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Theorem ofmul12 26690
Description: Function analog of mul12 9023. (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 730 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  A  e.  V )
2 simplr 731 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  F : A --> CC )
3 ffn 5427 . . 3  |-  ( F : A --> CC  ->  F  Fn  A )
42, 3syl 15 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  F  Fn  A )
5 simprl 732 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  G : A --> CC )
6 ffn 5427 . . . 4  |-  ( G : A --> CC  ->  G  Fn  A )
75, 6syl 15 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  G  Fn  A )
8 simprr 733 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  H : A --> CC )
9 ffn 5427 . . . 4  |-  ( H : A --> CC  ->  H  Fn  A )
108, 9syl 15 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> CC  /\  H : A
--> CC ) )  ->  H  Fn  A )
11 inidm 3412 . . 3  |-  ( A  i^i  A )  =  A
127, 10, 1, 1, 11offn 6131 . 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 6131 . . 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 6131 . 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 2317 . 2  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
16 eqidd 2317 . . 3  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( G `  x )  =  ( G `  x ) )
17 eqidd 2317 . . 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 6129 . 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 ) ) )
19 ffvelrn 5701 . . . . 5  |-  ( ( F : A --> CC  /\  x  e.  A )  ->  ( F `  x
)  e.  CC )
202, 19sylan 457 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( F `  x )  e.  CC )
21 ffvelrn 5701 . . . . 5  |-  ( ( G : A --> CC  /\  x  e.  A )  ->  ( G `  x
)  e.  CC )
225, 21sylan 457 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( G `  x )  e.  CC )
23 ffvelrn 5701 . . . . 5  |-  ( ( H : A --> CC  /\  x  e.  A )  ->  ( H `  x
)  e.  CC )
248, 23sylan 457 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> CC  /\  H : A --> CC ) )  /\  x  e.  A )  ->  ( H `  x )  e.  CC )
2520, 22, 24mul12d 9066 . . 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 )
) ) )
264, 10, 1, 1, 11, 15, 17ofval 6129 . . . 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 ) ) )
277, 13, 1, 1, 11, 16, 26ofval 6129 . . 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 )
) ) )
2825, 27eqtr4d 2351 . 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
) )
291, 4, 12, 14, 15, 18, 28offveq 6140 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 358    = wceq 1633    e. wcel 1701    Fn wfn 5287   -->wf 5288   ` cfv 5292  (class class class)co 5900    o Fcof 6118   CCcc 8780    x. cmul 8787
This theorem is referenced by:  expgrowth  26700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-mulcom 8846  ax-mulass 8848
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-of 6120
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