sylancr 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) -> ( 0s <_s w <-> ( 0s
122 simprrl 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) -> ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
123 122 adantr 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
124 1 peano2n0sd 
 |-  ( ph -> ( N +s 1s ) e. NN0_s )
125 n0bday 
 |-  ( ( N +s 1s ) e. NN0_s -> ( bday ` ( N +s 1s ) ) e. _om )
126 124 125 syl 
 |-  ( ph -> ( bday ` ( N +s 1s ) ) e. _om )
127 126 adantr 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) -> ( bday ` ( N +s 1s ) ) e. _om )
128 127 adantr 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` ( N +s 1s ) ) e. _om )
129 123 128 eqeltrd 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) e. _om )
130 oldfi 
 |-  ( ( bday ` w ) e. _om -> ( _Old ` ( bday ` w ) ) e. Fin )
131 129 130 syl 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( _Old ` ( bday ` w ) ) e. Fin )
132 leftssold 
 |-  ( _Left ` w ) C_ ( _Old ` ( bday ` w ) )
133 ssfi 
 |-  ( ( ( _Old ` ( bday ` w ) ) e. Fin /\ ( _Left ` w ) C_ ( _Old ` ( bday ` w ) ) ) -> ( _Left ` w ) e. Fin )
134 131 132 133 sylancl 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( _Left ` w ) e. Fin )
135 simplrl 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w e. No )
136 simpr 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s
137 135 136 0elleft 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s e. ( _Left ` w ) )
138 137 ne0d 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( _Left ` w ) =/= (/) )
139 leftssno 
 |-  ( _Left ` w ) C_ No
140 ltsso 
 |-
141 soss 
 |-  ( ( _Left ` w ) C_ No -> (
142 139 140 141 mp2 
 |-
143 fimax2g 
 |-  ( (  E. c e. ( _Left ` w ) A. e e. ( _Left ` w ) -. c
144 142 143 mp3an1 
 |-  ( ( ( _Left ` w ) e. Fin /\ ( _Left ` w ) =/= (/) ) -> E. c e. ( _Left ` w ) A. e e. ( _Left ` w ) -. c
145 leftno 
 |-  ( e e. ( _Left ` w ) -> e e. No )
146 leftno 
 |-  ( c e. ( _Left ` w ) -> c e. No )
147 lenlts 
 |-  ( ( e e. No /\ c e. No ) -> ( e <_s c <-> -. c
148 145 146 147 syl2anr 
 |-  ( ( c e. ( _Left ` w ) /\ e e. ( _Left ` w ) ) -> ( e <_s c <-> -. c
149 148 ralbidva 
 |-  ( c e. ( _Left ` w ) -> ( A. e e. ( _Left ` w ) e <_s c <-> A. e e. ( _Left ` w ) -. c
150 149 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
151 rightssold 
 |-  ( _Right ` w ) C_ ( _Old ` ( bday ` w ) )
152 ssfi 
 |-  ( ( ( _Old ` ( bday ` w ) ) e. Fin /\ ( _Right ` w ) C_ ( _Old ` ( bday ` w ) ) ) -> ( _Right ` w ) e. Fin )
153 131 151 152 sylancl 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( _Right ` w ) e. Fin )
154 153 adantr 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( _Right ` w ) e. Fin )
155 135 adantr 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w e. No )
156 simprrr 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) -> w =/= ( N +s 1s ) )
157 156 adantr 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w =/= ( N +s 1s ) )
158 157 adantr 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w =/= ( N +s 1s ) )
159 158 neneqd 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. w = ( N +s 1s ) )
160 simpr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w e. On_s )
161 124 ad4antr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) e. NN0_s )
162 n0on 
 |-  ( ( N +s 1s ) e. NN0_s -> ( N +s 1s ) e. On_s )
163 161 162 syl 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) e. On_s )
164 123 adantr 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
165 164 adantr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
166 bday11on 
 |-  ( ( w e. On_s /\ ( N +s 1s ) e. On_s /\ ( bday ` w ) = ( bday ` ( N +s 1s ) ) ) -> w = ( N +s 1s ) )
167 160 163 165 166 syl3anc 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( N +s 1s ) )
168 159 167 mtand 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. w e. On_s )
169 elons 
 |-  ( w e. On_s <-> ( w e. No /\ ( _Right ` w ) = (/) ) )
170 169 notbii 
 |-  ( -. w e. On_s <-> -. ( w e. No /\ ( _Right ` w ) = (/) ) )
171 imnan 
 |-  ( ( w e. No -> -. ( _Right ` w ) = (/) ) <-> -. ( w e. No /\ ( _Right ` w ) = (/) ) )
172 170 171 bitr4i 
 |-  ( -. w e. On_s <-> ( w e. No -> -. ( _Right ` w ) = (/) ) )
173 168 172 sylib 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( w e. No -> -. ( _Right ` w ) = (/) ) )
174 155 173 mpd 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. ( _Right ` w ) = (/) )
175 174 neqned 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( _Right ` w ) =/= (/) )
176 rightssno 
 |-  ( _Right ` w ) C_ No
177 soss 
 |-  ( ( _Right ` w ) C_ No -> (
178 176 140 177 mp2 
 |-
179 fimin2g 
 |-  ( (  E. d e. ( _Right ` w ) A. f e. ( _Right ` w ) -. f
180 178 179 mp3an1 
 |-  ( ( ( _Right ` w ) e. Fin /\ ( _Right ` w ) =/= (/) ) -> E. d e. ( _Right ` w ) A. f e. ( _Right ` w ) -. f
181 rightno 
 |-  ( d e. ( _Right ` w ) -> d e. No )
182 rightno 
 |-  ( f e. ( _Right ` w ) -> f e. No )
183 lenlts 
 |-  ( ( d e. No /\ f e. No ) -> ( d <_s f <-> -. f
184 181 182 183 syl2an 
 |-  ( ( d e. ( _Right ` w ) /\ f e. ( _Right ` w ) ) -> ( d <_s f <-> -. f
185 184 ralbidva 
 |-  ( d e. ( _Right ` w ) -> ( A. f e. ( _Right ` w ) d <_s f <-> A. f e. ( _Right ` w ) -. f
186 185 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
187 simp2l 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. ( _Left ` w ) )
188 simp2r 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  A. e e. ( _Left ` w ) e <_s c )
189 simp3l 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. ( _Right ` w ) )
190 simp3r 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  A. f e. ( _Right ` w ) d <_s f )
191 187 188 189 190 cutminmax 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
192 simpl2l 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. ( _Left ` w ) )
193 132 192 sselid 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. ( _Old ` ( bday ` w ) ) )
194 192 leftnod 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
195 oldbday 
 |-  ( ( ( bday ` w ) e. On /\ c e. No ) -> ( c e. ( _Old ` ( bday ` w ) ) <-> ( bday ` c ) e. ( bday ` w ) ) )
196 7 194 195 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 ) ) )
197 193 196 mpbid 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` c ) e. ( bday ` w ) )
198 123 3ad2ant1 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
199 198 adantr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
200 1 adantr 
 |-  ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) -> N e. NN0_s )
201 200 adantr 
 |-  ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. NN0_s )
202 201 3ad2ant1 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. NN0_s )
203 202 adantr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. NN0_s )
204 203 3 syl 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` ( N +s 1s ) ) = suc ( bday ` N ) )
205 199 204 eqtrd 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = suc ( bday ` N ) )
206 197 205 eleqtrd 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` c ) e. suc ( bday ` N ) )
207 bdayon 
 |-  ( bday ` c ) e. On
208 onsssuc 
 |-  ( ( ( bday ` c ) e. On /\ ( bday ` N ) e. On ) -> ( ( bday ` c ) C_ ( bday ` N ) <-> ( bday ` c ) e. suc ( bday ` N ) ) )
209 207 8 208 mp2an 
 |-  ( ( bday ` c ) C_ ( bday ` N ) <-> ( bday ` c ) e. suc ( bday ` N ) )
210 206 209 sylibr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` c ) C_ ( bday ` N ) )
211 breq1 
 |-  ( e = 0s -> ( e <_s c <-> 0s <_s c ) )
212 simpl2r 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  A. e e. ( _Left ` w ) e <_s c )
213 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
214 213 137 syl 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s e. ( _Left ` w ) )
215 211 212 214 rspcdva 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s <_s c )
216 simp1ll 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
217 216 adantr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
218 n0on 
 |-  ( N e. NN0_s -> N e. On_s )
219 1 218 syl 
 |-  ( ph -> N e. On_s )
220 217 219 syl 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. On_s )
221 simpl3l 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. ( _Right ` w ) )
222 151 221 sselid 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. ( _Old ` ( bday ` w ) ) )
223 oldbdayim 
 |-  ( d e. ( _Old ` ( bday ` w ) ) -> ( bday ` d ) e. ( bday ` w ) )
224 222 223 syl 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` d ) e. ( bday ` w ) )
225 224 205 eleqtrd 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` d ) e. suc ( bday ` N ) )
226 bdayon 
 |-  ( bday ` d ) e. On
227 onsssuc 
 |-  ( ( ( bday ` d ) e. On /\ ( bday ` N ) e. On ) -> ( ( bday ` d ) C_ ( bday ` N ) <-> ( bday ` d ) e. suc ( bday ` N ) ) )
228 226 8 227 mp2an 
 |-  ( ( bday ` d ) C_ ( bday ` N ) <-> ( bday ` d ) e. suc ( bday ` N ) )
229 225 228 sylibr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` d ) C_ ( bday ` N ) )
230 221 rightnod 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
231 madebday 
 |-  ( ( ( bday ` N ) e. On /\ d e. No ) -> ( d e. ( _Made ` ( bday ` N ) ) <-> ( bday ` d ) C_ ( bday ` N ) ) )
232 8 230 231 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 ) ) )
233 229 232 mpbird 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. ( _Made ` ( bday ` N ) ) )
234 onsbnd 
 |-  ( ( N e. On_s /\ d e. ( _Made ` ( bday ` N ) ) ) -> d <_s N )
235 220 233 234 syl2anc 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d <_s N )
236 203 n0nod 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. No )
237 230 236 lesnltd 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( d <_s N <-> -. N
238 235 237 mpbid 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. N
239 lltr 
 |-  ( _Left ` w ) <
240 239 a1i 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( _Left ` w ) <
241 240 187 189 sltssepcd 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
242 241 adantr 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
243 breq1 
 |-  ( c = N -> ( c  N
244 242 243 syl5ibcom 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( c = N -> N
245 238 244 mtod 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. c = N )
246 fveq2 
 |-  ( z = c -> ( bday ` z ) = ( bday ` c ) )
247 246 sseq1d 
 |-  ( z = c -> ( ( bday ` z ) C_ ( bday ` N ) <-> ( bday ` c ) C_ ( bday ` N ) ) )
248 breq2 
 |-  ( z = c -> ( 0s <_s z <-> 0s <_s c ) )
249 247 248 anbi12d 
 |-  ( z = c -> ( ( ( bday ` z ) C_ ( bday ` N ) /\ 0s <_s z ) <-> ( ( bday ` c ) C_ ( bday ` N ) /\ 0s <_s c ) ) )
250 eqeq1 
 |-  ( z = c -> ( z = N <-> c = N ) )
251 eqeq1 
 |-  ( z = c -> ( z = ( x +s ( y /su ( 2s ^su p ) ) ) <-> c = ( x +s ( y /su ( 2s ^su p ) ) ) ) )
252 251 3anbi1d 
 |-  ( z = c -> ( ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  ( c = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
253 252 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
254 253 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
255 oveq1 
 |-  ( x = g -> ( x +s ( y /su ( 2s ^su p ) ) ) = ( g +s ( y /su ( 2s ^su p ) ) ) )
256 255 eqeq2d 
 |-  ( x = g -> ( c = ( x +s ( y /su ( 2s ^su p ) ) ) <-> c = ( g +s ( y /su ( 2s ^su p ) ) ) ) )
257 oveq1 
 |-  ( x = g -> ( x +s p ) = ( g +s p ) )
258 257 breq1d 
 |-  ( x = g -> ( ( x +s p )  ( g +s p )
259 256 258 3anbi13d 
 |-  ( x = g -> ( ( c = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  ( c = ( g +s ( y /su ( 2s ^su p ) ) ) /\ y
260 259 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
261 oveq1 
 |-  ( y = h -> ( y /su ( 2s ^su p ) ) = ( h /su ( 2s ^su p ) ) )
262 261 oveq2d 
 |-  ( y = h -> ( g +s ( y /su ( 2s ^su p ) ) ) = ( g +s ( h /su ( 2s ^su p ) ) ) )
263 262 eqeq2d 
 |-  ( y = h -> ( c = ( g +s ( y /su ( 2s ^su p ) ) ) <-> c = ( g +s ( h /su ( 2s ^su p ) ) ) ) )
264 breq1 
 |-  ( y = h -> ( y  h
265 263 264 3anbi12d 
 |-  ( y = h -> ( ( c = ( g +s ( y /su ( 2s ^su p ) ) ) /\ y  ( c = ( g +s ( h /su ( 2s ^su p ) ) ) /\ h
266 265 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
267 oveq2 
 |-  ( p = i -> ( 2s ^su p ) = ( 2s ^su i ) )
268 267 oveq2d 
 |-  ( p = i -> ( h /su ( 2s ^su p ) ) = ( h /su ( 2s ^su i ) ) )
269 268 oveq2d 
 |-  ( p = i -> ( g +s ( h /su ( 2s ^su p ) ) ) = ( g +s ( h /su ( 2s ^su i ) ) ) )
270 269 eqeq2d 
 |-  ( p = i -> ( c = ( g +s ( h /su ( 2s ^su p ) ) ) <-> c = ( g +s ( h /su ( 2s ^su i ) ) ) ) )
271 267 breq2d 
 |-  ( p = i -> ( h  h
272 oveq2 
 |-  ( p = i -> ( g +s p ) = ( g +s i ) )
273 272 breq1d 
 |-  ( p = i -> ( ( g +s p )  ( g +s i )
274 270 271 273 3anbi123d 
 |-  ( p = i -> ( ( c = ( g +s ( h /su ( 2s ^su p ) ) ) /\ h  ( c = ( g +s ( h /su ( 2s ^su i ) ) ) /\ h
275 274 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
276 266 275 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
277 260 276 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
278 254 277 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
279 250 278 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
280 249 279 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
281 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
282 281 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
283 282 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
284 283 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
285 280 284 194 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
286 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
287 245 285 286 sylsyld 
 |-  ( ( ( ( ( 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
288 simp3l1 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
289 288 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
290 289 n0nod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
291 simp3l3 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
292 291 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
293 n0addscl 
 |-  ( ( g e. NN0_s /\ i e. NN0_s ) -> ( g +s i ) e. NN0_s )
294 289 292 293 syl2anc 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i ) e. NN0_s )
295 294 n0nod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i ) e. No )
296 216 3ad2ant1 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
297 296 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
298 297 32 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. No )
299 n0sge0 
 |-  ( i e. NN0_s -> 0s <_s i )
300 292 299 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s <_s i )
301 292 n0nod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. No )
302 290 301 addsge01d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 0s <_s i <-> g <_s ( g +s i ) ) )
303 300 302 mpbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g <_s ( g +s i ) )
304 simp3r3 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i )
305 304 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i )
306 290 295 298 303 305 leltstrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g
307 297 1 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. NN0_s )
308 n0ltsp1le 
 |-  ( ( g e. NN0_s /\ N e. NN0_s ) -> ( g  ( g +s 1s ) <_s N ) )
309 289 307 308 syl2anc 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g  ( g +s 1s ) <_s N ) )
310 306 309 mpbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) <_s N )
311 ltsirr 
 |-  ( N e. No -> -. N
312 298 311 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. N
313 289 peano2n0sd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. NN0_s )
314 313 n0nod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. No )
315 314 298 ltsnled 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s )  -. N <_s ( g +s 1s ) ) )
316 296 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
317 124 n0nod 
 |-  ( ph -> ( N +s 1s ) e. No )
318 316 317 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) e. No )
319 316 32 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. No )
320 52 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  2s e. No )
321 319 320 subscld 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N -s 2s ) e. No )
322 1no 
 |-  1s e. No
323 322 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. No )
324 321 323 323 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 ) ) )
325 1p1e2s 
 |-  ( 1s +s 1s ) = 2s
326 325 oveq2i 
 |-  ( ( N -s 2s ) +s ( 1s +s 1s ) ) = ( ( N -s 2s ) +s 2s )
327 npcans 
 |-  ( ( N e. No /\ 2s e. No ) -> ( ( N -s 2s ) +s 2s ) = N )
328 319 52 327 sylancl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 2s ) +s 2s ) = N )
329 326 328 eqtrid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 2s ) +s ( 1s +s 1s ) ) = N )
330 324 329 eqtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( N -s 2s ) +s 1s ) +s 1s ) = N )
331 330 319 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 )
332 321 323 addscld 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 2s ) +s 1s ) e. No )
333 198 3ad2ant1 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
334 333 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
335 simpl2 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
336 187 leftnod 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
337 336 3ad2ant1 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
338 337 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
339 288 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
340 339 peano2n0sd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. NN0_s )
341 340 n0nod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. No )
342 subscl 
 |-  ( ( N e. No /\ 1s e. No ) -> ( N -s 1s ) e. No )
343 32 322 342 sylancl 
 |-  ( ph -> ( N -s 1s ) e. No )
344 316 343 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N -s 1s ) e. No )
345 simp3r1 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
346 345 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
347 simp3r2 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
348 347 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
349 291 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
350 expscl 
 |-  ( ( 2s e. No /\ i e. NN0_s ) -> ( 2s ^su i ) e. No )
351 52 349 350 sylancr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) e. No )
352 351 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 ) )
353 348 352 breqtrrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
354 simp3l2 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
355 354 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
356 355 n0nod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. No )
357 356 323 349 pw2ltdivmuls2d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h /su ( 2s ^su i ) )  h
358 353 357 mpbird 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) )
359 356 349 pw2divscld 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) ) e. No )
360 339 n0nod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
361 359 323 360 ltadds2d 
 |-  ( ( ( ( ( ( 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 ) ) )
362 358 361 mpbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( h /su ( 2s ^su i ) ) )
363 346 362 eqbrtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
364 n0ltsp1le 
 |-  ( ( ( g +s 1s ) e. NN0_s /\ N e. NN0_s ) -> ( ( g +s 1s )  ( ( g +s 1s ) +s 1s ) <_s N ) )
365 313 307 364 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 ) )
366 365 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 ) )
367 366 impr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s ) +s 1s ) <_s N )
368 npcans 
 |-  ( ( N e. No /\ 1s e. No ) -> ( ( N -s 1s ) +s 1s ) = N )
369 319 322 368 sylancl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 1s ) +s 1s ) = N )
370 367 369 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 ) )
371 341 344 323 leadds1d 
 |-  ( ( ( ( ( ( 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 ) ) )
372 370 371 mpbird 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) <_s ( N -s 1s ) )
373 338 341 344 363 372 ltlestrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
374 325 oveq2i 
 |-  ( N -s ( 1s +s 1s ) ) = ( N -s 2s )
375 374 oveq1i 
 |-  ( ( N -s ( 1s +s 1s ) ) +s 1s ) = ( ( N -s 2s ) +s 1s )
376 319 323 323 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 ) ) )
377 376 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 ) )
378 npcans 
 |-  ( ( ( N -s 1s ) e. No /\ 1s e. No ) -> ( ( ( N -s 1s ) -s 1s ) +s 1s ) = ( N -s 1s ) )
379 344 322 378 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 ) )
380 377 379 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 ) )
381 375 380 eqtr3id 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 2s ) +s 1s ) = ( N -s 1s ) )
382 373 381 breqtrrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
383 338 332 382 sltssn 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { c } <
384 189 rightnod 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
385 384 3ad2ant1 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
386 385 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
387 simprl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = N )
388 387 oveq1d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( d -s 1s ) = ( N -s 1s ) )
389 386 ltsm1d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( d -s 1s )
390 388 389 eqbrtrrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N -s 1s )
391 381 390 eqbrtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 2s ) +s 1s )
392 332 386 391 sltssn 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { ( ( N -s 2s ) +s 1s ) } <
393 335 332 383 392 sltsbday 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) C_ ( bday ` ( ( N -s 2s ) +s 1s ) ) )
394 334 393 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 ) ) )
395 124 162 syl 
 |-  ( ph -> ( N +s 1s ) e. On_s )
396 316 395 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) e. On_s )
397 319 323 320 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 ) )
398 n0sge0 
 |-  ( g e. NN0_s -> 0s <_s g )
399 339 398 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s <_s g )
400 323 360 addsge01d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 0s <_s g <-> 1s <_s ( 1s +s g ) ) )
401 399 400 mpbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s <_s ( 1s +s g ) )
402 360 323 addscomd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) = ( 1s +s g ) )
403 401 402 breqtrrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s <_s ( g +s 1s ) )
404 simprr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s )
405 323 341 319 403 404 leltstrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s
406 323 319 405 ltlesd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s <_s N )
407 323 319 323 leadds1d 
 |-  ( ( ( ( ( ( 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 ) ) )
408 406 407 mpbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 1s +s 1s ) <_s ( N +s 1s ) )
409 325 408 eqbrtrrid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  2s <_s ( N +s 1s ) )
410 2nns 
 |-  2s e. NN_s
411 nnn0s 
 |-  ( 2s e. NN_s -> 2s e. NN0_s )
412 410 411 ax-mp 
 |-  2s e. NN0_s
413 296 124 syl 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) e. NN0_s )
414 413 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) e. NN0_s )
415 n0subs 
 |-  ( ( 2s e. NN0_s /\ ( N +s 1s ) e. NN0_s ) -> ( 2s <_s ( N +s 1s ) <-> ( ( N +s 1s ) -s 2s ) e. NN0_s ) )
416 412 414 415 sylancr 
 |-  ( ( ( ( ( ( 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 ) )
417 409 416 mpbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N +s 1s ) -s 2s ) e. NN0_s )
418 397 417 eqeltrrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 2s ) +s 1s ) e. NN0_s )
419 n0on 
 |-  ( ( ( N -s 2s ) +s 1s ) e. NN0_s -> ( ( N -s 2s ) +s 1s ) e. On_s )
420 418 419 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 2s ) +s 1s ) e. On_s )
421 396 420 onlesd 
 |-  ( ( ( ( ( ( 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 ) ) ) )
422 394 421 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 ) )
423 332 ltsp1d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N -s 2s ) +s 1s )
424 318 332 331 422 423 leltstrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s )
425 318 331 424 ltlesd 
 |-  ( ( ( ( ( ( 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 ) )
426 425 330 breqtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) <_s N )
427 316 1 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. NN0_s )
428 n0ltsp1le 
 |-  ( ( N e. NN0_s /\ N e. NN0_s ) -> ( N  ( N +s 1s ) <_s N ) )
429 427 427 428 syl2anc 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N  ( N +s 1s ) <_s N ) )
430 426 429 mpbird 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N
431 430 expr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s )  N
432 315 431 sylbird 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( -. N <_s ( g +s 1s ) -> N
433 312 432 mt3d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N <_s ( g +s 1s ) )
434 314 298 lestri3d 
 |-  ( ( ( ( ( ( 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 ) ) ) )
435 310 433 434 mpbir2and 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) = N )
436 304 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i )
437 simprr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) = N )
438 436 437 breqtrrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i )
439 291 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
440 439 n0nod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. No )
441 322 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. No )
442 288 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
443 442 n0nod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
444 440 441 443 ltadds2d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( i  ( g +s i )
445 438 444 mpbird 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i
446 n0lts1e0 
 |-  ( i e. NN0_s -> ( i  i = 0s ) )
447 439 446 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( i  i = 0s ) )
448 445 447 mpbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i = 0s )
449 345 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
450 347 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
451 oveq2 
 |-  ( i = 0s -> ( 2s ^su i ) = ( 2s ^su 0s ) )
452 451 adantl 
 |-  ( ( ( d = N /\ ( g +s 1s ) = N ) /\ i = 0s ) -> ( 2s ^su i ) = ( 2s ^su 0s ) )
453 452 adantl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) = ( 2s ^su 0s ) )
454 453 54 eqtrdi 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) = 1s )
455 450 454 breqtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
456 354 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
457 n0lts1e0 
 |-  ( h e. NN0_s -> ( h  h = 0s ) )
458 456 457 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h  h = 0s ) )
459 455 458 mpbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h = 0s )
460 459 454 oveq12d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) ) = ( 0s /su 1s ) )
461 460 59 eqtrdi 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) ) = 0s )
462 461 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 ) )
463 288 n0nod 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
464 463 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
465 464 addsridd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 0s ) = g )
466 449 462 465 3eqtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = g )
467 simpl2 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
468 54 oveq2i 
 |-  ( g /su ( 2s ^su 0s ) ) = ( g /su 1s )
469 463 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
470 469 divs1d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g /su 1s ) = g )
471 simprr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = g )
472 470 471 eqtr4d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g /su 1s ) = c )
473 468 472 eqtrid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g /su ( 2s ^su 0s ) ) = c )
474 473 sneqd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { ( g /su ( 2s ^su 0s ) ) } = { c } )
475 54 oveq2i 
 |-  ( ( g +s 1s ) /su ( 2s ^su 0s ) ) = ( ( g +s 1s ) /su 1s )
476 simpllr 
 |-  ( ( ( ( d = N /\ ( g +s 1s ) = N ) /\ i = 0s ) /\ c = g ) -> ( g +s 1s ) = N )
477 476 adantl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) = N )
478 288 peano2n0sd 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. NN0_s )
479 478 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. NN0_s )
480 479 n0nod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. No )
481 480 divs1d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s ) /su 1s ) = ( g +s 1s ) )
482 simplll 
 |-  ( ( ( ( d = N /\ ( g +s 1s ) = N ) /\ i = 0s ) /\ c = g ) -> d = N )
483 482 adantl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = N )
484 477 481 483 3eqtr4d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s ) /su 1s ) = d )
485 475 484 eqtrid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s ) /su ( 2s ^su 0s ) ) = d )
486 485 sneqd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { ( ( g +s 1s ) /su ( 2s ^su 0s ) ) } = { d } )
487 474 486 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 } ) )
488 288 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
489 488 n0zsd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. ZZ_s )
490 29 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s e. NN0_s )
491 489 490 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 ) ) ) )
492 467 487 491 3eqtr2d 
 |-  ( ( ( ( ( ( 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 ) ) ) )
493 mulscl 
 |-  ( ( 2s e. No /\ g e. No ) -> ( 2s x.s g ) e. No )
494 52 469 493 sylancr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s g ) e. No )
495 322 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. No )
496 addslid 
 |-  ( 1s e. No -> ( 0s +s 1s ) = 1s )
497 322 496 ax-mp 
 |-  ( 0s +s 1s ) = 1s
498 1n0s 
 |-  1s e. NN0_s
499 497 498 eqeltri 
 |-  ( 0s +s 1s ) e. NN0_s
500 499 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 0s +s 1s ) e. NN0_s )
501 494 495 500 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 ) ) ) ) )
502 exps1 
 |-  ( 2s e. No -> ( 2s ^su 1s ) = 2s )
503 52 502 ax-mp 
 |-  ( 2s ^su 1s ) = 2s
504 503 oveq1i 
 |-  ( ( 2s ^su 1s ) x.s g ) = ( 2s x.s g )
505 504 oveq1i 
 |-  ( ( ( 2s ^su 1s ) x.s g ) /su ( 2s ^su ( 0s +s 1s ) ) ) = ( ( 2s x.s g ) /su ( 2s ^su ( 0s +s 1s ) ) )
506 498 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. NN0_s )
507 469 490 506 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 ) ) ) )
508 468 507 470 3eqtr3a 
 |-  ( ( ( ( ( ( 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 )
509 505 508 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 )
510 497 oveq2i 
 |-  ( 2s ^su ( 0s +s 1s ) ) = ( 2s ^su 1s )
511 510 503 eqtri 
 |-  ( 2s ^su ( 0s +s 1s ) ) = 2s
512 511 oveq2i 
 |-  ( 1s /su ( 2s ^su ( 0s +s 1s ) ) ) = ( 1s /su 2s )
513 512 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 ) )
514 509 513 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 ) ) )
515 501 514 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 ) ) )
516 515 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 ) ) ) )
517 288 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
518 498 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. NN0_s )
519 simprr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( g +s ( 1s /su 2s ) ) )
520 ltadds1 
 |-  ( ( 0s e. No /\ 1s e. No /\ 1s e. No ) -> ( 0s  ( 0s +s 1s )
521 57 322 322 520 mp3an 
 |-  ( 0s  ( 0s +s 1s )
522 36 521 mpbi 
 |-  ( 0s +s 1s )
523 522 497 325 3brtr3i 
 |-  1s
524 523 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s
525 simp-4r 
 |-  ( ( ( ( ( d = N /\ ( g +s 1s ) = N ) /\ i = 0s ) /\ c = g ) /\ w = ( g +s ( 1s /su 2s ) ) ) -> ( g +s 1s ) = N )
526 525 adantl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) = N )
527 296 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
528 527 32 syl 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. No )
529 528 ltsp1d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N
530 526 529 eqbrtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s )
531 oveq1 
 |-  ( a = g -> ( a +s ( b /su ( 2s ^su q ) ) ) = ( g +s ( b /su ( 2s ^su q ) ) ) )
532 531 eqeq2d 
 |-  ( a = g -> ( w = ( a +s ( b /su ( 2s ^su q ) ) ) <-> w = ( g +s ( b /su ( 2s ^su q ) ) ) ) )
533 oveq1 
 |-  ( a = g -> ( a +s q ) = ( g +s q ) )
534 533 breq1d 
 |-  ( a = g -> ( ( a +s q )  ( g +s q )
535 532 534 3anbi13d 
 |-  ( a = g -> ( ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b  ( w = ( g +s ( b /su ( 2s ^su q ) ) ) /\ b
536 oveq1 
 |-  ( b = 1s -> ( b /su ( 2s ^su q ) ) = ( 1s /su ( 2s ^su q ) ) )
537 536 oveq2d 
 |-  ( b = 1s -> ( g +s ( b /su ( 2s ^su q ) ) ) = ( g +s ( 1s /su ( 2s ^su q ) ) ) )
538 537 eqeq2d 
 |-  ( b = 1s -> ( w = ( g +s ( b /su ( 2s ^su q ) ) ) <-> w = ( g +s ( 1s /su ( 2s ^su q ) ) ) ) )
539 breq1 
 |-  ( b = 1s -> ( b  1s
540 538 539 3anbi12d 
 |-  ( b = 1s -> ( ( w = ( g +s ( b /su ( 2s ^su q ) ) ) /\ b  ( w = ( g +s ( 1s /su ( 2s ^su q ) ) ) /\ 1s
541 oveq2 
 |-  ( q = 1s -> ( 2s ^su q ) = ( 2s ^su 1s ) )
542 541 503 eqtrdi 
 |-  ( q = 1s -> ( 2s ^su q ) = 2s )
543 542 oveq2d 
 |-  ( q = 1s -> ( 1s /su ( 2s ^su q ) ) = ( 1s /su 2s ) )
544 543 oveq2d 
 |-  ( q = 1s -> ( g +s ( 1s /su ( 2s ^su q ) ) ) = ( g +s ( 1s /su 2s ) ) )
545 544 eqeq2d 
 |-  ( q = 1s -> ( w = ( g +s ( 1s /su ( 2s ^su q ) ) ) <-> w = ( g +s ( 1s /su 2s ) ) ) )
546 542 breq2d 
 |-  ( q = 1s -> ( 1s  1s
547 oveq2 
 |-  ( q = 1s -> ( g +s q ) = ( g +s 1s ) )
548 547 breq1d 
 |-  ( q = 1s -> ( ( g +s q )  ( g +s 1s )
549 545 546 548 3anbi123d 
 |-  ( q = 1s -> ( ( w = ( g +s ( 1s /su ( 2s ^su q ) ) ) /\ 1s  ( w = ( g +s ( 1s /su 2s ) ) /\ 1s
550 535 540 549 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
551 517 518 518 519 524 530 550 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
552 551 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
553 516 552 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
554 492 553 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
555 554 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
556 466 555 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
557 556 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
558 448 557 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
559 558 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
560 435 559 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
561 560 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
562 simprr1 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( j +s ( k /su ( 2s ^su l ) ) ) )
563 simprr2 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k
564 simprl3 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. NN0_s )
565 expscl 
 |-  ( ( 2s e. No /\ l e. NN0_s ) -> ( 2s ^su l ) e. No )
566 52 564 565 sylancr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su l ) e. No )
567 566 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 ) )
568 563 567 breqtrrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k
569 simprl2 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k e. NN0_s )
570 569 n0nod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k e. No )
571 322 a1i 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. No )
572 570 571 564 pw2ltdivmuls2d 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( k /su ( 2s ^su l ) )  k
573 568 572 mpbird 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( k /su ( 2s ^su l ) )
574 570 564 pw2divscld 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( k /su ( 2s ^su l ) ) e. No )
575 simprl1 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j e. NN0_s )
576 575 n0nod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j e. No )
577 574 571 576 ltadds2d 
 |-  ( ( ( ( ( ( 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 ) ) )
578 573 577 mpbid 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s ( k /su ( 2s ^su l ) ) )
579 562 578 eqbrtrd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d
580 288 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
581 580 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
582 581 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
583 582 addsridd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 0s ) = g )
584 354 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
585 584 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
586 n0sge0 
 |-  ( h e. NN0_s -> 0s <_s h )
587 585 586 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s <_s h )
588 585 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. No )
589 291 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
590 589 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
591 588 590 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 ) ) ) )
592 57 a1i 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s e. No )
593 588 590 pw2divscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) ) e. No )
594 592 593 582 leadds2d 
 |-  ( ( ( ( ( ( ( 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 ) ) ) ) )
595 591 594 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 ) ) ) ) )
596 587 595 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 ) ) ) )
597 583 596 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 ) ) ) )
598 345 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
599 598 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
600 597 599 breqtrrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g <_s c )
601 580 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
602 601 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
603 337 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
604 603 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
605 385 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
606 605 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
607 simprr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g <_s c )
608 241 3ad2ant1 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
609 608 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
610 609 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
611 602 604 606 607 610 leltstrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g
612 580 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
613 612 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
614 605 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
615 575 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j e. NN0_s )
616 615 peano2n0sd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s 1s ) e. NN0_s )
617 616 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s 1s ) e. No )
618 simprr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g
619 simprll 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d
620 613 614 617 618 619 ltstrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g
621 n0lesltp1 
 |-  ( ( g e. NN0_s /\ j e. NN0_s ) -> ( g <_s j <-> g
622 612 615 621 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g <_s j <-> g
623 620 622 mpbird 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g <_s j )
624 623 expr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g  g <_s j ) )
625 611 624 mpd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g <_s j )
626 576 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j e. No )
627 602 626 lesloed 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g <_s j <-> ( g
628 575 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j e. NN0_s )
629 n0ltsp1le 
 |-  ( ( g e. NN0_s /\ j e. NN0_s ) -> ( g  ( g +s 1s ) <_s j ) )
630 601 628 629 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g  ( g +s 1s ) <_s j ) )
631 630 biimpd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g  ( g +s 1s ) <_s j ) )
632 631 impr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) <_s j )
633 478 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. NN0_s )
634 633 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. NN0_s )
635 634 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. No )
636 575 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j e. NN0_s )
637 636 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j e. No )
638 605 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
639 simprr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) <_s j )
640 569 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k e. NN0_s )
641 n0sge0 
 |-  ( k e. NN0_s -> 0s <_s k )
642 640 641 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s <_s k )
643 640 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k e. No )
644 564 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. NN0_s )
645 643 644 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 ) ) ) )
646 643 644 pw2divscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( k /su ( 2s ^su l ) ) e. No )
647 637 646 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 ) ) ) ) )
648 645 647 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 ) ) ) ) )
649 642 648 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 ) ) ) )
650 562 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( j +s ( k /su ( 2s ^su l ) ) ) )
651 649 650 breqtrrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j <_s d )
652 635 637 638 639 651 lestrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) <_s d )
653 575 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j e. NN0_s )
654 564 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. NN0_s )
655 n0addscl 
 |-  ( ( j e. NN0_s /\ l e. NN0_s ) -> ( j +s l ) e. NN0_s )
656 653 654 655 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s l ) e. NN0_s )
657 656 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s l ) e. No )
658 296 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
659 658 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
660 659 32 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. No )
661 659 317 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) e. No )
662 simprr3 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s l )
663 662 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s l )
664 660 ltsp1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N
665 657 660 661 663 664 ltstrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s l )
666 657 661 ltsnled 
 |-  ( ( ( ( ( ( ( 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 ) ) )
667 665 666 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. ( N +s 1s ) <_s ( j +s l ) )
668 633 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. NN0_s )
669 668 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. No )
670 605 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
671 669 670 ltsnled 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s 1s )  -. d <_s ( g +s 1s ) ) )
672 661 adantrr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) e. No )
673 576 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j e. No )
674 657 adantrr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s l ) e. No )
675 633 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. NN0_s )
676 675 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. No )
677 333 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
678 677 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
679 simpll2 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
680 603 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
681 598 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
682 347 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
683 682 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
684 589 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
685 52 684 350 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) e. No )
686 685 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 ) )
687 683 686 breqtrrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
688 584 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
689 688 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. No )
690 322 a1i 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. No )
691 689 690 684 pw2ltdivmuls2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h /su ( 2s ^su i ) )  h
692 687 691 mpbird 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) )
693 689 684 pw2divscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) ) e. No )
694 580 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
695 694 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
696 693 690 695 ltadds2d 
 |-  ( ( ( ( ( ( ( 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 ) ) )
697 692 696 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( h /su ( 2s ^su i ) ) )
698 681 697 eqbrtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
699 680 676 698 sltssn 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { c } <
700 605 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
701 simprr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s )
702 676 700 701 sltssn 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { ( g +s 1s ) } <
703 679 676 699 702 sltsbday 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) C_ ( bday ` ( g +s 1s ) ) )
704 678 703 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 ) ) )
705 658 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
706 705 395 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) e. On_s )
707 n0on 
 |-  ( ( g +s 1s ) e. NN0_s -> ( g +s 1s ) e. On_s )
708 675 707 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) e. On_s )
709 706 708 onlesd 
 |-  ( ( ( ( ( ( ( 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 ) ) ) )
710 704 709 mpbird 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) <_s ( g +s 1s ) )
711 simpllr 
 |-  ( ( ( ( ( ( d  ( g +s 1s ) <_s j )
712 711 adantl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) <_s j )
713 672 676 673 710 712 lestrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) <_s j )
714 564 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. NN0_s )
715 n0sge0 
 |-  ( l e. NN0_s -> 0s <_s l )
716 714 715 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s <_s l )
717 714 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. No )
718 673 717 addsge01d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 0s <_s l <-> j <_s ( j +s l ) ) )
719 716 718 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  j <_s ( j +s l ) )
720 672 673 674 713 719 lestrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) <_s ( j +s l ) )
721 720 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 ) ) )
722 671 721 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 ) ) )
723 667 722 mt3d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d <_s ( g +s 1s ) )
724 simprr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s 1s ) <_s d )
725 670 669 lestri3d 
 |-  ( ( ( ( ( ( ( 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 ) ) )
726 723 724 725 mpbir2and 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( g +s 1s ) )
727 682 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
728 584 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
729 589 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
730 n0expscl 
 |-  ( ( 2s e. NN0_s /\ i e. NN0_s ) -> ( 2s ^su i ) e. NN0_s )
731 412 729 730 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) e. NN0_s )
732 n0ltsp1le 
 |-  ( ( h e. NN0_s /\ ( 2s ^su i ) e. NN0_s ) -> ( h  ( h +s 1s ) <_s ( 2s ^su i ) ) )
733 728 731 732 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h  ( h +s 1s ) <_s ( 2s ^su i ) ) )
734 727 733 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s ) <_s ( 2s ^su i ) )
735 354 peano2n0sd 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s ) e. NN0_s )
736 735 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s ) e. NN0_s )
737 736 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s ) e. NN0_s )
738 737 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s ) e. No )
739 731 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) e. No )
740 738 739 lesloed 
 |-  ( ( ( ( ( ( ( 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 )
741 658 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
742 32 317 ltsnled 
 |-  ( ph -> ( N  -. ( N +s 1s ) <_s N ) )
743 38 742 mpbid 
 |-  ( ph -> -. ( N +s 1s ) <_s N )
744 741 743 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. ( N +s 1s ) <_s N )
745 677 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
746 simpll2 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
747 580 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
748 747 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
749 736 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s ) e. NN0_s )
750 749 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s ) e. No )
751 589 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
752 750 751 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 )
753 748 752 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 )
754 603 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
755 598 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
756 584 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
757 756 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. No )
758 757 ltsp1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
759 757 750 751 pw2ltsdiv1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h  ( h /su ( 2s ^su i ) )
760 758 759 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) )
761 757 751 pw2divscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) ) e. No )
762 761 752 748 ltadds2d 
 |-  ( ( ( ( ( ( ( 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 ) ) )
763 760 762 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( h /su ( 2s ^su i ) ) )
764 755 763 eqbrtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
765 754 753 764 sltssn 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { c } <
766 605 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
767 simprr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s )
768 52 751 350 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) e. No )
769 768 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 ) )
770 767 769 breqtrrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s )
771 322 a1i 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. No )
772 750 771 751 pw2ltdivmuls2d 
 |-  ( ( ( ( ( ( ( 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 )
773 752 771 748 ltadds2d 
 |-  ( ( ( ( ( ( ( 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 ) ) )
774 772 773 bitr3d 
 |-  ( ( ( ( ( ( ( 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 ) ) )
775 770 774 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 ) ) )
776 simprlr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( g +s 1s ) )
777 775 776 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 ) ) )
778 753 766 777 sltssn 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { ( g +s ( ( h +s 1s ) /su ( 2s ^su i ) ) ) } <
779 746 753 765 778 sltsbday 
 |-  ( ( ( ( ( ( ( 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 ) ) ) ) )
780 745 779 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 ) ) ) ) )
781 658 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
782 781 1 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. NN0_s )
783 304 adantr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i )
784 783 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i )
785 782 747 749 751 767 784 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 ) )
786 780 785 sstrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` ( N +s 1s ) ) C_ ( bday ` N ) )
787 219 395 onltsd 
 |-  ( ph -> ( N  ( bday ` N ) e. ( bday ` ( N +s 1s ) ) ) )
788 781 787 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N  ( bday ` N ) e. ( bday ` ( N +s 1s ) ) ) )
789 788 notbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( -. N  -. ( bday ` N ) e. ( bday ` ( N +s 1s ) ) ) )
790 317 32 lesnltd 
 |-  ( ph -> ( ( N +s 1s ) <_s N <-> -. N
791 781 790 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( N +s 1s ) <_s N <-> -. N
792 bdayon 
 |-  ( bday ` ( N +s 1s ) ) e. On
793 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 ) ) ) )
794 792 8 793 mp2an 
 |-  ( ( bday ` ( N +s 1s ) ) C_ ( bday ` N ) <-> -. ( bday ` N ) e. ( bday ` ( N +s 1s ) ) )
795 794 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 ) ) ) )
796 789 791 795 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 ) ) )
797 786 796 mpbird 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) <_s N )
798 797 expr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h +s 1s )  ( N +s 1s ) <_s N ) )
799 744 798 mtod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. ( h +s 1s )
800 orel1 
 |-  ( -. ( h +s 1s )  ( ( ( h +s 1s )  ( h +s 1s ) = ( 2s ^su i ) ) )
801 799 800 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 ) ) )
802 580 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
803 584 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
804 n0mulscl 
 |-  ( ( 2s e. NN0_s /\ h e. NN0_s ) -> ( 2s x.s h ) e. NN0_s )
805 412 803 804 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s h ) e. NN0_s )
806 805 peano2n0sd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s h ) +s 1s ) e. NN0_s )
807 589 peano2n0sd 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( i +s 1s ) e. NN0_s )
808 807 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( i +s 1s ) e. NN0_s )
809 simpll2 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
810 802 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
811 589 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
812 810 811 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 )
813 812 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 ) ) ) )
814 52 589 350 sylancr 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) e. No )
815 814 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) e. No )
816 815 810 mulscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su i ) x.s g ) e. No )
817 584 n0nod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. No )
818 817 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. No )
819 816 818 811 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 ) ) ) )
820 598 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
821 813 819 820 3eqtr4rd 
 |-  ( ( ( ( ( ( ( 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 ) ) )
822 821 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 ) ) } )
823 simprr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s ) = ( 2s ^su i ) )
824 823 815 eqeltrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h +s 1s ) e. No )
825 816 824 811 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 ) ) ) )
826 823 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 ) ) )
827 811 pw2divsidd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su i ) /su ( 2s ^su i ) ) = 1s )
828 826 827 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( h +s 1s ) /su ( 2s ^su i ) ) = 1s )
829 812 828 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 ) )
830 825 829 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 ) )
831 322 a1i 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. No )
832 816 818 831 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 ) ) )
833 832 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 ) ) )
834 simprlr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( g +s 1s ) )
835 830 833 834 3eqtr4rd 
 |-  ( ( ( ( ( ( ( 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 ) ) )
836 835 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 ) ) } )
837 822 836 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 ) ) } ) )
838 412 811 730 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) e. NN0_s )
839 n0mulscl 
 |-  ( ( ( 2s ^su i ) e. NN0_s /\ g e. NN0_s ) -> ( ( 2s ^su i ) x.s g ) e. NN0_s )
840 838 802 839 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 )
841 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 )
842 840 803 841 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 )
843 842 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 )
844 843 811 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 ) ) ) )
845 52 a1i 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  2s e. No )
846 845 816 818 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 ) ) )
847 expsp1 
 |-  ( ( 2s e. No /\ i e. NN0_s ) -> ( 2s ^su ( i +s 1s ) ) = ( ( 2s ^su i ) x.s 2s ) )
848 52 811 847 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 ) )
849 815 845 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 ) ) )
850 848 849 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 ) ) )
851 850 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 ) )
852 845 815 810 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 ) ) )
853 851 852 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 ) )
854 853 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 ) ) )
855 846 854 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 ) ) )
856 855 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 ) )
857 856 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 ) ) ) )
858 844 857 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( { ( ( ( ( 2s ^su i ) x.s g ) +s h ) /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 ) ) ) )
859 809 837 858 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 ) ) ) )
860 expscl 
 |-  ( ( 2s e. No /\ ( i +s 1s ) e. NN0_s ) -> ( 2s ^su ( i +s 1s ) ) e. No )
861 52 808 860 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( i +s 1s ) ) e. No )
862 861 810 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 )
863 805 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s h ) e. No )
864 862 863 831 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 ) ) )
865 864 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 ) ) ) )
866 806 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s h ) +s 1s ) e. No )
867 862 866 808 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 ) ) ) ) )
868 810 808 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 )
869 868 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 ) ) ) ) )
870 867 869 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 ) ) ) ) )
871 859 865 870 3eqtrd 
 |-  ( ( ( ( ( ( ( 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 ) ) ) ) )
872 831 845 863 ltadds2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 1s  ( ( 2s x.s h ) +s 1s )
873 523 872 mpbii 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s h ) +s 1s )
874 823 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 ) ) )
875 845 818 831 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 ) ) )
876 mulsrid 
 |-  ( 2s e. No -> ( 2s x.s 1s ) = 2s )
877 52 876 ax-mp 
 |-  ( 2s x.s 1s ) = 2s
878 877 oveq2i 
 |-  ( ( 2s x.s h ) +s ( 2s x.s 1s ) ) = ( ( 2s x.s h ) +s 2s )
879 875 878 eqtrdi 
 |-  ( ( ( ( ( ( ( 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 ) )
880 849 874 879 3eqtr2d 
 |-  ( ( ( ( ( ( ( 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 ) )
881 848 880 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 ) )
882 873 881 breqtrrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s h ) +s 1s )
883 811 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. No )
884 810 883 831 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 ) ) )
885 783 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i )
886 810 883 addscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i ) e. No )
887 658 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
888 887 32 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. No )
889 886 888 831 ltadds1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s i )  ( ( g +s i ) +s 1s )
890 885 889 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s i ) +s 1s )
891 884 890 eqbrtrrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( i +s 1s ) )
892 oveq1 
 |-  ( b = ( ( 2s x.s h ) +s 1s ) -> ( b /su ( 2s ^su q ) ) = ( ( ( 2s x.s h ) +s 1s ) /su ( 2s ^su q ) ) )
893 892 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 ) ) ) )
894 893 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 ) ) ) ) )
895 breq1 
 |-  ( b = ( ( 2s x.s h ) +s 1s ) -> ( b  ( ( 2s x.s h ) +s 1s )
896 894 895 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 )
897 oveq2 
 |-  ( q = ( i +s 1s ) -> ( 2s ^su q ) = ( 2s ^su ( i +s 1s ) ) )
898 897 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 ) ) ) )
899 898 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 ) ) ) ) )
900 899 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 ) ) ) ) ) )
901 897 breq2d 
 |-  ( q = ( i +s 1s ) -> ( ( ( 2s x.s h ) +s 1s )  ( ( 2s x.s h ) +s 1s )
902 oveq2 
 |-  ( q = ( i +s 1s ) -> ( g +s q ) = ( g +s ( i +s 1s ) ) )
903 902 breq1d 
 |-  ( q = ( i +s 1s ) -> ( ( g +s q )  ( g +s ( i +s 1s ) )
904 900 901 903 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 )
905 535 896 904 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
906 802 806 808 871 882 891 905 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
907 906 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
908 801 907 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
909 740 908 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
910 734 909 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
911 910 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
912 911 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
913 726 912 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
914 913 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
915 652 914 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
916 915 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
917 632 916 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
918 917 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
919 609 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
920 598 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
921 562 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( j +s ( k /su ( 2s ^su l ) ) ) )
922 simprr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g = j )
923 922 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 ) ) ) )
924 921 923 eqtr4d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( g +s ( k /su ( 2s ^su l ) ) ) )
925 919 920 924 3brtr3d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( h /su ( 2s ^su i ) ) )
926 817 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. No )
927 589 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
928 926 927 pw2divscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) ) e. No )
929 570 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k e. No )
930 564 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. NN0_s )
931 929 930 pw2divscld 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( k /su ( 2s ^su l ) ) e. No )
932 580 n0nod 
 |-  ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
933 932 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
934 928 931 933 ltadds2d 
 |-  ( ( ( ( ( ( ( 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 ) ) )
935 925 934 mpbird 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) )
936 584 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
937 564 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. NN0_s )
938 589 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
939 n0subs 
 |-  ( ( l e. NN0_s /\ i e. NN0_s ) -> ( l <_s i <-> ( i -s l ) e. NN0_s ) )
940 937 938 939 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 ) )
941 940 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 ) )
942 941 impr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( i -s l ) e. NN0_s )
943 n0expscl 
 |-  ( ( 2s e. NN0_s /\ ( i -s l ) e. NN0_s ) -> ( 2s ^su ( i -s l ) ) e. NN0_s )
944 412 942 943 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( i -s l ) ) e. NN0_s )
945 569 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k e. NN0_s )
946 n0mulscl 
 |-  ( ( ( 2s ^su ( i -s l ) ) e. NN0_s /\ k e. NN0_s ) -> ( ( 2s ^su ( i -s l ) ) x.s k ) e. NN0_s )
947 944 945 946 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 )
948 589 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
949 simprlr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) )
950 945 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k e. No )
951 564 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. NN0_s )
952 950 951 942 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 ) ) ) ) )
953 951 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. No )
954 942 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( i -s l ) e. No )
955 953 954 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 ) )
956 948 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. No )
957 npcans 
 |-  ( ( i e. No /\ l e. No ) -> ( ( i -s l ) +s l ) = i )
958 956 953 957 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( i -s l ) +s l ) = i )
959 955 958 eqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( l +s ( i -s l ) ) = i )
960 959 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 ) )
961 960 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 ) ) )
962 952 961 eqtr2d 
 |-  ( ( ( ( ( ( ( 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 ) ) )
963 949 962 breqtrrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) )
964 936 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. No )
965 947 n0nod 
 |-  ( ( ( ( ( ( ( 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 )
966 964 965 948 pw2ltsdiv1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h  ( h /su ( 2s ^su i ) )
967 963 966 mpbird 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
968 682 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
969 563 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k
970 n0expscl 
 |-  ( ( 2s e. NN0_s /\ l e. NN0_s ) -> ( 2s ^su l ) e. NN0_s )
971 412 951 970 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su l ) e. NN0_s )
972 971 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su l ) e. No )
973 944 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( i -s l ) ) e. No )
974 nnsgt0 
 |-  ( 2s e. NN_s -> 0s
975 410 974 ax-mp 
 |-  0s
976 expsgt0 
 |-  ( ( 2s e. No /\ ( i -s l ) e. NN0_s /\ 0s  0s
977 52 975 976 mp3an13 
 |-  ( ( i -s l ) e. NN0_s -> 0s
978 942 977 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s
979 950 972 973 978 ltmuls2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( k  ( ( 2s ^su ( i -s l ) ) x.s k )
980 969 979 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( i -s l ) ) x.s k )
981 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 ) ) )
982 52 942 951 981 mp3an2i 
 |-  ( ( ( ( ( ( ( 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 ) ) )
983 958 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 ) )
984 982 983 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 ) )
985 980 984 breqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( i -s l ) ) x.s k )
986 967 968 985 3jca 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h
987 598 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
988 562 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( j +s ( k /su ( 2s ^su l ) ) ) )
989 simpllr 
 |-  ( ( ( ( ( d  g = j )
990 989 adantl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g = j )
991 990 962 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 ) ) ) )
992 988 991 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 ) ) ) )
993 783 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s i )
994 987 992 993 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 )
995 breq1 
 |-  ( m = h -> ( m  h
996 breq1 
 |-  ( m = h -> ( m  h
997 995 996 3anbi12d 
 |-  ( m = h -> ( ( m  ( h
998 oveq1 
 |-  ( m = h -> ( m /su ( 2s ^su o ) ) = ( h /su ( 2s ^su o ) ) )
999 998 oveq2d 
 |-  ( m = h -> ( g +s ( m /su ( 2s ^su o ) ) ) = ( g +s ( h /su ( 2s ^su o ) ) ) )
1000 999 eqeq2d 
 |-  ( m = h -> ( c = ( g +s ( m /su ( 2s ^su o ) ) ) <-> c = ( g +s ( h /su ( 2s ^su o ) ) ) ) )
1001 1000 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 )
1002 997 1001 anbi12d 
 |-  ( m = h -> ( ( ( m  ( ( h
1003 breq2 
 |-  ( n = ( ( 2s ^su ( i -s l ) ) x.s k ) -> ( h  h
1004 breq1 
 |-  ( n = ( ( 2s ^su ( i -s l ) ) x.s k ) -> ( n  ( ( 2s ^su ( i -s l ) ) x.s k )
1005 1003 1004 3anbi13d 
 |-  ( n = ( ( 2s ^su ( i -s l ) ) x.s k ) -> ( ( h  ( h
1006 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 ) ) )
1007 1006 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 ) ) ) )
1008 1007 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 ) ) ) ) )
1009 1008 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 )
1010 1005 1009 anbi12d 
 |-  ( n = ( ( 2s ^su ( i -s l ) ) x.s k ) -> ( ( ( h  ( ( h
1011 oveq2 
 |-  ( o = i -> ( 2s ^su o ) = ( 2s ^su i ) )
1012 1011 breq2d 
 |-  ( o = i -> ( h  h
1013 1011 breq2d 
 |-  ( o = i -> ( ( ( 2s ^su ( i -s l ) ) x.s k )  ( ( 2s ^su ( i -s l ) ) x.s k )
1014 1012 1013 3anbi23d 
 |-  ( o = i -> ( ( h  ( h
1015 1011 oveq2d 
 |-  ( o = i -> ( h /su ( 2s ^su o ) ) = ( h /su ( 2s ^su i ) ) )
1016 1015 oveq2d 
 |-  ( o = i -> ( g +s ( h /su ( 2s ^su o ) ) ) = ( g +s ( h /su ( 2s ^su i ) ) ) )
1017 1016 eqeq2d 
 |-  ( o = i -> ( c = ( g +s ( h /su ( 2s ^su o ) ) ) <-> c = ( g +s ( h /su ( 2s ^su i ) ) ) ) )
1018 1011 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 ) ) )
1019 1018 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 ) ) ) )
1020 1019 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 ) ) ) ) )
1021 oveq2 
 |-  ( o = i -> ( g +s o ) = ( g +s i ) )
1022 1021 breq1d 
 |-  ( o = i -> ( ( g +s o )  ( g +s i )
1023 1017 1020 1022 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 )
1024 1014 1023 anbi12d 
 |-  ( o = i -> ( ( ( h  ( ( h
1025 1002 1010 1024 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
1026 936 947 948 986 994 1025 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
1027 1026 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
1028 n0subs 
 |-  ( ( i e. NN0_s /\ l e. NN0_s ) -> ( i <_s l <-> ( l -s i ) e. NN0_s ) )
1029 938 937 1028 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 ) )
1030 1029 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 ) )
1031 1030 impr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( l -s i ) e. NN0_s )
1032 n0expscl 
 |-  ( ( 2s e. NN0_s /\ ( l -s i ) e. NN0_s ) -> ( 2s ^su ( l -s i ) ) e. NN0_s )
1033 412 1031 1032 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( l -s i ) ) e. NN0_s )
1034 584 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. NN0_s )
1035 n0mulscl 
 |-  ( ( ( 2s ^su ( l -s i ) ) e. NN0_s /\ h e. NN0_s ) -> ( ( 2s ^su ( l -s i ) ) x.s h ) e. NN0_s )
1036 1033 1034 1035 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 )
1037 569 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k e. NN0_s )
1038 564 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. NN0_s )
1039 1034 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h e. No )
1040 589 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. NN0_s )
1041 1039 1040 1031 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 ) ) ) ) )
1042 1040 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. No )
1043 1031 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( l -s i ) e. No )
1044 1042 1043 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 ) )
1045 1044 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 ) ) )
1046 1045 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 ) ) ) )
1047 1038 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. No )
1048 npcans 
 |-  ( ( l e. No /\ i e. No ) -> ( ( l -s i ) +s i ) = l )
1049 1047 1042 1048 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( l -s i ) +s i ) = l )
1050 1049 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 ) )
1051 1050 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 ) ) )
1052 1041 1046 1051 3eqtrd 
 |-  ( ( ( ( ( ( ( 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 ) ) )
1053 simprlr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h /su ( 2s ^su i ) )
1054 1052 1053 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 ) )
1055 expscl 
 |-  ( ( 2s e. No /\ ( l -s i ) e. NN0_s ) -> ( 2s ^su ( l -s i ) ) e. No )
1056 52 1031 1055 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( l -s i ) ) e. No )
1057 1056 1039 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 )
1058 1037 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k e. No )
1059 1057 1058 1038 pw2ltsdiv1d 
 |-  ( ( ( ( ( ( ( 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 ) )
1060 1054 1059 mpbird 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( l -s i ) ) x.s h )
1061 682 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  h
1062 52 1040 350 sylancr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su i ) e. No )
1063 expsgt0 
 |-  ( ( 2s e. No /\ ( l -s i ) e. NN0_s /\ 0s  0s
1064 52 975 1063 mp3an13 
 |-  ( ( l -s i ) e. NN0_s -> 0s
1065 1031 1064 syl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s
1066 1039 1062 1056 1065 ltmuls2d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( h  ( ( 2s ^su ( l -s i ) ) x.s h )
1067 1061 1066 mpbid 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( l -s i ) ) x.s h )
1068 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 ) ) )
1069 52 1031 1040 1068 mp3an2i 
 |-  ( ( ( ( ( ( ( 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 ) ) )
1070 1069 1050 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 ) )
1071 1067 1070 breqtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su ( l -s i ) ) x.s h )
1072 563 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  k
1073 1060 1071 1072 3jca 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( ( 2s ^su ( l -s i ) ) x.s h )
1074 598 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( h /su ( 2s ^su i ) ) ) )
1075 1052 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 ) ) ) )
1076 1074 1075 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 ) ) ) )
1077 562 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( j +s ( k /su ( 2s ^su l ) ) ) )
1078 simpllr 
 |-  ( ( ( ( ( d  g = j )
1079 1078 adantl 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g = j )
1080 1079 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 ) ) ) )
1081 1077 1080 eqtr4d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( g +s ( k /su ( 2s ^su l ) ) ) )
1082 1079 oveq1d 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s l ) = ( j +s l ) )
1083 662 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( j +s l )
1084 1082 1083 eqbrtrd 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s l )
1085 1076 1081 1084 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 )
1086 breq1 
 |-  ( m = ( ( 2s ^su ( l -s i ) ) x.s h ) -> ( m  ( ( 2s ^su ( l -s i ) ) x.s h )
1087 breq1 
 |-  ( m = ( ( 2s ^su ( l -s i ) ) x.s h ) -> ( m  ( ( 2s ^su ( l -s i ) ) x.s h )
1088 1086 1087 3anbi12d 
 |-  ( m = ( ( 2s ^su ( l -s i ) ) x.s h ) -> ( ( m  ( ( ( 2s ^su ( l -s i ) ) x.s h )
1089 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 ) ) )
1090 1089 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 ) ) ) )
1091 1090 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 ) ) ) ) )
1092 1091 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 )
1093 1088 1092 anbi12d 
 |-  ( m = ( ( 2s ^su ( l -s i ) ) x.s h ) -> ( ( ( m  ( ( ( ( 2s ^su ( l -s i ) ) x.s h )
1094 breq2 
 |-  ( n = k -> ( ( ( 2s ^su ( l -s i ) ) x.s h )  ( ( 2s ^su ( l -s i ) ) x.s h )
1095 breq1 
 |-  ( n = k -> ( n  k
1096 1094 1095 3anbi13d 
 |-  ( n = k -> ( ( ( ( 2s ^su ( l -s i ) ) x.s h )  ( ( ( 2s ^su ( l -s i ) ) x.s h )
1097 oveq1 
 |-  ( n = k -> ( n /su ( 2s ^su o ) ) = ( k /su ( 2s ^su o ) ) )
1098 1097 oveq2d 
 |-  ( n = k -> ( g +s ( n /su ( 2s ^su o ) ) ) = ( g +s ( k /su ( 2s ^su o ) ) ) )
1099 1098 eqeq2d 
 |-  ( n = k -> ( d = ( g +s ( n /su ( 2s ^su o ) ) ) <-> d = ( g +s ( k /su ( 2s ^su o ) ) ) ) )
1100 1099 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 )
1101 1096 1100 anbi12d 
 |-  ( n = k -> ( ( ( ( ( 2s ^su ( l -s i ) ) x.s h )  ( ( ( ( 2s ^su ( l -s i ) ) x.s h )
1102 oveq2 
 |-  ( o = l -> ( 2s ^su o ) = ( 2s ^su l ) )
1103 1102 breq2d 
 |-  ( o = l -> ( ( ( 2s ^su ( l -s i ) ) x.s h )  ( ( 2s ^su ( l -s i ) ) x.s h )
1104 1102 breq2d 
 |-  ( o = l -> ( k  k
1105 1103 1104 3anbi23d 
 |-  ( o = l -> ( ( ( ( 2s ^su ( l -s i ) ) x.s h )  ( ( ( 2s ^su ( l -s i ) ) x.s h )
1106 1102 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 ) ) )
1107 1106 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 ) ) ) )
1108 1107 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 ) ) ) ) )
1109 1102 oveq2d 
 |-  ( o = l -> ( k /su ( 2s ^su o ) ) = ( k /su ( 2s ^su l ) ) )
1110 1109 oveq2d 
 |-  ( o = l -> ( g +s ( k /su ( 2s ^su o ) ) ) = ( g +s ( k /su ( 2s ^su l ) ) ) )
1111 1110 eqeq2d 
 |-  ( o = l -> ( d = ( g +s ( k /su ( 2s ^su o ) ) ) <-> d = ( g +s ( k /su ( 2s ^su l ) ) ) ) )
1112 oveq2 
 |-  ( o = l -> ( g +s o ) = ( g +s l ) )
1113 1112 breq1d 
 |-  ( o = l -> ( ( g +s o )  ( g +s l )
1114 1108 1111 1113 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 )
1115 1105 1114 anbi12d 
 |-  ( o = l -> ( ( ( ( ( 2s ^su ( l -s i ) ) x.s h )  ( ( ( ( 2s ^su ( l -s i ) ) x.s h )
1116 1093 1101 1115 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
1117 1036 1037 1038 1073 1085 1116 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
1118 1117 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
1119 937 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  l e. No )
1120 938 n0nod 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  i e. No )
1121 lestric 
 |-  ( ( l e. No /\ i e. No ) -> ( l <_s i \/ i <_s l ) )
1122 1119 1120 1121 syl2anc 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( l <_s i \/ i <_s l ) )
1123 1027 1118 1122 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
1124 580 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
1125 1124 adantr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
1126 simprl1 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  m e. NN0_s )
1127 n0mulscl 
 |-  ( ( 2s e. NN0_s /\ m e. NN0_s ) -> ( 2s x.s m ) e. NN0_s )
1128 412 1126 1127 sylancr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s m ) e. NN0_s )
1129 1128 peano2n0sd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s m ) +s 1s ) e. NN0_s )
1130 simprl3 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  o e. NN0_s )
1131 1130 peano2n0sd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( o +s 1s ) e. NN0_s )
1132 simpll2 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
1133 1132 adantr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
1134 1125 n0nod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
1135 1134 1130 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 )
1136 1135 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 ) ) ) )
1137 n0expscl 
 |-  ( ( 2s e. NN0_s /\ o e. NN0_s ) -> ( 2s ^su o ) e. NN0_s )
1138 412 1130 1137 sylancr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su o ) e. NN0_s )
1139 n0mulscl 
 |-  ( ( ( 2s ^su o ) e. NN0_s /\ g e. NN0_s ) -> ( ( 2s ^su o ) x.s g ) e. NN0_s )
1140 1138 1125 1139 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 )
1141 1140 n0nod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s ^su o ) x.s g ) e. No )
1142 1126 n0nod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  m e. No )
1143 1141 1142 1130 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 ) ) ) )
1144 simprr1 
 |-  ( ( ( m e. NN0_s /\ n e. NN0_s /\ o e. NN0_s ) /\ ( ( m  c = ( g +s ( m /su ( 2s ^su o ) ) ) )
1145 1144 adantl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( m /su ( 2s ^su o ) ) ) )
1146 1136 1143 1145 3eqtr4rd 
 |-  ( ( ( ( ( ( ( ( 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 ) ) )
1147 1146 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 ) ) } )
1148 1126 peano2n0sd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s ) e. NN0_s )
1149 1148 n0nod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s ) e. No )
1150 1141 1149 1130 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 ) ) ) )
1151 1135 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 ) ) ) )
1152 1150 1151 eqtr2d 
 |-  ( ( ( ( ( ( ( ( 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 ) ) )
1153 simprr2 
 |-  ( ( ( m e. NN0_s /\ n e. NN0_s /\ o e. NN0_s ) /\ ( ( m  d = ( g +s ( n /su ( 2s ^su o ) ) ) )
1154 1153 adantl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( g +s ( n /su ( 2s ^su o ) ) ) )
1155 658 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
1156 1155 adantr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
1157 1156 743 syl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  -. ( N +s 1s ) <_s N )
1158 322 a1i 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s e. No )
1159 simprl2 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  n e. NN0_s )
1160 1159 n0nod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  n e. No )
1161 1160 1142 subscld 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( n -s m ) e. No )
1162 1158 1161 ltsnled 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 1s  -. ( n -s m ) <_s 1s ) )
1163 677 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
1164 1163 adantr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` w ) = ( bday ` ( N +s 1s ) ) )
1165 1132 adantr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w = ( { c } |s { d } ) )
1166 1124 adantr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. NN0_s )
1167 1166 n0nod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  g e. No )
1168 1126 adantrr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  m e. NN0_s )
1169 1168 peano2n0sd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s ) e. NN0_s )
1170 1169 n0nod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s ) e. No )
1171 1130 adantrr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  o e. NN0_s )
1172 1170 1171 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 )
1173 1167 1172 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 )
1174 603 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
1175 1174 adantr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c e. No )
1176 1144 ad2antrl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c = ( g +s ( m /su ( 2s ^su o ) ) ) )
1177 1142 adantrr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  m e. No )
1178 1177 ltsp1d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  m
1179 1177 1170 1171 pw2ltsdiv1d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m  ( m /su ( 2s ^su o ) )
1180 1177 1171 pw2divscld 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m /su ( 2s ^su o ) ) e. No )
1181 1180 1172 1167 ltadds2d 
 |-  ( ( ( ( ( ( ( ( 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 ) ) )
1182 1179 1181 bitrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m  ( g +s ( m /su ( 2s ^su o ) ) )
1183 1178 1182 mpbid 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( m /su ( 2s ^su o ) ) )
1184 1176 1183 eqbrtrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  c
1185 1175 1173 1184 sltssn 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { c } <
1186 605 adantr 
 |-  ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
1187 1186 adantr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d e. No )
1188 1142 1158 1160 ltaddsubs2d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( m +s 1s )  1s
1189 1188 biimprd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 1s  ( m +s 1s )
1190 1189 impr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s )
1191 1159 adantrr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  n e. NN0_s )
1192 1191 n0nod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  n e. No )
1193 1170 1192 1171 pw2ltsdiv1d 
 |-  ( ( ( ( ( ( ( ( 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 ) )
1194 1192 1171 pw2divscld 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( n /su ( 2s ^su o ) ) e. No )
1195 1172 1194 1167 ltadds2d 
 |-  ( ( ( ( ( ( ( ( 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 ) ) )
1196 1193 1195 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 ) ) )
1197 1190 1196 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 ) ) )
1198 1153 ad2antrl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  d = ( g +s ( n /su ( 2s ^su o ) ) ) )
1199 1197 1198 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 ) ) )
1200 1173 1187 1199 sltssn 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  { ( g +s ( ( m +s 1s ) /su ( 2s ^su o ) ) ) } <
1201 1165 1173 1185 1200 sltsbday 
 |-  ( ( ( ( ( ( ( ( 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 ) ) ) ) )
1202 1164 1201 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 ) ) ) ) )
1203 1155 adantr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ph )
1204 1203 1 syl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. NN0_s )
1205 expscl 
 |-  ( ( 2s e. No /\ o e. NN0_s ) -> ( 2s ^su o ) e. No )
1206 52 1130 1205 sylancr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su o ) e. No )
1207 1206 adantrr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su o ) e. No )
1208 simprl1 
 |-  ( ( ( m e. NN0_s /\ n e. NN0_s /\ o e. NN0_s ) /\ ( ( m  m
1209 1208 ad2antrl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  m
1210 n0ltsp1le 
 |-  ( ( m e. NN0_s /\ n e. NN0_s ) -> ( m  ( m +s 1s ) <_s n ) )
1211 1168 1191 1210 syl2anc 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m  ( m +s 1s ) <_s n ) )
1212 1209 1211 mpbid 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s ) <_s n )
1213 simprl3 
 |-  ( ( ( m e. NN0_s /\ n e. NN0_s /\ o e. NN0_s ) /\ ( ( m  n
1214 1213 ad2antrl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  n
1215 1170 1192 1207 1212 1214 leltstrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s )
1216 simprr3 
 |-  ( ( ( m e. NN0_s /\ n e. NN0_s /\ o e. NN0_s ) /\ ( ( m  ( g +s o )
1217 1216 ad2antrl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s o )
1218 1204 1166 1169 1171 1215 1217 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 ) )
1219 1202 1218 sstrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` ( N +s 1s ) ) C_ ( bday ` N ) )
1220 395 219 onlesd 
 |-  ( ph -> ( ( N +s 1s ) <_s N <-> ( bday ` ( N +s 1s ) ) C_ ( bday ` N ) ) )
1221 1203 1220 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 ) ) )
1222 1219 1221 mpbird 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( N +s 1s ) <_s N )
1223 1222 expr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 1s  ( N +s 1s ) <_s N ) )
1224 1162 1223 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 ) )
1225 1157 1224 mt3d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( n -s m ) <_s 1s )
1226 1208 adantl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  m
1227 npcans 
 |-  ( ( n e. No /\ 1s e. No ) -> ( ( n -s 1s ) +s 1s ) = n )
1228 1160 322 1227 sylancl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( n -s 1s ) +s 1s ) = n )
1229 1228 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 ) )
1230 1160 1158 subscld 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( n -s 1s ) e. No )
1231 1142 1230 1158 leadds1d 
 |-  ( ( ( ( ( ( ( ( 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 ) ) )
1232 1126 1159 1210 syl2anc 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m  ( m +s 1s ) <_s n ) )
1233 1229 1231 1232 3bitr4rd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m  m <_s ( n -s 1s ) ) )
1234 1142 1160 1158 lesubsd 
 |-  ( ( ( ( ( ( ( ( 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 ) ) )
1235 1233 1234 bitrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m  1s <_s ( n -s m ) ) )
1236 1226 1235 mpbid 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  1s <_s ( n -s m ) )
1237 1161 1158 lestri3d 
 |-  ( ( ( ( ( ( ( ( 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 ) ) ) )
1238 1225 1236 1237 mpbir2and 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( n -s m ) = 1s )
1239 1160 1142 1158 subaddsd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( n -s m ) = 1s <-> ( m +s 1s ) = n ) )
1240 1238 1239 mpbid 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s ) = n )
1241 1240 eqcomd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  n = ( m +s 1s ) )
1242 1241 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 ) ) )
1243 1242 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 ) ) ) )
1244 1154 1243 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 ) ) ) )
1245 1141 1142 1158 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 ) ) )
1246 1245 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 ) ) )
1247 1152 1244 1246 3eqtr4d 
 |-  ( ( ( ( ( ( ( ( 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 ) ) )
1248 1247 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 ) ) } )
1249 1147 1248 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 ) ) } ) )
1250 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 )
1251 1140 1126 1250 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 )
1252 1251 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 )
1253 1252 1130 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 ) ) ) )
1254 1133 1249 1253 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 ) ) ) )
1255 52 a1i 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  2s e. No )
1256 1255 1141 1142 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 ) ) )
1257 expsp1 
 |-  ( ( 2s e. No /\ o e. NN0_s ) -> ( 2s ^su ( o +s 1s ) ) = ( ( 2s ^su o ) x.s 2s ) )
1258 52 1130 1257 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 ) )
1259 1206 1255 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 ) ) )
1260 1258 1259 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 ) ) )
1261 1260 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 ) )
1262 1255 1206 1134 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 ) ) )
1263 1261 1262 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 ) ) )
1264 1263 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 ) ) )
1265 1256 1264 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 ) ) )
1266 1265 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 ) )
1267 n0expscl 
 |-  ( ( 2s e. NN0_s /\ ( o +s 1s ) e. NN0_s ) -> ( 2s ^su ( o +s 1s ) ) e. NN0_s )
1268 412 1131 1267 sylancr 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( o +s 1s ) ) e. NN0_s )
1269 n0mulscl 
 |-  ( ( ( 2s ^su ( o +s 1s ) ) e. NN0_s /\ g e. NN0_s ) -> ( ( 2s ^su ( o +s 1s ) ) x.s g ) e. NN0_s )
1270 1268 1125 1269 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 )
1271 1270 n0nod 
 |-  ( ( ( ( ( ( ( ( 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 )
1272 1128 n0nod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s m ) e. No )
1273 1271 1272 1158 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 ) ) )
1274 1266 1273 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 ) ) )
1275 1274 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 ) ) ) )
1276 1254 1275 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 ) ) ) )
1277 1129 n0nod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s m ) +s 1s ) e. No )
1278 1271 1277 1131 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 ) ) ) ) )
1279 1134 1131 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 )
1280 1279 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 ) ) ) ) )
1281 1276 1278 1280 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 ) ) ) ) )
1282 n0mulscl 
 |-  ( ( 2s e. NN0_s /\ ( m +s 1s ) e. NN0_s ) -> ( 2s x.s ( m +s 1s ) ) e. NN0_s )
1283 412 1148 1282 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 )
1284 1283 n0nod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s x.s ( m +s 1s ) ) e. No )
1285 1268 n0nod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 2s ^su ( o +s 1s ) ) e. No )
1286 1158 1255 1272 ltadds2d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( 1s  ( ( 2s x.s m ) +s 1s )
1287 523 1286 mpbii 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s m ) +s 1s )
1288 1255 1142 1158 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 ) ) )
1289 877 oveq2i 
 |-  ( ( 2s x.s m ) +s ( 2s x.s 1s ) ) = ( ( 2s x.s m ) +s 2s )
1290 1288 1289 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 ) )
1291 1287 1290 breqtrrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s m ) +s 1s )
1292 simprl2 
 |-  ( ( ( m e. NN0_s /\ n e. NN0_s /\ o e. NN0_s ) /\ ( ( m  m
1293 1292 adantl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  m
1294 n0ltsp1le 
 |-  ( ( m e. NN0_s /\ ( 2s ^su o ) e. NN0_s ) -> ( m  ( m +s 1s ) <_s ( 2s ^su o ) ) )
1295 1126 1138 1294 syl2anc 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m  ( m +s 1s ) <_s ( 2s ^su o ) ) )
1296 1293 1295 mpbid 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( m +s 1s ) <_s ( 2s ^su o ) )
1297 975 a1i 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s
1298 1149 1206 1255 1297 lemuls2d 
 |-  ( ( ( ( ( ( ( ( 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 ) ) ) )
1299 1296 1298 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 ) ) )
1300 1299 1260 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 ) ) )
1301 1277 1284 1285 1291 1300 ltlestrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( 2s x.s m ) +s 1s )
1302 1130 n0nod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  o e. No )
1303 1134 1302 1158 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 ) ) )
1304 1216 adantl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s o )
1305 n0addscl 
 |-  ( ( g e. NN0_s /\ o e. NN0_s ) -> ( g +s o ) e. NN0_s )
1306 1125 1130 1305 syl2anc 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s o ) e. NN0_s )
1307 1306 n0nod 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s o ) e. No )
1308 1156 32 syl 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  N e. No )
1309 1307 1308 1158 ltadds1d 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s o )  ( ( g +s o ) +s 1s )
1310 1304 1309 mpbid 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( ( g +s o ) +s 1s )
1311 1303 1310 eqbrtrrd 
 |-  ( ( ( ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( g +s ( o +s 1s ) )
1312 oveq1 
 |-  ( b = ( ( 2s x.s m ) +s 1s ) -> ( b /su ( 2s ^su q ) ) = ( ( ( 2s x.s m ) +s 1s ) /su ( 2s ^su q ) ) )
1313 1312 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 ) ) ) )
1314 1313 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 ) ) ) ) )
1315 breq1 
 |-  ( b = ( ( 2s x.s m ) +s 1s ) -> ( b  ( ( 2s x.s m ) +s 1s )
1316 1314 1315 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 )
1317 oveq2 
 |-  ( q = ( o +s 1s ) -> ( 2s ^su q ) = ( 2s ^su ( o +s 1s ) ) )
1318 1317 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 ) ) ) )
1319 1318 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 ) ) ) ) )
1320 1319 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 ) ) ) ) ) )
1321 1317 breq2d 
 |-  ( q = ( o +s 1s ) -> ( ( ( 2s x.s m ) +s 1s )  ( ( 2s x.s m ) +s 1s )
1322 oveq2 
 |-  ( q = ( o +s 1s ) -> ( g +s q ) = ( g +s ( o +s 1s ) ) )
1323 1322 breq1d 
 |-  ( q = ( o +s 1s ) -> ( ( g +s q )  ( g +s ( o +s 1s ) )
1324 1320 1321 1323 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 )
1325 535 1316 1324 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
1326 1125 1129 1131 1281 1301 1311 1325 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
1327 1326 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
1328 1327 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
1329 1123 1328 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
1330 1329 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
1331 935 1330 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
1332 1331 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
1333 918 1332 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
1334 627 1333 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
1335 625 1334 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
1336 1335 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
1337 600 1336 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
1338 579 1337 mpdan 
 |-  ( ( ( ( ( ( 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
1339 1338 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
1340 1339 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
1341 229 3adant3 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  ( bday ` d ) C_ ( bday ` N ) )
1342 57 a1i 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s e. No )
1343 135 3ad2ant1 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  w e. No )
1344 simp1r 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s
1345 1343 1344 0elleft 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s e. ( _Left ` w ) )
1346 240 1345 189 sltssepcd 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s
1347 1342 384 1346 ltlesd 
 |-  ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s <_s d )
1348 1347 3ad2ant1 
 |-  ( ( ( ( ( ph /\ ( w e. No /\ ( ( bday ` w ) = ( bday ` ( N +s 1s ) ) /\ w =/= ( N +s 1s ) ) ) ) /\ 0s  0s <_s d )
1349 fveq2 
 |-  ( z = d -> ( bday ` z ) = ( bday ` d ) )
1350 1349 sseq1d 
 |-  ( z = d -> ( ( bday ` z ) C_ ( bday ` N ) <-> ( bday ` d ) C_ ( bday ` N ) ) )
1351 breq2 
 |-  ( z = d -> ( 0s <_s z <-> 0s <_s d ) )
1352 1350 1351 anbi12d 
 |-  ( z = d -> ( ( ( bday ` z ) C_ ( bday ` N ) /\ 0s <_s z ) <-> ( ( bday ` d ) C_ ( bday ` N ) /\ 0s <_s d ) ) )
1353 eqeq1 
 |-  ( z = d -> ( z = N <-> d = N ) )
1354 eqeq1 
 |-  ( z = d -> ( z = ( x +s ( y /su ( 2s ^su p ) ) ) <-> d = ( x +s ( y /su ( 2s ^su p ) ) ) ) )
1355 1354 3anbi1d 
 |-  ( z = d -> ( ( z = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  ( d = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y
1356 1355 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
1357 1356 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
1358 oveq1 
 |-  ( x = j -> ( x +s ( y /su ( 2s ^su p ) ) ) = ( j +s ( y /su ( 2s ^su p ) ) ) )
1359 1358 eqeq2d 
 |-  ( x = j -> ( d = ( x +s ( y /su ( 2s ^su p ) ) ) <-> d = ( j +s ( y /su ( 2s ^su p ) ) ) ) )
1360 oveq1 
 |-  ( x = j -> ( x +s p ) = ( j +s p ) )
1361 1360 breq1d 
 |-  ( x = j -> ( ( x +s p )  ( j +s p )
1362 1359 1361 3anbi13d 
 |-  ( x = j -> ( ( d = ( x +s ( y /su ( 2s ^su p ) ) ) /\ y  ( d = ( j +s ( y /su ( 2s ^su p ) ) ) /\ y
1363 1362 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
1364 oveq1 
 |-  ( y = k -> ( y /su ( 2s ^su p ) ) = ( k /su ( 2s ^su p ) ) )
1365 1364 oveq2d 
 |-  ( y = k -> ( j +s ( y /su ( 2s ^su p ) ) ) = ( j +s ( k /su ( 2s ^su p ) ) ) )
1366 1365 eqeq2d 
 |-  ( y = k -> ( d = ( j +s ( y /su ( 2s ^su p ) ) ) <-> d = ( j +s ( k /su ( 2s ^su p ) ) ) ) )
1367 breq1 
 |-  ( y = k -> ( y  k
1368 1366 1367 3anbi12d 
 |-  ( y = k -> ( ( d = ( j +s ( y /su ( 2s ^su p ) ) ) /\ y  ( d = ( j +s ( k /su ( 2s ^su p ) ) ) /\ k
1369 1368 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
1370 oveq2 
 |-  ( p = l -> ( 2s ^su p ) = ( 2s ^su l ) )
1371 1370 oveq2d 
 |-  ( p = l -> ( k /su ( 2s ^su p ) ) = ( k /su ( 2s ^su l ) ) )
1372 1371 oveq2d 
 |-  ( p = l -> ( j +s ( k /su ( 2s ^su p ) ) ) = ( j +s ( k /su ( 2s ^su l ) ) ) )
1373 1372 eqeq2d 
 |-  ( p = l -> ( d = ( j +s ( k /su ( 2s ^su p ) ) ) <-> d = ( j +s ( k /su ( 2s ^su l ) ) ) ) )
1374 1370 breq2d 
 |-  ( p = l -> ( k  k
1375 oveq2 
 |-  ( p = l -> ( j +s p ) = ( j +s l ) )
1376 1375 breq1d 
 |-  ( p = l -> ( ( j +s p )  ( j +s l )
1377 1373 1374 1376 3anbi123d 
 |-  ( p = l -> ( ( d = ( j +s ( k /su ( 2s ^su p ) ) ) /\ k  ( d = ( j +s ( k /su ( 2s ^su l ) ) ) /\ k
1378 1377 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
1379 1369 1378 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
1380 1363 1379 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
1381 1357 1380 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
1382 1353 1381 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
1383 1352 1382 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
1384 296 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
1385 1383 1384 385 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
1386 1341 1348 1385 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
1387 561 1340 1386 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
1388 1387 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
1389 1388 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
1390 1389 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
1391 287 1390 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
1392 210 215 1391 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
1393 191 1392 mpdan 
 |-  ( ( ( ( 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
1394 1393 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
1395 1394 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
1396 186 1395 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
1397 1396 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
1398 180 1397 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
1399 154 175 1398 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
1400 1399 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
1401 150 1400 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
1402 1401 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
1403 144 1402 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
1404 134 138 1403 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
1405 1404 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
1406 addslid 
 |-  ( 0s e. No -> ( 0s +s 0s ) = 0s )
1407 57 1406 ax-mp 
 |-  ( 0s +s 0s ) = 0s
1408 1407 eqcomi 
 |-  0s = ( 0s +s 0s )
1409 n0p1nns 
 |-  ( N e. NN0_s -> ( N +s 1s ) e. NN_s )
1410 1 1409 syl 
 |-  ( ph -> ( N +s 1s ) e. NN_s )
1411 nnsgt0 
 |-  ( ( N +s 1s ) e. NN_s -> 0s
1412 1410 1411 syl 
 |-  ( ph -> 0s
1413 29 29 29 3pm3.2i 
 |-  ( 0s e. NN0_s /\ 0s e. NN0_s /\ 0s e. NN0_s )
1414 oveq1 
 |-  ( a = 0s -> ( a +s ( b /su ( 2s ^su q ) ) ) = ( 0s +s ( b /su ( 2s ^su q ) ) ) )
1415 1414 eqeq2d 
 |-  ( a = 0s -> ( 0s = ( a +s ( b /su ( 2s ^su q ) ) ) <-> 0s = ( 0s +s ( b /su ( 2s ^su q ) ) ) ) )
1416 oveq1 
 |-  ( a = 0s -> ( a +s q ) = ( 0s +s q ) )
1417 1416 breq1d 
 |-  ( a = 0s -> ( ( a +s q )  ( 0s +s q )
1418 1415 1417 3anbi13d 
 |-  ( a = 0s -> ( ( 0s = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b  ( 0s = ( 0s +s ( b /su ( 2s ^su q ) ) ) /\ b
1419 46 oveq2d 
 |-  ( b = 0s -> ( 0s +s ( b /su ( 2s ^su q ) ) ) = ( 0s +s ( 0s /su ( 2s ^su q ) ) ) )
1420 1419 eqeq2d 
 |-  ( b = 0s -> ( 0s = ( 0s +s ( b /su ( 2s ^su q ) ) ) <-> 0s = ( 0s +s ( 0s /su ( 2s ^su q ) ) ) ) )
1421 1420 49 3anbi12d 
 |-  ( b = 0s -> ( ( 0s = ( 0s +s ( b /su ( 2s ^su q ) ) ) /\ b  ( 0s = ( 0s +s ( 0s /su ( 2s ^su q ) ) ) /\ 0s
1422 60 oveq2d 
 |-  ( q = 0s -> ( 0s +s ( 0s /su ( 2s ^su q ) ) ) = ( 0s +s 0s ) )
1423 1422 eqeq2d 
 |-  ( q = 0s -> ( 0s = ( 0s +s ( 0s /su ( 2s ^su q ) ) ) <-> 0s = ( 0s +s 0s ) ) )
1424 oveq2 
 |-  ( q = 0s -> ( 0s +s q ) = ( 0s +s 0s ) )
1425 1424 1407 eqtrdi 
 |-  ( q = 0s -> ( 0s +s q ) = 0s )
1426 1425 breq1d 
 |-  ( q = 0s -> ( ( 0s +s q )  0s
1427 1423 63 1426 3anbi123d 
 |-  ( q = 0s -> ( ( 0s = ( 0s +s ( 0s /su ( 2s ^su q ) ) ) /\ 0s  ( 0s = ( 0s +s 0s ) /\ 0s
1428 1418 1421 1427 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
1429 1413 1428 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
1430 1408 36 1412 1429 mp3an12i 
 |-  ( 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
1431 eqeq1 
 |-  ( 0s = w -> ( 0s = ( a +s ( b /su ( 2s ^su q ) ) ) <-> w = ( a +s ( b /su ( 2s ^su q ) ) ) ) )
1432 1431 3anbi1d 
 |-  ( 0s = w -> ( ( 0s = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b  ( w = ( a +s ( b /su ( 2s ^su q ) ) ) /\ b
1433 1432 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
1434 1433 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
1435 1430 1434 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
1436 1435 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
1437 1405 1436 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
1438 121 1437 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
1439 1438 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
1440 1439 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
1441 1440 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
1442 1441 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
1443 1442 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
1444 118 1443 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
1445 1444 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
1446 1445 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
1447 1446 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
1448 117 1447 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
1449 116 1448 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
1450 14 1449 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
1451 6 1450 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
1452 1451 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
1453 1452 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