Metamath Proof Explorer

Theorem ovolscalem1

Description: Lemma for ovolsca . (Contributed by Mario Carneiro, 6-Apr-2015)

Ref Expression
Hypotheses ovolsca.1 
|- ( ph -> A C_ RR )
ovolsca.2 
|- ( ph -> C e. RR+ )
ovolsca.3 
|- ( ph -> B = { x e. RR | ( C x. x ) e. A } )
ovolsca.4 
|- ( ph -> ( vol* ` A ) e. RR )
ovolsca.5 
|- S = seq 1 ( + , ( ( abs o. - ) o. F ) )
ovolsca.6 
|- G = ( n e. NN |-> <. ( ( 1st ` ( F ` n ) ) / C ) , ( ( 2nd ` ( F ` n ) ) / C ) >. )
ovolsca.7 
|- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )
ovolsca.8 
|- ( ph -> A C_ U. ran ( (,) o. F ) )
ovolsca.9 
|- ( ph -> R e. RR+ )
ovolsca.10 
|- ( ph -> sup ( ran S , RR* , < ) <_ ( ( vol* ` A ) + ( C x. R ) ) )
Assertion ovolscalem1 
|- ( ph -> ( vol* ` B ) <_ ( ( ( vol* ` A ) / C ) + R ) )

Proof

Step Hyp Ref Expression
1 ovolsca.1 
 |-  ( ph -> A C_ RR )
2 ovolsca.2 
 |-  ( ph -> C e. RR+ )
3 ovolsca.3 
 |-  ( ph -> B = { x e. RR | ( C x. x ) e. A } )
4 ovolsca.4 
 |-  ( ph -> ( vol* ` A ) e. RR )
5 ovolsca.5 
 |-  S = seq 1 ( + , ( ( abs o. - ) o. F ) )
6 ovolsca.6 
 |-  G = ( n e. NN |-> <. ( ( 1st ` ( F ` n ) ) / C ) , ( ( 2nd ` ( F ` n ) ) / C ) >. )
7 ovolsca.7 
 |-  ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )
8 ovolsca.8 
 |-  ( ph -> A C_ U. ran ( (,) o. F ) )
9 ovolsca.9 
 |-  ( ph -> R e. RR+ )
10 ovolsca.10 
 |-  ( ph -> sup ( ran S , RR* , < ) <_ ( ( vol* ` A ) + ( C x. R ) ) )
11 ssrab2 
 |-  { x e. RR | ( C x. x ) e. A } C_ RR
12 3 11 eqsstrdi 
 |-  ( ph -> B C_ RR )
13 ovolcl 
 |-  ( B C_ RR -> ( vol* ` B ) e. RR* )
14 12 13 syl 
 |-  ( ph -> ( vol* ` B ) e. RR* )
15 ovolfcl 
 |-  ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) e. RR /\ ( 2nd ` ( F ` n ) ) e. RR /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) )
16 7 15 sylan 
 |-  ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) e. RR /\ ( 2nd ` ( F ` n ) ) e. RR /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) )
17 16 simp3d 
 |-  ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) )
18 16 simp1d 
 |-  ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. RR )
19 16 simp2d 
 |-  ( ( ph /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. RR )
20 2 rpregt0d 
 |-  ( ph -> ( C e. RR /\ 0 < C ) )
21 20 adantr 
 |-  ( ( ph /\ n e. NN ) -> ( C e. RR /\ 0 < C ) )
22 lediv1 
 |-  ( ( ( 1st ` ( F ` n ) ) e. RR /\ ( 2nd ` ( F ` n ) ) e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) <-> ( ( 1st ` ( F ` n ) ) / C ) <_ ( ( 2nd ` ( F ` n ) ) / C ) ) )
23 18 19 21 22 syl3anc 
 |-  ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) <-> ( ( 1st ` ( F ` n ) ) / C ) <_ ( ( 2nd ` ( F ` n ) ) / C ) ) )
24 17 23 mpbid 
 |-  ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) / C ) <_ ( ( 2nd ` ( F ` n ) ) / C ) )
25 df-br 
 |-  ( ( ( 1st ` ( F ` n ) ) / C ) <_ ( ( 2nd ` ( F ` n ) ) / C ) <-> <. ( ( 1st ` ( F ` n ) ) / C ) , ( ( 2nd ` ( F ` n ) ) / C ) >. e. <_ )
26 24 25 sylib 
 |-  ( ( ph /\ n e. NN ) -> <. ( ( 1st ` ( F ` n ) ) / C ) , ( ( 2nd ` ( F ` n ) ) / C ) >. e. <_ )
27 2 adantr 
 |-  ( ( ph /\ n e. NN ) -> C e. RR+ )
28 18 27 rerpdivcld 
 |-  ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) / C ) e. RR )
29 19 27 rerpdivcld 
 |-  ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( F ` n ) ) / C ) e. RR )
30 28 29 opelxpd 
 |-  ( ( ph /\ n e. NN ) -> <. ( ( 1st ` ( F ` n ) ) / C ) , ( ( 2nd ` ( F ` n ) ) / C ) >. e. ( RR X. RR ) )
31 26 30 elind 
 |-  ( ( ph /\ n e. NN ) -> <. ( ( 1st ` ( F ` n ) ) / C ) , ( ( 2nd ` ( F ` n ) ) / C ) >. e. ( <_ i^i ( RR X. RR ) ) )
32 31 6 fmptd 
 |-  ( ph -> G : NN --> ( <_ i^i ( RR X. RR ) ) )
33 eqid 
 |-  ( ( abs o. - ) o. G ) = ( ( abs o. - ) o. G )
34 eqid 
 |-  seq 1 ( + , ( ( abs o. - ) o. G ) ) = seq 1 ( + , ( ( abs o. - ) o. G ) )
35 33 34 ovolsf 
 |-  ( G : NN --> ( <_ i^i ( RR X. RR ) ) -> seq 1 ( + , ( ( abs o. - ) o. G ) ) : NN --> ( 0 [,) +oo ) )
36 32 35 syl 
 |-  ( ph -> seq 1 ( + , ( ( abs o. - ) o. G ) ) : NN --> ( 0 [,) +oo ) )
37 36 frnd 
 |-  ( ph -> ran seq 1 ( + , ( ( abs o. - ) o. G ) ) C_ ( 0 [,) +oo ) )
38 icossxr 
 |-  ( 0 [,) +oo ) C_ RR*
39 37 38 sstrdi 
 |-  ( ph -> ran seq 1 ( + , ( ( abs o. - ) o. G ) ) C_ RR* )
40 supxrcl 
 |-  ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) C_ RR* -> sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) e. RR* )
41 39 40 syl 
 |-  ( ph -> sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) e. RR* )
42 4 2 rerpdivcld 
 |-  ( ph -> ( ( vol* ` A ) / C ) e. RR )
43 9 rpred 
 |-  ( ph -> R e. RR )
44 42 43 readdcld 
 |-  ( ph -> ( ( ( vol* ` A ) / C ) + R ) e. RR )
45 44 rexrd 
 |-  ( ph -> ( ( ( vol* ` A ) / C ) + R ) e. RR* )
46 3 eleq2d 
 |-  ( ph -> ( y e. B <-> y e. { x e. RR | ( C x. x ) e. A } ) )
47 oveq2 
 |-  ( x = y -> ( C x. x ) = ( C x. y ) )
48 47 eleq1d 
 |-  ( x = y -> ( ( C x. x ) e. A <-> ( C x. y ) e. A ) )
49 48 elrab 
 |-  ( y e. { x e. RR | ( C x. x ) e. A } <-> ( y e. RR /\ ( C x. y ) e. A ) )
50 46 49 bitrdi 
 |-  ( ph -> ( y e. B <-> ( y e. RR /\ ( C x. y ) e. A ) ) )
51 breq2 
 |-  ( x = ( C x. y ) -> ( ( 1st ` ( F ` n ) ) < x <-> ( 1st ` ( F ` n ) ) < ( C x. y ) ) )
52 breq1 
 |-  ( x = ( C x. y ) -> ( x < ( 2nd ` ( F ` n ) ) <-> ( C x. y ) < ( 2nd ` ( F ` n ) ) ) )
53 51 52 anbi12d 
 |-  ( x = ( C x. y ) -> ( ( ( 1st ` ( F ` n ) ) < x /\ x < ( 2nd ` ( F ` n ) ) ) <-> ( ( 1st ` ( F ` n ) ) < ( C x. y ) /\ ( C x. y ) < ( 2nd ` ( F ` n ) ) ) ) )
54 53 rexbidv 
 |-  ( x = ( C x. y ) -> ( E. n e. NN ( ( 1st ` ( F ` n ) ) < x /\ x < ( 2nd ` ( F ` n ) ) ) <-> E. n e. NN ( ( 1st ` ( F ` n ) ) < ( C x. y ) /\ ( C x. y ) < ( 2nd ` ( F ` n ) ) ) ) )
55 ovolfioo 
 |-  ( ( A C_ RR /\ F : NN --> ( <_ i^i ( RR X. RR ) ) ) -> ( A C_ U. ran ( (,) o. F ) <-> A. x e. A E. n e. NN ( ( 1st ` ( F ` n ) ) < x /\ x < ( 2nd ` ( F ` n ) ) ) ) )
56 1 7 55 syl2anc 
 |-  ( ph -> ( A C_ U. ran ( (,) o. F ) <-> A. x e. A E. n e. NN ( ( 1st ` ( F ` n ) ) < x /\ x < ( 2nd ` ( F ` n ) ) ) ) )
57 8 56 mpbid 
 |-  ( ph -> A. x e. A E. n e. NN ( ( 1st ` ( F ` n ) ) < x /\ x < ( 2nd ` ( F ` n ) ) ) )
58 57 adantr 
 |-  ( ( ph /\ ( y e. RR /\ ( C x. y ) e. A ) ) -> A. x e. A E. n e. NN ( ( 1st ` ( F ` n ) ) < x /\ x < ( 2nd ` ( F ` n ) ) ) )
59 simprr 
 |-  ( ( ph /\ ( y e. RR /\ ( C x. y ) e. A ) ) -> ( C x. y ) e. A )
60 54 58 59 rspcdva 
 |-  ( ( ph /\ ( y e. RR /\ ( C x. y ) e. A ) ) -> E. n e. NN ( ( 1st ` ( F ` n ) ) < ( C x. y ) /\ ( C x. y ) < ( 2nd ` ( F ` n ) ) ) )
61 opex 
 |-  <. ( ( 1st ` ( F ` n ) ) / C ) , ( ( 2nd ` ( F ` n ) ) / C ) >. e. _V
62 6 fvmpt2 
 |-  ( ( n e. NN /\ <. ( ( 1st ` ( F ` n ) ) / C ) , ( ( 2nd ` ( F ` n ) ) / C ) >. e. _V ) -> ( G ` n ) = <. ( ( 1st ` ( F ` n ) ) / C ) , ( ( 2nd ` ( F ` n ) ) / C ) >. )
63 61 62 mpan2 
 |-  ( n e. NN -> ( G ` n ) = <. ( ( 1st ` ( F ` n ) ) / C ) , ( ( 2nd ` ( F ` n ) ) / C ) >. )
64 63 fveq2d 
 |-  ( n e. NN -> ( 1st ` ( G ` n ) ) = ( 1st ` <. ( ( 1st ` ( F ` n ) ) / C ) , ( ( 2nd ` ( F ` n ) ) / C ) >. ) )
65 ovex 
 |-  ( ( 1st ` ( F ` n ) ) / C ) e. _V
66 ovex 
 |-  ( ( 2nd ` ( F ` n ) ) / C ) e. _V
67 65 66 op1st 
 |-  ( 1st ` <. ( ( 1st ` ( F ` n ) ) / C ) , ( ( 2nd ` ( F ` n ) ) / C ) >. ) = ( ( 1st ` ( F ` n ) ) / C )
68 64 67 eqtrdi 
 |-  ( n e. NN -> ( 1st ` ( G ` n ) ) = ( ( 1st ` ( F ` n ) ) / C ) )
69 68 adantl 
 |-  ( ( ( ph /\ ( y e. RR /\ ( C x. y ) e. A ) ) /\ n e. NN ) -> ( 1st ` ( G ` n ) ) = ( ( 1st ` ( F ` n ) ) / C ) )
70 69 breq1d 
 |-  ( ( ( ph /\ ( y e. RR /\ ( C x. y ) e. A ) ) /\ n e. NN ) -> ( ( 1st ` ( G ` n ) ) < y <-> ( ( 1st ` ( F ` n ) ) / C ) < y ) )
71 18 adantlr 
 |-  ( ( ( ph /\ ( y e. RR /\ ( C x. y ) e. A ) ) /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. RR )
72 simplrl 
 |-  ( ( ( ph /\ ( y e. RR /\ ( C x. y ) e. A ) ) /\ n e. NN ) -> y e. RR )
73 21 adantlr 
 |-  ( ( ( ph /\ ( y e. RR /\ ( C x. y ) e. A ) ) /\ n e. NN ) -> ( C e. RR /\ 0 < C ) )
74 ltdivmul 
 |-  ( ( ( 1st ` ( F ` n ) ) e. RR /\ y e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( ( 1st ` ( F ` n ) ) / C ) < y <-> ( 1st ` ( F ` n ) ) < ( C x. y ) ) )
75 71 72 73 74 syl3anc 
 |-  ( ( ( ph /\ ( y e. RR /\ ( C x. y ) e. A ) ) /\ n e. NN ) -> ( ( ( 1st ` ( F ` n ) ) / C ) < y <-> ( 1st ` ( F ` n ) ) < ( C x. y ) ) )
76 70 75 bitr2d 
 |-  ( ( ( ph /\ ( y e. RR /\ ( C x. y ) e. A ) ) /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) < ( C x. y ) <-> ( 1st ` ( G ` n ) ) < y ) )
77 19 adantlr 
 |-  ( ( ( ph /\ ( y e. RR /\ ( C x. y ) e. A ) ) /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. RR )
78 ltmuldiv2 
 |-  ( ( y e. RR /\ ( 2nd ` ( F ` n ) ) e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. y ) < ( 2nd ` ( F ` n ) ) <-> y < ( ( 2nd ` ( F ` n ) ) / C ) ) )
79 72 77 73 78 syl3anc 
 |-  ( ( ( ph /\ ( y e. RR /\ ( C x. y ) e. A ) ) /\ n e. NN ) -> ( ( C x. y ) < ( 2nd ` ( F ` n ) ) <-> y < ( ( 2nd ` ( F ` n ) ) / C ) ) )
80 63 fveq2d 
 |-  ( n e. NN -> ( 2nd ` ( G ` n ) ) = ( 2nd ` <. ( ( 1st ` ( F ` n ) ) / C ) , ( ( 2nd ` ( F ` n ) ) / C ) >. ) )
81 65 66 op2nd 
 |-  ( 2nd ` <. ( ( 1st ` ( F ` n ) ) / C ) , ( ( 2nd ` ( F ` n ) ) / C ) >. ) = ( ( 2nd ` ( F ` n ) ) / C )
82 80 81 eqtrdi 
 |-  ( n e. NN -> ( 2nd ` ( G ` n ) ) = ( ( 2nd ` ( F ` n ) ) / C ) )
83 82 adantl 
 |-  ( ( ( ph /\ ( y e. RR /\ ( C x. y ) e. A ) ) /\ n e. NN ) -> ( 2nd ` ( G ` n ) ) = ( ( 2nd ` ( F ` n ) ) / C ) )
84 83 breq2d 
 |-  ( ( ( ph /\ ( y e. RR /\ ( C x. y ) e. A ) ) /\ n e. NN ) -> ( y < ( 2nd ` ( G ` n ) ) <-> y < ( ( 2nd ` ( F ` n ) ) / C ) ) )
85 79 84 bitr4d 
 |-  ( ( ( ph /\ ( y e. RR /\ ( C x. y ) e. A ) ) /\ n e. NN ) -> ( ( C x. y ) < ( 2nd ` ( F ` n ) ) <-> y < ( 2nd ` ( G ` n ) ) ) )
86 76 85 anbi12d 
 |-  ( ( ( ph /\ ( y e. RR /\ ( C x. y ) e. A ) ) /\ n e. NN ) -> ( ( ( 1st ` ( F ` n ) ) < ( C x. y ) /\ ( C x. y ) < ( 2nd ` ( F ` n ) ) ) <-> ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) ) )
87 86 rexbidva 
 |-  ( ( ph /\ ( y e. RR /\ ( C x. y ) e. A ) ) -> ( E. n e. NN ( ( 1st ` ( F ` n ) ) < ( C x. y ) /\ ( C x. y ) < ( 2nd ` ( F ` n ) ) ) <-> E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) ) )
88 60 87 mpbid 
 |-  ( ( ph /\ ( y e. RR /\ ( C x. y ) e. A ) ) -> E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) )
89 88 ex 
 |-  ( ph -> ( ( y e. RR /\ ( C x. y ) e. A ) -> E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) ) )
90 50 89 sylbid 
 |-  ( ph -> ( y e. B -> E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) ) )
91 90 ralrimiv 
 |-  ( ph -> A. y e. B E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) )
92 ovolfioo 
 |-  ( ( B C_ RR /\ G : NN --> ( <_ i^i ( RR X. RR ) ) ) -> ( B C_ U. ran ( (,) o. G ) <-> A. y e. B E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) ) )
93 12 32 92 syl2anc 
 |-  ( ph -> ( B C_ U. ran ( (,) o. G ) <-> A. y e. B E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) ) )
94 91 93 mpbird 
 |-  ( ph -> B C_ U. ran ( (,) o. G ) )
95 34 ovollb 
 |-  ( ( G : NN --> ( <_ i^i ( RR X. RR ) ) /\ B C_ U. ran ( (,) o. G ) ) -> ( vol* ` B ) <_ sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) )
96 32 94 95 syl2anc 
 |-  ( ph -> ( vol* ` B ) <_ sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) )
97 fzfid 
 |-  ( ( ph /\ k e. NN ) -> ( 1 ... k ) e. Fin )
98 2 rpcnd 
 |-  ( ph -> C e. CC )
99 98 adantr 
 |-  ( ( ph /\ k e. NN ) -> C e. CC )
100 simpl 
 |-  ( ( ph /\ k e. NN ) -> ph )
101 elfznn 
 |-  ( n e. ( 1 ... k ) -> n e. NN )
102 19 18 resubcld 
 |-  ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) e. RR )
103 100 101 102 syl2an 
 |-  ( ( ( ph /\ k e. NN ) /\ n e. ( 1 ... k ) ) -> ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) e. RR )
104 103 recnd 
 |-  ( ( ( ph /\ k e. NN ) /\ n e. ( 1 ... k ) ) -> ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) e. CC )
105 2 rpne0d 
 |-  ( ph -> C =/= 0 )
106 105 adantr 
 |-  ( ( ph /\ k e. NN ) -> C =/= 0 )
107 97 99 104 106 fsumdivc 
 |-  ( ( ph /\ k e. NN ) -> ( sum_ n e. ( 1 ... k ) ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) / C ) = sum_ n e. ( 1 ... k ) ( ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) / C ) )
108 82 68 oveq12d 
 |-  ( n e. NN -> ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) = ( ( ( 2nd ` ( F ` n ) ) / C ) - ( ( 1st ` ( F ` n ) ) / C ) ) )
109 108 adantl 
 |-  ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) = ( ( ( 2nd ` ( F ` n ) ) / C ) - ( ( 1st ` ( F ` n ) ) / C ) ) )
110 33 ovolfsval 
 |-  ( ( G : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( ( ( abs o. - ) o. G ) ` n ) = ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) )
111 32 110 sylan 
 |-  ( ( ph /\ n e. NN ) -> ( ( ( abs o. - ) o. G ) ` n ) = ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) )
112 19 recnd 
 |-  ( ( ph /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. CC )
113 18 recnd 
 |-  ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. CC )
114 2 rpcnne0d 
 |-  ( ph -> ( C e. CC /\ C =/= 0 ) )
115 114 adantr 
 |-  ( ( ph /\ n e. NN ) -> ( C e. CC /\ C =/= 0 ) )
116 divsubdir 
 |-  ( ( ( 2nd ` ( F ` n ) ) e. CC /\ ( 1st ` ( F ` n ) ) e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) / C ) = ( ( ( 2nd ` ( F ` n ) ) / C ) - ( ( 1st ` ( F ` n ) ) / C ) ) )
117 112 113 115 116 syl3anc 
 |-  ( ( ph /\ n e. NN ) -> ( ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) / C ) = ( ( ( 2nd ` ( F ` n ) ) / C ) - ( ( 1st ` ( F ` n ) ) / C ) ) )
118 109 111 117 3eqtr4d 
 |-  ( ( ph /\ n e. NN ) -> ( ( ( abs o. - ) o. G ) ` n ) = ( ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) / C ) )
119 100 101 118 syl2an 
 |-  ( ( ( ph /\ k e. NN ) /\ n e. ( 1 ... k ) ) -> ( ( ( abs o. - ) o. G ) ` n ) = ( ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) / C ) )
120 simpr 
 |-  ( ( ph /\ k e. NN ) -> k e. NN )
121 nnuz 
 |-  NN = ( ZZ>= ` 1 )
122 120 121 eleqtrdi 
 |-  ( ( ph /\ k e. NN ) -> k e. ( ZZ>= ` 1 ) )
123 102 27 rerpdivcld 
 |-  ( ( ph /\ n e. NN ) -> ( ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) / C ) e. RR )
124 123 recnd 
 |-  ( ( ph /\ n e. NN ) -> ( ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) / C ) e. CC )
125 100 101 124 syl2an 
 |-  ( ( ( ph /\ k e. NN ) /\ n e. ( 1 ... k ) ) -> ( ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) / C ) e. CC )
126 119 122 125 fsumser 
 |-  ( ( ph /\ k e. NN ) -> sum_ n e. ( 1 ... k ) ( ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) / C ) = ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` k ) )
127 107 126 eqtrd 
 |-  ( ( ph /\ k e. NN ) -> ( sum_ n e. ( 1 ... k ) ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) / C ) = ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` k ) )
128 eqid 
 |-  ( ( abs o. - ) o. F ) = ( ( abs o. - ) o. F )
129 128 5 ovolsf 
 |-  ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> S : NN --> ( 0 [,) +oo ) )
130 7 129 syl 
 |-  ( ph -> S : NN --> ( 0 [,) +oo ) )
131 130 frnd 
 |-  ( ph -> ran S C_ ( 0 [,) +oo ) )
132 131 38 sstrdi 
 |-  ( ph -> ran S C_ RR* )
133 2 9 rpmulcld 
 |-  ( ph -> ( C x. R ) e. RR+ )
134 133 rpred 
 |-  ( ph -> ( C x. R ) e. RR )
135 4 134 readdcld 
 |-  ( ph -> ( ( vol* ` A ) + ( C x. R ) ) e. RR )
136 135 rexrd 
 |-  ( ph -> ( ( vol* ` A ) + ( C x. R ) ) e. RR* )
137 supxrleub 
 |-  ( ( ran S C_ RR* /\ ( ( vol* ` A ) + ( C x. R ) ) e. RR* ) -> ( sup ( ran S , RR* , < ) <_ ( ( vol* ` A ) + ( C x. R ) ) <-> A. x e. ran S x <_ ( ( vol* ` A ) + ( C x. R ) ) ) )
138 132 136 137 syl2anc 
 |-  ( ph -> ( sup ( ran S , RR* , < ) <_ ( ( vol* ` A ) + ( C x. R ) ) <-> A. x e. ran S x <_ ( ( vol* ` A ) + ( C x. R ) ) ) )
139 10 138 mpbid 
 |-  ( ph -> A. x e. ran S x <_ ( ( vol* ` A ) + ( C x. R ) ) )
140 130 ffnd 
 |-  ( ph -> S Fn NN )
141 breq1 
 |-  ( x = ( S ` k ) -> ( x <_ ( ( vol* ` A ) + ( C x. R ) ) <-> ( S ` k ) <_ ( ( vol* ` A ) + ( C x. R ) ) ) )
142 141 ralrn 
 |-  ( S Fn NN -> ( A. x e. ran S x <_ ( ( vol* ` A ) + ( C x. R ) ) <-> A. k e. NN ( S ` k ) <_ ( ( vol* ` A ) + ( C x. R ) ) ) )
143 140 142 syl 
 |-  ( ph -> ( A. x e. ran S x <_ ( ( vol* ` A ) + ( C x. R ) ) <-> A. k e. NN ( S ` k ) <_ ( ( vol* ` A ) + ( C x. R ) ) ) )
144 139 143 mpbid 
 |-  ( ph -> A. k e. NN ( S ` k ) <_ ( ( vol* ` A ) + ( C x. R ) ) )
145 144 r19.21bi 
 |-  ( ( ph /\ k e. NN ) -> ( S ` k ) <_ ( ( vol* ` A ) + ( C x. R ) ) )
146 7 adantr 
 |-  ( ( ph /\ k e. NN ) -> F : NN --> ( <_ i^i ( RR X. RR ) ) )
147 128 ovolfsval 
 |-  ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( ( ( abs o. - ) o. F ) ` n ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
148 146 101 147 syl2an 
 |-  ( ( ( ph /\ k e. NN ) /\ n e. ( 1 ... k ) ) -> ( ( ( abs o. - ) o. F ) ` n ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
149 148 122 104 fsumser 
 |-  ( ( ph /\ k e. NN ) -> sum_ n e. ( 1 ... k ) ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) = ( seq 1 ( + , ( ( abs o. - ) o. F ) ) ` k ) )
150 5 fveq1i 
 |-  ( S ` k ) = ( seq 1 ( + , ( ( abs o. - ) o. F ) ) ` k )
151 149 150 eqtr4di 
 |-  ( ( ph /\ k e. NN ) -> sum_ n e. ( 1 ... k ) ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) = ( S ` k ) )
152 42 recnd 
 |-  ( ph -> ( ( vol* ` A ) / C ) e. CC )
153 9 rpcnd 
 |-  ( ph -> R e. CC )
154 98 152 153 adddid 
 |-  ( ph -> ( C x. ( ( ( vol* ` A ) / C ) + R ) ) = ( ( C x. ( ( vol* ` A ) / C ) ) + ( C x. R ) ) )
155 4 recnd 
 |-  ( ph -> ( vol* ` A ) e. CC )
156 155 98 105 divcan2d 
 |-  ( ph -> ( C x. ( ( vol* ` A ) / C ) ) = ( vol* ` A ) )
157 156 oveq1d 
 |-  ( ph -> ( ( C x. ( ( vol* ` A ) / C ) ) + ( C x. R ) ) = ( ( vol* ` A ) + ( C x. R ) ) )
158 154 157 eqtrd 
 |-  ( ph -> ( C x. ( ( ( vol* ` A ) / C ) + R ) ) = ( ( vol* ` A ) + ( C x. R ) ) )
159 158 adantr 
 |-  ( ( ph /\ k e. NN ) -> ( C x. ( ( ( vol* ` A ) / C ) + R ) ) = ( ( vol* ` A ) + ( C x. R ) ) )
160 145 151 159 3brtr4d 
 |-  ( ( ph /\ k e. NN ) -> sum_ n e. ( 1 ... k ) ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) <_ ( C x. ( ( ( vol* ` A ) / C ) + R ) ) )
161 97 103 fsumrecl 
 |-  ( ( ph /\ k e. NN ) -> sum_ n e. ( 1 ... k ) ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) e. RR )
162 44 adantr 
 |-  ( ( ph /\ k e. NN ) -> ( ( ( vol* ` A ) / C ) + R ) e. RR )
163 20 adantr 
 |-  ( ( ph /\ k e. NN ) -> ( C e. RR /\ 0 < C ) )
164 ledivmul 
 |-  ( ( sum_ n e. ( 1 ... k ) ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) e. RR /\ ( ( ( vol* ` A ) / C ) + R ) e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( sum_ n e. ( 1 ... k ) ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) / C ) <_ ( ( ( vol* ` A ) / C ) + R ) <-> sum_ n e. ( 1 ... k ) ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) <_ ( C x. ( ( ( vol* ` A ) / C ) + R ) ) ) )
165 161 162 163 164 syl3anc 
 |-  ( ( ph /\ k e. NN ) -> ( ( sum_ n e. ( 1 ... k ) ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) / C ) <_ ( ( ( vol* ` A ) / C ) + R ) <-> sum_ n e. ( 1 ... k ) ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) <_ ( C x. ( ( ( vol* ` A ) / C ) + R ) ) ) )
166 160 165 mpbird 
 |-  ( ( ph /\ k e. NN ) -> ( sum_ n e. ( 1 ... k ) ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) / C ) <_ ( ( ( vol* ` A ) / C ) + R ) )
167 127 166 eqbrtrrd 
 |-  ( ( ph /\ k e. NN ) -> ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` k ) <_ ( ( ( vol* ` A ) / C ) + R ) )
168 167 ralrimiva 
 |-  ( ph -> A. k e. NN ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` k ) <_ ( ( ( vol* ` A ) / C ) + R ) )
169 36 ffnd 
 |-  ( ph -> seq 1 ( + , ( ( abs o. - ) o. G ) ) Fn NN )
170 breq1 
 |-  ( y = ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` k ) -> ( y <_ ( ( ( vol* ` A ) / C ) + R ) <-> ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` k ) <_ ( ( ( vol* ` A ) / C ) + R ) ) )
171 170 ralrn 
 |-  ( seq 1 ( + , ( ( abs o. - ) o. G ) ) Fn NN -> ( A. y e. ran seq 1 ( + , ( ( abs o. - ) o. G ) ) y <_ ( ( ( vol* ` A ) / C ) + R ) <-> A. k e. NN ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` k ) <_ ( ( ( vol* ` A ) / C ) + R ) ) )
172 169 171 syl 
 |-  ( ph -> ( A. y e. ran seq 1 ( + , ( ( abs o. - ) o. G ) ) y <_ ( ( ( vol* ` A ) / C ) + R ) <-> A. k e. NN ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` k ) <_ ( ( ( vol* ` A ) / C ) + R ) ) )
173 168 172 mpbird 
 |-  ( ph -> A. y e. ran seq 1 ( + , ( ( abs o. - ) o. G ) ) y <_ ( ( ( vol* ` A ) / C ) + R ) )
174 supxrleub 
 |-  ( ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) C_ RR* /\ ( ( ( vol* ` A ) / C ) + R ) e. RR* ) -> ( sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) <_ ( ( ( vol* ` A ) / C ) + R ) <-> A. y e. ran seq 1 ( + , ( ( abs o. - ) o. G ) ) y <_ ( ( ( vol* ` A ) / C ) + R ) ) )
175 39 45 174 syl2anc 
 |-  ( ph -> ( sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) <_ ( ( ( vol* ` A ) / C ) + R ) <-> A. y e. ran seq 1 ( + , ( ( abs o. - ) o. G ) ) y <_ ( ( ( vol* ` A ) / C ) + R ) ) )
176 173 175 mpbird 
 |-  ( ph -> sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) <_ ( ( ( vol* ` A ) / C ) + R ) )
177 14 41 45 96 176 xrletrd 
 |-  ( ph -> ( vol* ` B ) <_ ( ( ( vol* ` A ) / C ) + R ) )