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Theorem ofmul12 38524
Description: Function analogue of mul12 10202. (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 790 . 2 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> CC /\ H : A --> CC ) ) -> A e. V )
2 simplr 792 . . 3 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> CC /\ H : A --> CC ) ) -> F : A --> CC )
3 ffn 6045 . . 3 |- ( F : A --> CC -> F Fn A )
42, 3syl 17 . 2 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> CC /\ H : A --> CC ) ) -> F Fn A )
5 simprl 794 . . . 4 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> CC /\ H : A --> CC ) ) -> G : A --> CC )
6 ffn 6045 . . . 4 |- ( G : A --> CC -> G Fn A )
75, 6syl 17 . . 3 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> CC /\ H : A --> CC ) ) -> G Fn A )
8 simprr 796 . . . 4 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> CC /\ H : A --> CC ) ) -> H : A --> CC )
9 ffn 6045 . . . 4 |- ( H : A --> CC -> H Fn A )
108, 9syl 17 . . 3 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> CC /\ H : A --> CC ) ) -> H Fn A )
11 inidm 3822 . . 3 |- ( A i^i A ) = A
127, 10, 1, 1, 11offn 6908 . 2 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> CC /\ H : A --> CC ) ) -> ( G oF x. H ) Fn A )
134, 10, 1, 1, 11offn 6908 . . 3 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> CC /\ H : A --> CC ) ) -> ( F oF x. H ) Fn A )
147, 13, 1, 1, 11offn 6908 . 2 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> CC /\ H : A --> CC ) ) -> ( G oF x. ( F oF x. H ) ) Fn A )
15 eqidd 2623 . 2 |- ( ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> CC /\ H : A --> CC ) ) /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
16 eqidd 2623 . . 3 |- ( ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> CC /\ H : A --> CC ) ) /\ x e. A ) -> ( G ` x ) = ( G ` x ) )
17 eqidd 2623 . . 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 6906 . 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 6359 . . . 4 |- ( ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> CC /\ H : A --> CC ) ) /\ x e. A ) -> ( F ` x ) e. CC )
205ffvelrnda 6359 . . . 4 |- ( ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> CC /\ H : A --> CC ) ) /\ x e. A ) -> ( G ` x ) e. CC )
218ffvelrnda 6359 . . . 4 |- ( ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> CC /\ H : A --> CC ) ) /\ x e. A ) -> ( H ` x ) e. CC )
2219, 20, 21mul12d 10245 . . 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 6906 . . . 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 6906 . . 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 2659 . 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 6918 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 384   = wceq 1483   e. wcel 1990   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   x. cmul 9941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-mulcom 10000  ax-mulass 10002
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897
This theorem is referenced by:  expgrowth  38534
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