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Theorem ofmul12 29524
Description: Function analog of mul12 9531. (Contributed by Steve Rodriguez, 13-Nov-2015.)
Assertion
Ref Expression
ofmul12  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  -> 
( F  oF  x.  ( G  oF  x.  H )
)  =  ( G  oF  x.  ( F  oF  x.  H
) ) )

Proof of Theorem ofmul12
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpll 748 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  A  e.  V )
2 simplr 749 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  F : A --> CC )
3 ffn 5556 . . 3  |-  ( F : A --> CC  ->  F  Fn  A )
42, 3syl 16 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  F  Fn  A )
5 simprl 750 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  G : A --> CC )
6 ffn 5556 . . . 4  |-  ( G : A --> CC  ->  G  Fn  A )
75, 6syl 16 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  G  Fn  A )
8 simprr 751 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  H : A --> CC )
9 ffn 5556 . . . 4  |-  ( H : A --> CC  ->  H  Fn  A )
108, 9syl 16 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  H  Fn  A )
11 inidm 3556 . . 3  |-  ( A  i^i  A )  =  A
127, 10, 1, 1, 11offn 6330 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  -> 
( G  oF  x.  H )  Fn  A )
134, 10, 1, 1, 11offn 6330 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  -> 
( F  oF  x.  H )  Fn  A )
147, 13, 1, 1, 11offn 6330 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  -> 
( G  oF  x.  ( F  oF  x.  H )
)  Fn  A )
15 eqidd 2442 . 2  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
16 eqidd 2442 . . 3  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( G `  x )  =  ( G `  x ) )
17 eqidd 2442 . . 3  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( H `  x )  =  ( H `  x ) )
187, 10, 1, 1, 11, 16, 17ofval 6328 . 2  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  (
( G  oF  x.  H ) `  x )  =  ( ( G `  x
)  x.  ( H `
 x ) ) )
192ffvelrnda 5840 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( F `  x )  e.  CC )
205ffvelrnda 5840 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( G `  x )  e.  CC )
218ffvelrnda 5840 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( H `  x )  e.  CC )
2219, 20, 21mul12d 9574 . . 3  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  (
( F `  x
)  x.  ( ( G `  x )  x.  ( H `  x ) ) )  =  ( ( G `
 x )  x.  ( ( F `  x )  x.  ( H `  x )
) ) )
234, 10, 1, 1, 11, 15, 17ofval 6328 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  (
( F  oF  x.  H ) `  x )  =  ( ( F `  x
)  x.  ( H `
 x ) ) )
247, 13, 1, 1, 11, 16, 23ofval 6328 . . 3  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  (
( G  oF  x.  ( F  oF  x.  H )
) `  x )  =  ( ( G `
 x )  x.  ( ( F `  x )  x.  ( H `  x )
) ) )
2522, 24eqtr4d 2476 . 2  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  (
( F `  x
)  x.  ( ( G `  x )  x.  ( H `  x ) ) )  =  ( ( G  oF  x.  ( F  oF  x.  H
) ) `  x
) )
261, 4, 12, 14, 15, 18, 25offveq 6340 1  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  -> 
( F  oF  x.  ( G  oF  x.  H )
)  =  ( G  oF  x.  ( F  oF  x.  H
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317   CCcc 9276    x. cmul 9283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-mulcom 9342  ax-mulass 9344
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319
This theorem is referenced by:  expgrowth  29534
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