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Theorem ofmul12 27519
Description: Function analog of mul12 9232. (Contributed by Steve Rodriguez, 13-Nov-2015.)
Assertion
Ref Expression
ofmul12  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  -> 
( F  o F  x.  ( G  o F  x.  H )
)  =  ( G  o F  x.  ( F  o F  x.  H
) ) )

Proof of Theorem ofmul12
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpll 731 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  A  e.  V )
2 simplr 732 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  F : A --> CC )
3 ffn 5591 . . 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 733 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  G : A --> CC )
6 ffn 5591 . . . 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 734 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  H : A --> CC )
9 ffn 5591 . . . 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 3550 . . 3  |-  ( A  i^i  A )  =  A
127, 10, 1, 1, 11offn 6316 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  -> 
( G  o F  x.  H )  Fn  A )
134, 10, 1, 1, 11offn 6316 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  -> 
( F  o F  x.  H )  Fn  A )
147, 13, 1, 1, 11offn 6316 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  -> 
( G  o F  x.  ( F  o F  x.  H )
)  Fn  A )
15 eqidd 2437 . 2  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
16 eqidd 2437 . . 3  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( G `  x )  =  ( G `  x ) )
17 eqidd 2437 . . 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 6314 . 2  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  (
( G  o F  x.  H ) `  x )  =  ( ( G `  x
)  x.  ( H `
 x ) ) )
192ffvelrnda 5870 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( F `  x )  e.  CC )
205ffvelrnda 5870 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( G `  x )  e.  CC )
218ffvelrnda 5870 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( H `  x )  e.  CC )
2219, 20, 21mul12d 9275 . . 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 6314 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  (
( F  o F  x.  H ) `  x )  =  ( ( F `  x
)  x.  ( H `
 x ) ) )
247, 13, 1, 1, 11, 16, 23ofval 6314 . . 3  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  (
( G  o F  x.  ( F  o F  x.  H )
) `  x )  =  ( ( G `
 x )  x.  ( ( F `  x )  x.  ( H `  x )
) ) )
2522, 24eqtr4d 2471 . 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  o F  x.  ( F  o F  x.  H
) ) `  x
) )
261, 4, 12, 14, 15, 18, 25offveq 6325 1  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  -> 
( F  o F  x.  ( G  o F  x.  H )
)  =  ( G  o F  x.  ( F  o F  x.  H
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303   CCcc 8988    x. cmul 8995
This theorem is referenced by:  expgrowth  27529
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-mulcom 9054  ax-mulass 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305
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