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Theorem imo72b2 36663
Description: IMO 1972 B2. (14th International Mathemahics Olympiad in Poland, problem B2). (Contributed by Stanislas Polu, 9-Mar-2020.)
Hypotheses
Ref Expression
imo72b2.1  |-  ( ph  ->  F : RR --> RR )
imo72b2.2  |-  ( ph  ->  G : RR --> RR )
imo72b2.4  |-  ( ph  ->  B  e.  RR )
imo72b2.5  |-  ( ph  ->  A. u  e.  RR  A. v  e.  RR  (
( F `  (
u  +  v ) )  +  ( F `
 ( u  -  v ) ) )  =  ( 2  x.  ( ( F `  u )  x.  ( G `  v )
) ) )
imo72b2.6  |-  ( ph  ->  A. y  e.  RR  ( abs `  ( F `
 y ) )  <_  1 )
imo72b2.7  |-  ( ph  ->  E. x  e.  RR  ( F `  x )  =/=  0 )
Assertion
Ref Expression
imo72b2  |-  ( ph  ->  ( abs `  ( G `  B )
)  <_  1 )
Distinct variable groups:    u, B, v    x, B    y, B    u, F, v    x, F   
y, F    u, G, v    x, G    y, G    ph, u, v    ph, x    ph, y, u

Proof of Theorem imo72b2
Dummy variables  c 
t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imo72b2.2 . . . . 5  |-  ( ph  ->  G : RR --> RR )
2 imo72b2.4 . . . . 5  |-  ( ph  ->  B  e.  RR )
31, 2ffvelrnd 6046 . . . 4  |-  ( ph  ->  ( G `  B
)  e.  RR )
43recnd 9695 . . 3  |-  ( ph  ->  ( G `  B
)  e.  CC )
54abscld 13547 . 2  |-  ( ph  ->  ( abs `  ( G `  B )
)  e.  RR )
6 1red 9684 . 2  |-  ( ph  ->  1  e.  RR )
7 simpr 467 . . 3  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  1  <  ( abs `  ( G `
 B ) ) )
81adantr 471 . . . . . . 7  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  G : RR
--> RR )
92adantr 471 . . . . . . 7  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  B  e.  RR )
108, 9ffvelrnd 6046 . . . . . 6  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( G `  B )  e.  RR )
1110recnd 9695 . . . . 5  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( G `  B )  e.  CC )
1211abscld 13547 . . . 4  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( abs `  ( G `  B
) )  e.  RR )
136adantr 471 . . . 4  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  1  e.  RR )
14 ax-resscn 9622 . . . . . . . . 9  |-  RR  C_  CC
15 imaco 5359 . . . . . . . . . . . 12  |-  ( ( abs  o.  F )
" RR )  =  ( abs " ( F " RR ) )
1615eqcomi 2471 . . . . . . . . . . 11  |-  ( abs " ( F " RR ) )  =  ( ( abs  o.  F
) " RR )
17 imassrn 5198 . . . . . . . . . . . . 13  |-  ( ( abs  o.  F )
" RR )  C_  ran  ( abs  o.  F
)
1817a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( ( abs  o.  F ) " RR )  C_  ran  ( abs  o.  F ) )
19 imo72b2.1 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : RR --> RR )
2019adantr 471 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  F : RR
--> RR )
21 absf 13449 . . . . . . . . . . . . . . . 16  |-  abs : CC
--> RR
2221a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  abs : CC --> RR )
2314a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  RR  C_  CC )
2422, 23fssresd 5773 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( abs  |`  RR ) : RR --> RR )
2520, 24fco2d 36646 . . . . . . . . . . . . 13  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( abs  o.  F ) : RR --> RR )
26 frn 5758 . . . . . . . . . . . . 13  |-  ( ( abs  o.  F ) : RR --> RR  ->  ran  ( abs  o.  F
)  C_  RR )
2725, 26syl 17 . . . . . . . . . . . 12  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ran  ( abs 
o.  F )  C_  RR )
2818, 27sstrd 3454 . . . . . . . . . . 11  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( ( abs  o.  F ) " RR )  C_  RR )
2916, 28syl5eqss 3488 . . . . . . . . . 10  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( abs " ( F " RR ) )  C_  RR )
30 0re 9669 . . . . . . . . . . . . . . . 16  |-  0  e.  RR
3130ne0ii 3750 . . . . . . . . . . . . . . 15  |-  RR  =/=  (/)
3231a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  RR  =/=  (/) )
3332, 25wnefimgd 36645 . . . . . . . . . . . . 13  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( ( abs  o.  F ) " RR )  =/=  (/) )
3433necomd 2691 . . . . . . . . . . . 12  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  (/)  =/=  (
( abs  o.  F
) " RR ) )
3516a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( abs " ( F " RR ) )  =  ( ( abs  o.  F
) " RR ) )
3634, 35neeqtrrd 2710 . . . . . . . . . . 11  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  (/)  =/=  ( abs " ( F " RR ) ) )
3736necomd 2691 . . . . . . . . . 10  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( abs " ( F " RR ) )  =/=  (/) )
38 simpr 467 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  c  =  1 )  -> 
c  =  1 )
3938breq2d 4428 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  c  =  1 )  -> 
( t  <_  c  <->  t  <_  1 ) )
4039ralbidv 2839 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  c  =  1 )  -> 
( A. t  e.  ( abs " ( F " RR ) ) t  <_  c  <->  A. t  e.  ( abs " ( F " RR ) ) t  <_  1 ) )
41 imo72b2.6 . . . . . . . . . . . . 13  |-  ( ph  ->  A. y  e.  RR  ( abs `  ( F `
 y ) )  <_  1 )
4219, 41extoimad 36652 . . . . . . . . . . . 12  |-  ( ph  ->  A. t  e.  ( abs " ( F
" RR ) ) t  <_  1 )
4342adantr 471 . . . . . . . . . . 11  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  A. t  e.  ( abs " ( F " RR ) ) t  <_  1 )
4413, 40, 43rspcedvd 3167 . . . . . . . . . 10  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  E. c  e.  RR  A. t  e.  ( abs " ( F " RR ) ) t  <_  c )
4529, 37, 44suprcld 36648 . . . . . . . . 9  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  sup (
( abs " ( F " RR ) ) ,  RR ,  <  )  e.  RR )
4614, 45sseldi 3442 . . . . . . . 8  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  sup (
( abs " ( F " RR ) ) ,  RR ,  <  )  e.  CC )
4714, 12sseldi 3442 . . . . . . . 8  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( abs `  ( G `  B
) )  e.  CC )
4846, 47mulcomd 9690 . . . . . . 7  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( sup ( ( abs " ( F " RR ) ) ,  RR ,  <  )  x.  ( abs `  ( G `  B )
) )  =  ( ( abs `  ( G `  B )
)  x.  sup (
( abs " ( F " RR ) ) ,  RR ,  <  ) ) )
4930a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  0  e.  RR )
50 0lt1 10164 . . . . . . . . . . . . 13  |-  0  <  1
5150a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  0  <  1 )
5249, 13, 12, 51, 7lttrd 9822 . . . . . . . . . . 11  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  0  <  ( abs `  ( G `
 B ) ) )
5352gt0ne0d 10206 . . . . . . . . . 10  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( abs `  ( G `  B
) )  =/=  0
)
5445, 12, 53redivcld 10463 . . . . . . . . 9  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( sup ( ( abs " ( F " RR ) ) ,  RR ,  <  )  /  ( abs `  ( G `  B )
) )  e.  RR )
5520adantr 471 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  F : RR --> RR )
568adantr 471 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  G : RR --> RR )
57 simpr 467 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  u  e.  RR )
589adantr 471 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  B  e.  RR )
59 simpr 467 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  v  =  B )  ->  v  =  B )
6059oveq2d 6331 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  v  =  B )  ->  (
u  +  v )  =  ( u  +  B ) )
6160fveq2d 5892 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  v  =  B )  ->  ( F `  ( u  +  v ) )  =  ( F `  ( u  +  B
) ) )
6259oveq2d 6331 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  v  =  B )  ->  (
u  -  v )  =  ( u  -  B ) )
6362fveq2d 5892 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  v  =  B )  ->  ( F `  ( u  -  v ) )  =  ( F `  ( u  -  B
) ) )
6461, 63oveq12d 6333 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  v  =  B )  ->  (
( F `  (
u  +  v ) )  +  ( F `
 ( u  -  v ) ) )  =  ( ( F `
 ( u  +  B ) )  +  ( F `  (
u  -  B ) ) ) )
6559fveq2d 5892 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  v  =  B )  ->  ( G `  v )  =  ( G `  B ) )
6665oveq2d 6331 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  v  =  B )  ->  (
( F `  u
)  x.  ( G `
 v ) )  =  ( ( F `
 u )  x.  ( G `  B
) ) )
6766oveq2d 6331 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  v  =  B )  ->  (
2  x.  ( ( F `  u )  x.  ( G `  v ) ) )  =  ( 2  x.  ( ( F `  u )  x.  ( G `  B )
) ) )
6864, 67eqeq12d 2477 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  v  =  B )  ->  (
( ( F `  ( u  +  v
) )  +  ( F `  ( u  -  v ) ) )  =  ( 2  x.  ( ( F `
 u )  x.  ( G `  v
) ) )  <->  ( ( F `  ( u  +  B ) )  +  ( F `  (
u  -  B ) ) )  =  ( 2  x.  ( ( F `  u )  x.  ( G `  B ) ) ) ) )
6968ralbidv 2839 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  v  =  B )  ->  ( A. u  e.  RR  ( ( F `  ( u  +  v
) )  +  ( F `  ( u  -  v ) ) )  =  ( 2  x.  ( ( F `
 u )  x.  ( G `  v
) ) )  <->  A. u  e.  RR  ( ( F `
 ( u  +  B ) )  +  ( F `  (
u  -  B ) ) )  =  ( 2  x.  ( ( F `  u )  x.  ( G `  B ) ) ) ) )
70 imo72b2.5 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. u  e.  RR  A. v  e.  RR  (
( F `  (
u  +  v ) )  +  ( F `
 ( u  -  v ) ) )  =  ( 2  x.  ( ( F `  u )  x.  ( G `  v )
) ) )
71 ralcom2 2967 . . . . . . . . . . . . . . . . . 18  |-  ( A. u  e.  RR  A. v  e.  RR  ( ( F `
 ( u  +  v ) )  +  ( F `  (
u  -  v ) ) )  =  ( 2  x.  ( ( F `  u )  x.  ( G `  v ) ) )  ->  A. v  e.  RR  A. u  e.  RR  (
( F `  (
u  +  v ) )  +  ( F `
 ( u  -  v ) ) )  =  ( 2  x.  ( ( F `  u )  x.  ( G `  v )
) ) )
7271a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( A. u  e.  RR  A. v  e.  RR  ( ( F `
 ( u  +  v ) )  +  ( F `  (
u  -  v ) ) )  =  ( 2  x.  ( ( F `  u )  x.  ( G `  v ) ) )  ->  A. v  e.  RR  A. u  e.  RR  (
( F `  (
u  +  v ) )  +  ( F `
 ( u  -  v ) ) )  =  ( 2  x.  ( ( F `  u )  x.  ( G `  v )
) ) ) )
7372imp 435 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  A. u  e.  RR  A. v  e.  RR  ( ( F `
 ( u  +  v ) )  +  ( F `  (
u  -  v ) ) )  =  ( 2  x.  ( ( F `  u )  x.  ( G `  v ) ) ) )  ->  A. v  e.  RR  A. u  e.  RR  ( ( F `
 ( u  +  v ) )  +  ( F `  (
u  -  v ) ) )  =  ( 2  x.  ( ( F `  u )  x.  ( G `  v ) ) ) )
7470, 73mpdan 679 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. v  e.  RR  A. u  e.  RR  (
( F `  (
u  +  v ) )  +  ( F `
 ( u  -  v ) ) )  =  ( 2  x.  ( ( F `  u )  x.  ( G `  v )
) ) )
7569, 2, 74rspcdvinvd 36662 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. u  e.  RR  ( ( F `  ( u  +  B
) )  +  ( F `  ( u  -  B ) ) )  =  ( 2  x.  ( ( F `
 u )  x.  ( G `  B
) ) ) )
7675r19.21bi 2769 . . . . . . . . . . . . 13  |-  ( (
ph  /\  u  e.  RR )  ->  ( ( F `  ( u  +  B ) )  +  ( F `  ( u  -  B
) ) )  =  ( 2  x.  (
( F `  u
)  x.  ( G `
 B ) ) ) )
7776adantlr 726 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  (
( F `  (
u  +  B ) )  +  ( F `
 ( u  -  B ) ) )  =  ( 2  x.  ( ( F `  u )  x.  ( G `  B )
) ) )
7841ad2antrr 737 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  A. y  e.  RR  ( abs `  ( F `  y )
)  <_  1 )
7955, 56, 57, 58, 77, 78imo72b2lem0 36653 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  (
( abs `  ( F `  u )
)  x.  ( abs `  ( G `  B
) ) )  <_  sup ( ( abs " ( F " RR ) ) ,  RR ,  <  ) )
80 0xr 9713 . . . . . . . . . . . . 13  |-  0  e.  RR*
8180a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  0  e.  RR* )
82 1re 9668 . . . . . . . . . . . . . 14  |-  1  e.  RR
8382rexri 9719 . . . . . . . . . . . . 13  |-  1  e.  RR*
8483a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  1  e.  RR* )
8512adantr 471 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  ( abs `  ( G `  B ) )  e.  RR )
8685rexrd 9716 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  ( abs `  ( G `  B ) )  e. 
RR* )
8750a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  0  <  1 )
88 simplr 767 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  1  <  ( abs `  ( G `  B )
) )
8981, 84, 86, 87, 88xrlttrd 11485 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  0  <  ( abs `  ( G `  B )
) )
9020ffvelrnda 6045 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  ( F `  u )  e.  RR )
9190recnd 9695 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  ( F `  u )  e.  CC )
9291abscld 13547 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  ( abs `  ( F `  u ) )  e.  RR )
9345adantr 471 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  sup ( ( abs " ( F " RR ) ) ,  RR ,  <  )  e.  RR )
9479, 89, 85, 92, 93lemuldiv3d 36660 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  ( abs `  ( G `  B )
) )  /\  u  e.  RR )  ->  ( abs `  ( F `  u ) )  <_ 
( sup ( ( abs " ( F
" RR ) ) ,  RR ,  <  )  /  ( abs `  ( G `  B )
) ) )
9594ralrimiva 2814 . . . . . . . . 9  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  A. u  e.  RR  ( abs `  ( F `  u )
)  <_  ( sup ( ( abs " ( F " RR ) ) ,  RR ,  <  )  /  ( abs `  ( G `  B )
) ) )
9620, 54, 95imo72b2lem2 36655 . . . . . . . 8  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  sup (
( abs " ( F " RR ) ) ,  RR ,  <  )  <_  ( sup (
( abs " ( F " RR ) ) ,  RR ,  <  )  /  ( abs `  ( G `  B )
) ) )
9796, 52, 12, 45, 45lemuldiv4d 36661 . . . . . . 7  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( sup ( ( abs " ( F " RR ) ) ,  RR ,  <  )  x.  ( abs `  ( G `  B )
) )  <_  sup ( ( abs " ( F " RR ) ) ,  RR ,  <  ) )
9848, 97eqbrtrrd 4439 . . . . . 6  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( ( abs `  ( G `  B ) )  x. 
sup ( ( abs " ( F " RR ) ) ,  RR ,  <  ) )  <_  sup ( ( abs " ( F " RR ) ) ,  RR ,  <  ) )
99 imo72b2.7 . . . . . . . 8  |-  ( ph  ->  E. x  e.  RR  ( F `  x )  =/=  0 )
10099adantr 471 . . . . . . 7  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  E. x  e.  RR  ( F `  x )  =/=  0
)
10141adantr 471 . . . . . . 7  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  A. y  e.  RR  ( abs `  ( F `  y )
)  <_  1 )
10220, 100, 101imo72b2lem1 36659 . . . . . 6  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  0  <  sup ( ( abs " ( F " RR ) ) ,  RR ,  <  ) )
10398, 102, 45, 12, 45lemuldiv3d 36660 . . . . 5  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( abs `  ( G `  B
) )  <_  ( sup ( ( abs " ( F " RR ) ) ,  RR ,  <  )  /  sup ( ( abs " ( F
" RR ) ) ,  RR ,  <  ) ) )
10423, 45sseldd 3445 . . . . . . 7  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  sup (
( abs " ( F " RR ) ) ,  RR ,  <  )  e.  CC )
105102gt0ne0d 10206 . . . . . . 7  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  sup (
( abs " ( F " RR ) ) ,  RR ,  <  )  =/=  0 )
106104, 105dividd 10409 . . . . . 6  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( sup ( ( abs " ( F " RR ) ) ,  RR ,  <  )  /  sup ( ( abs " ( F
" RR ) ) ,  RR ,  <  ) )  =  1 )
107106eqcomd 2468 . . . . 5  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  1  =  ( sup ( ( abs " ( F " RR ) ) ,  RR ,  <  )  /  sup ( ( abs " ( F " RR ) ) ,  RR ,  <  ) ) )
108103, 107breqtrrd 4443 . . . 4  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  ( abs `  ( G `  B
) )  <_  1
)
10912, 13, 108lensymd 9812 . . 3  |-  ( (
ph  /\  1  <  ( abs `  ( G `
 B ) ) )  ->  -.  1  <  ( abs `  ( G `  B )
) )
1107, 109pm2.65da 584 . 2  |-  ( ph  ->  -.  1  <  ( abs `  ( G `  B ) ) )
1115, 6, 110nltled 9811 1  |-  ( ph  ->  ( abs `  ( G `  B )
)  <_  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898    =/= wne 2633   A.wral 2749   E.wrex 2750    C_ wss 3416   (/)c0 3743   class class class wbr 4416   ran crn 4854   "cima 4856    o. ccom 4857   -->wf 5597   ` cfv 5601  (class class class)co 6315   supcsup 7980   CCcc 9563   RRcr 9564   0cc0 9565   1c1 9566    + caddc 9568    x. cmul 9570   RR*cxr 9700    < clt 9701    <_ cle 9702    - cmin 9886    / cdiv 10297   2c2 10687   abscabs 13346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-sup 7982  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-3 10697  df-n0 10899  df-z 10967  df-uz 11189  df-rp 11332  df-seq 12246  df-exp 12305  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348
This theorem is referenced by: (None)
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