Theorem cofuass 16549

Description: Functor composition is associative. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
cofuass.g	|- ( ph -> G e. ( C Func D ) )
cofuass.h	|- ( ph -> H e. ( D Func E ) )
cofuass.k	|- ( ph -> K e. ( E Func F ) )

Assertion

Ref	Expression
cofuass	|- ( ph -> ( ( K o.func H ) o.func G ) = ( K o.func ( H o.func G ) ) )

Proof of Theorem cofuass

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

Step	Hyp	Ref	Expression
1		coass 5654	. . . 4 |- ( ( ( 1st ` K ) o. ( 1st ` H ) ) o. ( 1st ` G ) ) = ( ( 1st ` K ) o. ( ( 1st ` H ) o. ( 1st ` G ) ) )
2		eqid 2622	. . . . . 6 |- ( Base ` D ) = ( Base ` D )
3		cofuass.h	. . . . . 6 |- ( ph -> H e. ( D Func E ) )
4		cofuass.k	. . . . . 6 |- ( ph -> K e. ( E Func F ) )
5	2, 3, 4	cofu1st 16543	. . . . 5 |- ( ph -> ( 1st ` ( K o.func H ) ) = ( ( 1st ` K ) o. ( 1st ` H ) ) )
6	5	coeq1d 5283	. . . 4 |- ( ph -> ( ( 1st ` ( K o.func H ) ) o. ( 1st ` G ) ) = ( ( ( 1st ` K ) o. ( 1st ` H ) ) o. ( 1st ` G ) ) )
7		eqid 2622	. . . . . 6 |- ( Base ` C ) = ( Base ` C )
8		cofuass.g	. . . . . 6 |- ( ph -> G e. ( C Func D ) )
9	7, 8, 3	cofu1st 16543	. . . . 5 |- ( ph -> ( 1st ` ( H o.func G ) ) = ( ( 1st ` H ) o. ( 1st ` G ) ) )
10	9	coeq2d 5284	. . . 4 |- ( ph -> ( ( 1st ` K ) o. ( 1st ` ( H o.func G ) ) ) = ( ( 1st ` K ) o. ( ( 1st ` H ) o. ( 1st ` G ) ) ) )
11	1, 6, 10	3eqtr4a 2682	. . 3 |- ( ph -> ( ( 1st ` ( K o.func H ) ) o. ( 1st ` G ) ) = ( ( 1st ` K ) o. ( 1st ` ( H o.func G ) ) ) )
12		coass 5654	. . . . 5 |- ( ( ( ( ( 1st ` H ) ` ( ( 1st ` G ) ` x ) ) ( 2nd ` K ) ( ( 1st ` H ) ` ( ( 1st ` G ) ` y ) ) ) o. ( ( ( 1st ` G ) ` x ) ( 2nd ` H ) ( ( 1st ` G ) ` y ) ) ) o. ( x ( 2nd ` G ) y ) ) = ( ( ( ( 1st ` H ) ` ( ( 1st ` G ) ` x ) ) ( 2nd ` K ) ( ( 1st ` H ) ` ( ( 1st ` G ) ` y ) ) ) o. ( ( ( ( 1st ` G ) ` x ) ( 2nd ` H ) ( ( 1st ` G ) ` y ) ) o. ( x ( 2nd ` G ) y ) ) )
13	3	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> H e. ( D Func E ) )
14	4	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> K e. ( E Func F ) )
15		relfunc 16522	. . . . . . . . . . 11 |- Rel ( C Func D )
16		1st2ndbr 7217	. . . . . . . . . . 11 |- ( ( Rel ( C Func D ) /\ G e. ( C Func D ) ) -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) )
17	15, 8, 16	sylancr 695	. . . . . . . . . 10 |- ( ph -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) )
18	17	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) )
19	7, 2, 18	funcf1 16526	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( 1st ` G ) : ( Base ` C ) --> ( Base ` D ) )
20		simp2 1062	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> x e. ( Base ` C ) )
21	19, 20	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( ( 1st ` G ) ` x ) e. ( Base ` D ) )
22		simp3 1063	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> y e. ( Base ` C ) )
23	19, 22	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( ( 1st ` G ) ` y ) e. ( Base ` D ) )
24	2, 13, 14, 21, 23	cofu2nd 16545	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( ( ( 1st ` G ) ` x ) ( 2nd ` ( K o.func H ) ) ( ( 1st ` G ) ` y ) ) = ( ( ( ( 1st ` H ) ` ( ( 1st ` G ) ` x ) ) ( 2nd ` K ) ( ( 1st ` H ) ` ( ( 1st ` G ) ` y ) ) ) o. ( ( ( 1st ` G ) ` x ) ( 2nd ` H ) ( ( 1st ` G ) ` y ) ) ) )
25	24	coeq1d 5283	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( ( ( ( 1st ` G ) ` x ) ( 2nd ` ( K o.func H ) ) ( ( 1st ` G ) ` y ) ) o. ( x ( 2nd ` G ) y ) ) = ( ( ( ( ( 1st ` H ) ` ( ( 1st ` G ) ` x ) ) ( 2nd ` K ) ( ( 1st ` H ) ` ( ( 1st ` G ) ` y ) ) ) o. ( ( ( 1st ` G ) ` x ) ( 2nd ` H ) ( ( 1st ` G ) ` y ) ) ) o. ( x ( 2nd ` G ) y ) ) )
26	8	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> G e. ( C Func D ) )
27	7, 26, 13, 20	cofu1 16544	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( ( 1st ` ( H o.func G ) ) ` x ) = ( ( 1st ` H ) ` ( ( 1st ` G ) ` x ) ) )
28	7, 26, 13, 22	cofu1 16544	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( ( 1st ` ( H o.func G ) ) ` y ) = ( ( 1st ` H ) ` ( ( 1st ` G ) ` y ) ) )
29	27, 28	oveq12d 6668	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( ( ( 1st ` ( H o.func G ) ) ` x ) ( 2nd ` K ) ( ( 1st ` ( H o.func G ) ) ` y ) ) = ( ( ( 1st ` H ) ` ( ( 1st ` G ) ` x ) ) ( 2nd ` K ) ( ( 1st ` H ) ` ( ( 1st ` G ) ` y ) ) ) )
30	7, 26, 13, 20, 22	cofu2nd 16545	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( x ( 2nd ` ( H o.func G ) ) y ) = ( ( ( ( 1st ` G ) ` x ) ( 2nd ` H ) ( ( 1st ` G ) ` y ) ) o. ( x ( 2nd ` G ) y ) ) )
31	29, 30	coeq12d 5286	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( ( ( ( 1st ` ( H o.func G ) ) ` x ) ( 2nd ` K ) ( ( 1st ` ( H o.func G ) ) ` y ) ) o. ( x ( 2nd ` ( H o.func G ) ) y ) ) = ( ( ( ( 1st ` H ) ` ( ( 1st ` G ) ` x ) ) ( 2nd ` K ) ( ( 1st ` H ) ` ( ( 1st ` G ) ` y ) ) ) o. ( ( ( ( 1st ` G ) ` x ) ( 2nd ` H ) ( ( 1st ` G ) ` y ) ) o. ( x ( 2nd ` G ) y ) ) ) )
32	12, 25, 31	3eqtr4a 2682	. . . 4 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( ( ( ( 1st ` G ) ` x ) ( 2nd ` ( K o.func H ) ) ( ( 1st ` G ) ` y ) ) o. ( x ( 2nd ` G ) y ) ) = ( ( ( ( 1st ` ( H o.func G ) ) ` x ) ( 2nd ` K ) ( ( 1st ` ( H o.func G ) ) ` y ) ) o. ( x ( 2nd ` ( H o.func G ) ) y ) ) )
33	32	mpt2eq3dva 6719	. . 3 |- ( ph -> ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( ( ( 1st ` G ) ` x ) ( 2nd ` ( K o.func H ) ) ( ( 1st ` G ) ` y ) ) o. ( x ( 2nd ` G ) y ) ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( ( ( 1st ` ( H o.func G ) ) ` x ) ( 2nd ` K ) ( ( 1st ` ( H o.func G ) ) ` y ) ) o. ( x ( 2nd ` ( H o.func G ) ) y ) ) ) )
34	11, 33	opeq12d 4410	. 2 |- ( ph -> <. ( ( 1st ` ( K o.func H ) ) o. ( 1st ` G ) ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( ( ( 1st ` G ) ` x ) ( 2nd ` ( K o.func H ) ) ( ( 1st ` G ) ` y ) ) o. ( x ( 2nd ` G ) y ) ) ) >. = <. ( ( 1st ` K ) o. ( 1st ` ( H o.func G ) ) ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( ( ( 1st ` ( H o.func G ) ) ` x ) ( 2nd ` K ) ( ( 1st ` ( H o.func G ) ) ` y ) ) o. ( x ( 2nd ` ( H o.func G ) ) y ) ) ) >. )
35	3, 4	cofucl 16548	. . 3 |- ( ph -> ( K o.func H ) e. ( D Func F ) )
36	7, 8, 35	cofuval 16542	. 2 |- ( ph -> ( ( K o.func H ) o.func G ) = <. ( ( 1st ` ( K o.func H ) ) o. ( 1st ` G ) ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( ( ( 1st ` G ) ` x ) ( 2nd ` ( K o.func H ) ) ( ( 1st ` G ) ` y ) ) o. ( x ( 2nd ` G ) y ) ) ) >. )
37	8, 3	cofucl 16548	. . 3 |- ( ph -> ( H o.func G ) e. ( C Func E ) )
38	7, 37, 4	cofuval 16542	. 2 |- ( ph -> ( K o.func ( H o.func G ) ) = <. ( ( 1st ` K ) o. ( 1st ` ( H o.func G ) ) ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( ( ( 1st ` ( H o.func G ) ) ` x ) ( 2nd ` K ) ( ( 1st ` ( H o.func G ) ) ` y ) ) o. ( x ( 2nd ` ( H o.func G ) ) y ) ) ) >. )
39	34, 36, 38	3eqtr4d 2666	1 |- ( ph -> ( ( K o.func H ) o.func G ) = ( K o.func ( H o.func G ) ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   o. ccom 5118   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Func cfunc 16514   o._func ccofu 16516

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-un 6949

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-rmo 2920  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-pw 4160  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-ixp 7909  df-cat 16329  df-cid 16330  df-func 16518  df-cofu 16520

This theorem is referenced by:  catccatid  16752