Theorem ofco 6559

Description: The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)

Hypotheses

Ref	Expression
ofco.1	|- ( ph -> F Fn A )
ofco.2	|- ( ph -> G Fn B )
ofco.3	|- ( ph -> H : D --> C )
ofco.4	|- ( ph -> A e. V )
ofco.5	|- ( ph -> B e. W )
ofco.6	|- ( ph -> D e. X )
ofco.7	|- ( A i^i B ) = C

Assertion

Ref	Expression
ofco	|- ( ph -> ( ( F oF R G ) o. H ) = ( ( F o. H ) oF R ( G o. H ) ) )

Proof of Theorem ofco

Dummy variables y x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofco.3	. . . 4 |- ( ph -> H : D --> C )
2	1	ffvelrnda 6032	. . 3 |- ( ( ph /\ x e. D ) -> ( H ` x ) e. C )
3	1	feqmptd 5926	. . 3 |- ( ph -> H = ( x e. D |-> ( H ` x ) ) )
4		ofco.1	. . . 4 |- ( ph -> F Fn A )
5		ofco.2	. . . 4 |- ( ph -> G Fn B )
6		ofco.4	. . . 4 |- ( ph -> A e. V )
7		ofco.5	. . . 4 |- ( ph -> B e. W )
8		ofco.7	. . . 4 |- ( A i^i B ) = C
9		eqidd 2458	. . . 4 |- ( ( ph /\ y e. A ) -> ( F ` y ) = ( F ` y ) )
10		eqidd 2458	. . . 4 |- ( ( ph /\ y e. B ) -> ( G ` y ) = ( G ` y ) )
11	4, 5, 6, 7, 8, 9, 10	offval 6546	. . 3 |- ( ph -> ( F oF R G ) = ( y e. C |-> ( ( F ` y ) R ( G ` y ) ) ) )
12		fveq2 5872	. . . 4 |- ( y = ( H ` x ) -> ( F ` y ) = ( F ` ( H ` x ) ) )
13		fveq2 5872	. . . 4 |- ( y = ( H ` x ) -> ( G ` y ) = ( G ` ( H ` x ) ) )
14	12, 13	oveq12d 6314	. . 3 |- ( y = ( H ` x ) -> ( ( F ` y ) R ( G ` y ) ) = ( ( F ` ( H ` x ) ) R ( G ` ( H ` x ) ) ) )
15	2, 3, 11, 14	fmptco 6065	. 2 |- ( ph -> ( ( F oF R G ) o. H ) = ( x e. D |-> ( ( F ` ( H ` x ) ) R ( G ` ( H ` x ) ) ) ) )
16		inss1 3714	. . . . . 6 |- ( A i^i B ) C_ A
17	8, 16	eqsstr3i 3530	. . . . 5 |- C C_ A
18		fss 5745	. . . . 5 |- ( ( H : D --> C /\ C C_ A ) -> H : D --> A )
19	1, 17, 18	sylancl 662	. . . 4 |- ( ph -> H : D --> A )
20		fnfco 5756	. . . 4 |- ( ( F Fn A /\ H : D --> A ) -> ( F o. H ) Fn D )
21	4, 19, 20	syl2anc 661	. . 3 |- ( ph -> ( F o. H ) Fn D )
22		inss2 3715	. . . . . 6 |- ( A i^i B ) C_ B
23	8, 22	eqsstr3i 3530	. . . . 5 |- C C_ B
24		fss 5745	. . . . 5 |- ( ( H : D --> C /\ C C_ B ) -> H : D --> B )
25	1, 23, 24	sylancl 662	. . . 4 |- ( ph -> H : D --> B )
26		fnfco 5756	. . . 4 |- ( ( G Fn B /\ H : D --> B ) -> ( G o. H ) Fn D )
27	5, 25, 26	syl2anc 661	. . 3 |- ( ph -> ( G o. H ) Fn D )
28		ofco.6	. . 3 |- ( ph -> D e. X )
29		inidm 3703	. . 3 |- ( D i^i D ) = D
30		ffn 5737	. . . . 5 |- ( H : D --> C -> H Fn D )
31	1, 30	syl 16	. . . 4 |- ( ph -> H Fn D )
32		fvco2 5948	. . . 4 |- ( ( H Fn D /\ x e. D ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) )
33	31, 32	sylan 471	. . 3 |- ( ( ph /\ x e. D ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) )
34		fvco2 5948	. . . 4 |- ( ( H Fn D /\ x e. D ) -> ( ( G o. H ) ` x ) = ( G ` ( H ` x ) ) )
35	31, 34	sylan 471	. . 3 |- ( ( ph /\ x e. D ) -> ( ( G o. H ) ` x ) = ( G ` ( H ` x ) ) )
36	21, 27, 28, 28, 29, 33, 35	offval 6546	. 2 |- ( ph -> ( ( F o. H ) oF R ( G o. H ) ) = ( x e. D |-> ( ( F ` ( H ` x ) ) R ( G ` ( H ` x ) ) ) ) )
37	15, 36	eqtr4d 2501	1 |- ( ph -> ( ( F oF R G ) o. H ) = ( ( F o. H ) oF R ( G o. H ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1395   e. wcel 1819   i^i cin 3470   C_ wss 3471   |-> cmpt 4515   o. ccom 5012   Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   oFcof 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6299  df-oprab 6300  df-mpt2 6301  df-of 6539

This theorem is referenced by:  gsumzaddlem  17060  gsumzaddlemOLD  17062  coe1add  18431  pf1ind  18517  mendring  31303