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Theorem ofmul12 31160
Description: Function analog of mul12 9757. (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 753 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  A  e.  V )
2 simplr 754 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  F : A --> CC )
3 ffn 5737 . . 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 755 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  G : A --> CC )
6 ffn 5737 . . . 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 756 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  H : A --> CC )
9 ffn 5737 . . . 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 3712 . . 3  |-  ( A  i^i  A )  =  A
127, 10, 1, 1, 11offn 6546 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  -> 
( G  oF  x.  H )  Fn  A )
134, 10, 1, 1, 11offn 6546 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  -> 
( F  oF  x.  H )  Fn  A )
147, 13, 1, 1, 11offn 6546 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  -> 
( G  oF  x.  ( F  oF  x.  H )
)  Fn  A )
15 eqidd 2468 . 2  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
16 eqidd 2468 . . 3  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( G `  x )  =  ( G `  x ) )
17 eqidd 2468 . . 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 6544 . 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 6032 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( F `  x )  e.  CC )
205ffvelrnda 6032 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( G `  x )  e.  CC )
218ffvelrnda 6032 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( H `  x )  e.  CC )
2219, 20, 21mul12d 9800 . . 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 6544 . . . 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 6544 . . 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 2511 . 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 6556 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 1379    e. wcel 1767    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    oFcof 6533   CCcc 9502    x. cmul 9509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-mulcom 9568  ax-mulass 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535
This theorem is referenced by:  expgrowth  31170
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