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Theorem imo72b2lem0 38174
Description: Lemma for imo72b2 38185. (Contributed by Stanislas Polu, 9-Mar-2020.)
Hypotheses
Ref Expression
imo72b2lem0.1  |-  ( ph  ->  F : RR --> RR )
imo72b2lem0.2  |-  ( ph  ->  G : RR --> RR )
imo72b2lem0.3  |-  ( ph  ->  A  e.  RR )
imo72b2lem0.4  |-  ( ph  ->  B  e.  RR )
imo72b2lem0.5  |-  ( ph  ->  ( ( F `  ( A  +  B
) )  +  ( F `  ( A  -  B ) ) )  =  ( 2  x.  ( ( F `
 A )  x.  ( G `  B
) ) ) )
imo72b2lem0.6  |-  ( ph  ->  A. y  e.  RR  ( abs `  ( F `
 y ) )  <_  1 )
Assertion
Ref Expression
imo72b2lem0  |-  ( ph  ->  ( ( abs `  ( F `  A )
)  x.  ( abs `  ( G `  B
) ) )  <_  sup ( ( abs " ( F " RR ) ) ,  RR ,  <  ) )
Distinct variable groups:    y, F    ph, y
Allowed substitution hints:    A( y)    B( y)    G( y)

Proof of Theorem imo72b2lem0
Dummy variables  c  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imo72b2lem0.1 . . . . . . . 8  |-  ( ph  ->  F : RR --> RR )
2 imo72b2lem0.3 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
31, 2ffvelrnd 6033 . . . . . . 7  |-  ( ph  ->  ( F `  A
)  e.  RR )
43recnd 9639 . . . . . 6  |-  ( ph  ->  ( F `  A
)  e.  CC )
54idi 2 . . . . 5  |-  ( ph  ->  ( F `  A
)  e.  CC )
6 imo72b2lem0.2 . . . . . . . 8  |-  ( ph  ->  G : RR --> RR )
7 imo72b2lem0.4 . . . . . . . 8  |-  ( ph  ->  B  e.  RR )
86, 7ffvelrnd 6033 . . . . . . 7  |-  ( ph  ->  ( G `  B
)  e.  RR )
98recnd 9639 . . . . . 6  |-  ( ph  ->  ( G `  B
)  e.  CC )
109idi 2 . . . . 5  |-  ( ph  ->  ( G `  B
)  e.  CC )
115, 10mulcld 9633 . . . 4  |-  ( ph  ->  ( ( F `  A )  x.  ( G `  B )
)  e.  CC )
1211abscld 13279 . . 3  |-  ( ph  ->  ( abs `  (
( F `  A
)  x.  ( G `
 B ) ) )  e.  RR )
13 imaco 5518 . . . . . 6  |-  ( ( abs  o.  F )
" RR )  =  ( abs " ( F " RR ) )
1413eqcomi 2470 . . . . 5  |-  ( abs " ( F " RR ) )  =  ( ( abs  o.  F
) " RR )
15 imassrn 5358 . . . . . . 7  |-  ( ( abs  o.  F )
" RR )  C_  ran  ( abs  o.  F
)
1615a1i 11 . . . . . 6  |-  ( ph  ->  ( ( abs  o.  F ) " RR )  C_  ran  ( abs 
o.  F ) )
17 absf 13182 . . . . . . . . . 10  |-  abs : CC
--> RR
1817a1i 11 . . . . . . . . 9  |-  ( ph  ->  abs : CC --> RR )
19 ax-resscn 9566 . . . . . . . . . 10  |-  RR  C_  CC
2019a1i 11 . . . . . . . . 9  |-  ( ph  ->  RR  C_  CC )
2118, 20fssresd 5758 . . . . . . . 8  |-  ( ph  ->  ( abs  |`  RR ) : RR --> RR )
221, 21fco2d 38167 . . . . . . 7  |-  ( ph  ->  ( abs  o.  F
) : RR --> RR )
23 frn 5743 . . . . . . 7  |-  ( ( abs  o.  F ) : RR --> RR  ->  ran  ( abs  o.  F
)  C_  RR )
2422, 23syl 16 . . . . . 6  |-  ( ph  ->  ran  ( abs  o.  F )  C_  RR )
2516, 24sstrd 3509 . . . . 5  |-  ( ph  ->  ( ( abs  o.  F ) " RR )  C_  RR )
2614, 25syl5eqss 3543 . . . 4  |-  ( ph  ->  ( abs " ( F " RR ) ) 
C_  RR )
27 0re 9613 . . . . . . . . . 10  |-  0  e.  RR
2827ne0ii 3800 . . . . . . . . 9  |-  RR  =/=  (/)
2928a1i 11 . . . . . . . 8  |-  ( ph  ->  RR  =/=  (/) )
3029, 22wnefimgd 38166 . . . . . . 7  |-  ( ph  ->  ( ( abs  o.  F ) " RR )  =/=  (/) )
3130necomd 2728 . . . . . 6  |-  ( ph  -> 
(/)  =/=  ( ( abs  o.  F ) " RR ) )
3214a1i 11 . . . . . 6  |-  ( ph  ->  ( abs " ( F " RR ) )  =  ( ( abs 
o.  F ) " RR ) )
3331, 32neeqtrrd 2757 . . . . 5  |-  ( ph  -> 
(/)  =/=  ( abs " ( F " RR ) ) )
3433necomd 2728 . . . 4  |-  ( ph  ->  ( abs " ( F " RR ) )  =/=  (/) )
35 1red 9628 . . . . 5  |-  ( ph  ->  1  e.  RR )
36 simpr 461 . . . . . . 7  |-  ( (
ph  /\  c  = 
1 )  ->  c  =  1 )
3736breq2d 4468 . . . . . 6  |-  ( (
ph  /\  c  = 
1 )  ->  (
x  <_  c  <->  x  <_  1 ) )
3837ralbidv 2896 . . . . 5  |-  ( (
ph  /\  c  = 
1 )  ->  ( A. x  e.  ( abs " ( F " RR ) ) x  <_ 
c  <->  A. x  e.  ( abs " ( F
" RR ) ) x  <_  1 ) )
39 imo72b2lem0.6 . . . . . 6  |-  ( ph  ->  A. y  e.  RR  ( abs `  ( F `
 y ) )  <_  1 )
401, 39extoimad 38173 . . . . 5  |-  ( ph  ->  A. x  e.  ( abs " ( F
" RR ) ) x  <_  1 )
4135, 38, 40rspcedvd 3215 . . . 4  |-  ( ph  ->  E. c  e.  RR  A. x  e.  ( abs " ( F " RR ) ) x  <_ 
c )
4226, 34, 41suprcld 38169 . . 3  |-  ( ph  ->  sup ( ( abs " ( F " RR ) ) ,  RR ,  <  )  e.  RR )
43 2re 10626 . . . 4  |-  2  e.  RR
4443a1i 11 . . 3  |-  ( ph  ->  2  e.  RR )
45 imo72b2lem0.5 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  ( A  +  B
) )  +  ( F `  ( A  -  B ) ) )  =  ( 2  x.  ( ( F `
 A )  x.  ( G `  B
) ) ) )
4645idi 2 . . . . . . . 8  |-  ( ph  ->  ( ( F `  ( A  +  B
) )  +  ( F `  ( A  -  B ) ) )  =  ( 2  x.  ( ( F `
 A )  x.  ( G `  B
) ) ) )
4746fveq2d 5876 . . . . . . 7  |-  ( ph  ->  ( abs `  (
( F `  ( A  +  B )
)  +  ( F `
 ( A  -  B ) ) ) )  =  ( abs `  ( 2  x.  (
( F `  A
)  x.  ( G `
 B ) ) ) ) )
48 2cnd 10629 . . . . . . . . 9  |-  ( ph  ->  2  e.  CC )
4948, 11mulcld 9633 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  (
( F `  A
)  x.  ( G `
 B ) ) )  e.  CC )
5049abscld 13279 . . . . . . 7  |-  ( ph  ->  ( abs `  (
2  x.  ( ( F `  A )  x.  ( G `  B ) ) ) )  e.  RR )
5147, 50eqeltrd 2545 . . . . . 6  |-  ( ph  ->  ( abs `  (
( F `  ( A  +  B )
)  +  ( F `
 ( A  -  B ) ) ) )  e.  RR )
521idi 2 . . . . . . . . . 10  |-  ( ph  ->  F : RR --> RR )
532idi 2 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR )
547idi 2 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR )
5553, 54readdcld 9640 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  B
)  e.  RR )
5652, 55ffvelrnd 6033 . . . . . . . . 9  |-  ( ph  ->  ( F `  ( A  +  B )
)  e.  RR )
5756recnd 9639 . . . . . . . 8  |-  ( ph  ->  ( F `  ( A  +  B )
)  e.  CC )
5857abscld 13279 . . . . . . 7  |-  ( ph  ->  ( abs `  ( F `  ( A  +  B ) ) )  e.  RR )
5953, 54resubcld 10008 . . . . . . . . . 10  |-  ( ph  ->  ( A  -  B
)  e.  RR )
6052, 59ffvelrnd 6033 . . . . . . . . 9  |-  ( ph  ->  ( F `  ( A  -  B )
)  e.  RR )
6160recnd 9639 . . . . . . . 8  |-  ( ph  ->  ( F `  ( A  -  B )
)  e.  CC )
6261abscld 13279 . . . . . . 7  |-  ( ph  ->  ( abs `  ( F `  ( A  -  B ) ) )  e.  RR )
6358, 62readdcld 9640 . . . . . 6  |-  ( ph  ->  ( ( abs `  ( F `  ( A  +  B ) ) )  +  ( abs `  ( F `  ( A  -  B ) ) ) )  e.  RR )
6444, 42remulcld 9641 . . . . . 6  |-  ( ph  ->  ( 2  x.  sup ( ( abs " ( F " RR ) ) ,  RR ,  <  ) )  e.  RR )
6557, 61abstrid 13299 . . . . . 6  |-  ( ph  ->  ( abs `  (
( F `  ( A  +  B )
)  +  ( F `
 ( A  -  B ) ) ) )  <_  ( ( abs `  ( F `  ( A  +  B
) ) )  +  ( abs `  ( F `  ( A  -  B ) ) ) ) )
661, 55fvco3d 38170 . . . . . . . . . 10  |-  ( ph  ->  ( ( abs  o.  F ) `  ( A  +  B )
)  =  ( abs `  ( F `  ( A  +  B )
) ) )
6755, 22wfximgfd 38171 . . . . . . . . . . 11  |-  ( ph  ->  ( ( abs  o.  F ) `  ( A  +  B )
)  e.  ( ( abs  o.  F )
" RR ) )
6832idi 2 . . . . . . . . . . 11  |-  ( ph  ->  ( abs " ( F " RR ) )  =  ( ( abs 
o.  F ) " RR ) )
6967, 68eleqtrrd 2548 . . . . . . . . . 10  |-  ( ph  ->  ( ( abs  o.  F ) `  ( A  +  B )
)  e.  ( abs " ( F " RR ) ) )
7066, 69eqeltrrd 2546 . . . . . . . . 9  |-  ( ph  ->  ( abs `  ( F `  ( A  +  B ) ) )  e.  ( abs " ( F " RR ) ) )
7126, 34, 41, 70suprubd 38168 . . . . . . . 8  |-  ( ph  ->  ( abs `  ( F `  ( A  +  B ) ) )  <_  sup ( ( abs " ( F " RR ) ) ,  RR ,  <  ) )
721, 59fvco3d 38170 . . . . . . . . . 10  |-  ( ph  ->  ( ( abs  o.  F ) `  ( A  -  B )
)  =  ( abs `  ( F `  ( A  -  B )
) ) )
7359, 22wfximgfd 38171 . . . . . . . . . . 11  |-  ( ph  ->  ( ( abs  o.  F ) `  ( A  -  B )
)  e.  ( ( abs  o.  F )
" RR ) )
7473, 32eleqtrrd 2548 . . . . . . . . . 10  |-  ( ph  ->  ( ( abs  o.  F ) `  ( A  -  B )
)  e.  ( abs " ( F " RR ) ) )
7572, 74eqeltrrd 2546 . . . . . . . . 9  |-  ( ph  ->  ( abs `  ( F `  ( A  -  B ) ) )  e.  ( abs " ( F " RR ) ) )
7626, 34, 41, 75suprubd 38168 . . . . . . . 8  |-  ( ph  ->  ( abs `  ( F `  ( A  -  B ) ) )  <_  sup ( ( abs " ( F " RR ) ) ,  RR ,  <  ) )
7758, 62, 42, 42, 71, 76le2addd 10191 . . . . . . 7  |-  ( ph  ->  ( ( abs `  ( F `  ( A  +  B ) ) )  +  ( abs `  ( F `  ( A  -  B ) ) ) )  <_  ( sup ( ( abs " ( F " RR ) ) ,  RR ,  <  )  +  sup ( ( abs " ( F
" RR ) ) ,  RR ,  <  ) ) )
7842recnd 9639 . . . . . . . . 9  |-  ( ph  ->  sup ( ( abs " ( F " RR ) ) ,  RR ,  <  )  e.  CC )
79782timesd 10802 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  sup ( ( abs " ( F " RR ) ) ,  RR ,  <  ) )  =  ( sup ( ( abs " ( F " RR ) ) ,  RR ,  <  )  +  sup ( ( abs " ( F
" RR ) ) ,  RR ,  <  ) ) )
8079eqcomd 2465 . . . . . . 7  |-  ( ph  ->  ( sup ( ( abs " ( F
" RR ) ) ,  RR ,  <  )  +  sup ( ( abs " ( F
" RR ) ) ,  RR ,  <  ) )  =  ( 2  x.  sup ( ( abs " ( F
" RR ) ) ,  RR ,  <  ) ) )
8180, 64eqeltrd 2545 . . . . . . 7  |-  ( ph  ->  ( sup ( ( abs " ( F
" RR ) ) ,  RR ,  <  )  +  sup ( ( abs " ( F
" RR ) ) ,  RR ,  <  ) )  e.  RR )
8277, 80, 63, 81leeq2d 38162 . . . . . 6  |-  ( ph  ->  ( ( abs `  ( F `  ( A  +  B ) ) )  +  ( abs `  ( F `  ( A  -  B ) ) ) )  <_  ( 2  x.  sup ( ( abs " ( F
" RR ) ) ,  RR ,  <  ) ) )
8351, 63, 64, 65, 82letrd 9756 . . . . 5  |-  ( ph  ->  ( abs `  (
( F `  ( A  +  B )
)  +  ( F `
 ( A  -  B ) ) ) )  <_  ( 2  x.  sup ( ( abs " ( F
" RR ) ) ,  RR ,  <  ) ) )
8483, 47, 51, 64leeq1d 38161 . . . 4  |-  ( ph  ->  ( abs `  (
2  x.  ( ( F `  A )  x.  ( G `  B ) ) ) )  <_  ( 2  x.  sup ( ( abs " ( F
" RR ) ) ,  RR ,  <  ) ) )
85 0le2 10647 . . . . . 6  |-  0  <_  2
8685a1i 11 . . . . 5  |-  ( ph  ->  0  <_  2 )
873idi 2 . . . . . 6  |-  ( ph  ->  ( F `  A
)  e.  RR )
888idi 2 . . . . . 6  |-  ( ph  ->  ( G `  B
)  e.  RR )
8987, 88remulcld 9641 . . . . 5  |-  ( ph  ->  ( ( F `  A )  x.  ( G `  B )
)  e.  RR )
9086, 44, 89absmulrposd 38163 . . . 4  |-  ( ph  ->  ( abs `  (
2  x.  ( ( F `  A )  x.  ( G `  B ) ) ) )  =  ( 2  x.  ( abs `  (
( F `  A
)  x.  ( G `
 B ) ) ) ) )
9184, 90, 50, 64leeq1d 38161 . . 3  |-  ( ph  ->  ( 2  x.  ( abs `  ( ( F `
 A )  x.  ( G `  B
) ) ) )  <_  ( 2  x. 
sup ( ( abs " ( F " RR ) ) ,  RR ,  <  ) ) )
92 2pos 10648 . . . 4  |-  0  <  2
9392a1i 11 . . 3  |-  ( ph  ->  0  <  2 )
9412, 42, 44, 91, 93wwlemuld 38160 . 2  |-  ( ph  ->  ( abs `  (
( F `  A
)  x.  ( G `
 B ) ) )  <_  sup (
( abs " ( F " RR ) ) ,  RR ,  <  ) )
954, 9absmuld 13297 . . 3  |-  ( ph  ->  ( abs `  (
( F `  A
)  x.  ( G `
 B ) ) )  =  ( ( abs `  ( F `
 A ) )  x.  ( abs `  ( G `  B )
) ) )
9695idi 2 . 2  |-  ( ph  ->  ( abs `  (
( F `  A
)  x.  ( G `
 B ) ) )  =  ( ( abs `  ( F `
 A ) )  x.  ( abs `  ( G `  B )
) ) )
9794, 96, 12, 42leeq1d 38161 1  |-  ( ph  ->  ( ( abs `  ( F `  A )
)  x.  ( abs `  ( G `  B
) ) )  <_  sup ( ( abs " ( F " RR ) ) ,  RR ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807    C_ wss 3471   (/)c0 3793   class class class wbr 4456   ran crn 5009   "cima 5011    o. ccom 5012   -->wf 5590   ` cfv 5594  (class class class)co 6296   supcsup 7918   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    < clt 9645    <_ cle 9646    - cmin 9824   2c2 10606   abscabs 13079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081
This theorem is referenced by:  imo72b2  38185
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