Theorem arwass 16724

Description: Associativity of composition in a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
arwlid.h	|- H = (Homa ` C )
arwlid.o	|- .x. = (compa ` C )
arwlid.a	|- .1. = (Ida ` C )
arwlid.f	|- ( ph -> F e. ( X H Y ) )
arwass.g	|- ( ph -> G e. ( Y H Z ) )
arwass.k	|- ( ph -> K e. ( Z H W ) )

Assertion

Ref	Expression
arwass	|- ( ph -> ( ( K .x. G ) .x. F ) = ( K .x. ( G .x. F ) ) )

Proof of Theorem arwass

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( Base ` C ) = ( Base ` C )
2		eqid 2622	. . . . 5 |- ( Hom ` C ) = ( Hom ` C )
3		eqid 2622	. . . . 5 |- (comp ` C ) = (comp ` C )
4		arwlid.f	. . . . . 6 |- ( ph -> F e. ( X H Y ) )
5		arwlid.h	. . . . . . 7 |- H = (Homa ` C )
6	5	homarcl 16678	. . . . . 6 |- ( F e. ( X H Y ) -> C e. Cat )
7	4, 6	syl 17	. . . . 5 |- ( ph -> C e. Cat )
8	5, 1	homarcl2 16685	. . . . . . 7 |- ( F e. ( X H Y ) -> ( X e. ( Base ` C ) /\ Y e. ( Base ` C ) ) )
9	4, 8	syl 17	. . . . . 6 |- ( ph -> ( X e. ( Base ` C ) /\ Y e. ( Base ` C ) ) )
10	9	simpld 475	. . . . 5 |- ( ph -> X e. ( Base ` C ) )
11	9	simprd 479	. . . . 5 |- ( ph -> Y e. ( Base ` C ) )
12		arwass.k	. . . . . . 7 |- ( ph -> K e. ( Z H W ) )
13	5, 1	homarcl2 16685	. . . . . . 7 |- ( K e. ( Z H W ) -> ( Z e. ( Base ` C ) /\ W e. ( Base ` C ) ) )
14	12, 13	syl 17	. . . . . 6 |- ( ph -> ( Z e. ( Base ` C ) /\ W e. ( Base ` C ) ) )
15	14	simpld 475	. . . . 5 |- ( ph -> Z e. ( Base ` C ) )
16	5, 2	homahom 16689	. . . . . 6 |- ( F e. ( X H Y ) -> ( 2nd ` F ) e. ( X ( Hom ` C ) Y ) )
17	4, 16	syl 17	. . . . 5 |- ( ph -> ( 2nd ` F ) e. ( X ( Hom ` C ) Y ) )
18		arwass.g	. . . . . 6 |- ( ph -> G e. ( Y H Z ) )
19	5, 2	homahom 16689	. . . . . 6 |- ( G e. ( Y H Z ) -> ( 2nd ` G ) e. ( Y ( Hom ` C ) Z ) )
20	18, 19	syl 17	. . . . 5 |- ( ph -> ( 2nd ` G ) e. ( Y ( Hom ` C ) Z ) )
21	14	simprd 479	. . . . 5 |- ( ph -> W e. ( Base ` C ) )
22	5, 2	homahom 16689	. . . . . 6 |- ( K e. ( Z H W ) -> ( 2nd ` K ) e. ( Z ( Hom ` C ) W ) )
23	12, 22	syl 17	. . . . 5 |- ( ph -> ( 2nd ` K ) e. ( Z ( Hom ` C ) W ) )
24	1, 2, 3, 7, 10, 11, 15, 17, 20, 21, 23	catass 16347	. . . 4 |- ( ph -> ( ( ( 2nd ` K ) ( <. Y , Z >. (comp ` C ) W ) ( 2nd ` G ) ) ( <. X , Y >. (comp ` C ) W ) ( 2nd ` F ) ) = ( ( 2nd ` K ) ( <. X , Z >. (comp ` C ) W ) ( ( 2nd ` G ) ( <. X , Y >. (comp ` C ) Z ) ( 2nd ` F ) ) ) )
25		arwlid.o	. . . . . 6 |- .x. = (compa ` C )
26	25, 5, 18, 12, 3	coa2 16719	. . . . 5 |- ( ph -> ( 2nd ` ( K .x. G ) ) = ( ( 2nd ` K ) ( <. Y , Z >. (comp ` C ) W ) ( 2nd ` G ) ) )
27	26	oveq1d 6665	. . . 4 |- ( ph -> ( ( 2nd ` ( K .x. G ) ) ( <. X , Y >. (comp ` C ) W ) ( 2nd ` F ) ) = ( ( ( 2nd ` K ) ( <. Y , Z >. (comp ` C ) W ) ( 2nd ` G ) ) ( <. X , Y >. (comp ` C ) W ) ( 2nd ` F ) ) )
28	25, 5, 4, 18, 3	coa2 16719	. . . . 5 |- ( ph -> ( 2nd ` ( G .x. F ) ) = ( ( 2nd ` G ) ( <. X , Y >. (comp ` C ) Z ) ( 2nd ` F ) ) )
29	28	oveq2d 6666	. . . 4 |- ( ph -> ( ( 2nd ` K ) ( <. X , Z >. (comp ` C ) W ) ( 2nd ` ( G .x. F ) ) ) = ( ( 2nd ` K ) ( <. X , Z >. (comp ` C ) W ) ( ( 2nd ` G ) ( <. X , Y >. (comp ` C ) Z ) ( 2nd ` F ) ) ) )
30	24, 27, 29	3eqtr4d 2666	. . 3 |- ( ph -> ( ( 2nd ` ( K .x. G ) ) ( <. X , Y >. (comp ` C ) W ) ( 2nd ` F ) ) = ( ( 2nd ` K ) ( <. X , Z >. (comp ` C ) W ) ( 2nd ` ( G .x. F ) ) ) )
31	30	oteq3d 4416	. 2 |- ( ph -> <. X , W , ( ( 2nd ` ( K .x. G ) ) ( <. X , Y >. (comp ` C ) W ) ( 2nd ` F ) ) >. = <. X , W , ( ( 2nd ` K ) ( <. X , Z >. (comp ` C ) W ) ( 2nd ` ( G .x. F ) ) ) >. )
32	25, 5, 18, 12	coahom 16720	. . 3 |- ( ph -> ( K .x. G ) e. ( Y H W ) )
33	25, 5, 4, 32, 3	coaval 16718	. 2 |- ( ph -> ( ( K .x. G ) .x. F ) = <. X , W , ( ( 2nd ` ( K .x. G ) ) ( <. X , Y >. (comp ` C ) W ) ( 2nd ` F ) ) >. )
34	25, 5, 4, 18	coahom 16720	. . 3 |- ( ph -> ( G .x. F ) e. ( X H Z ) )
35	25, 5, 34, 12, 3	coaval 16718	. 2 |- ( ph -> ( K .x. ( G .x. F ) ) = <. X , W , ( ( 2nd ` K ) ( <. X , Z >. (comp ` C ) W ) ( 2nd ` ( G .x. F ) ) ) >. )
36	31, 33, 35	3eqtr4d 2666	1 |- ( ph -> ( ( K .x. G ) .x. F ) = ( K .x. ( G .x. F ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   <.cotp 4185   ` cfv 5888  (class class class)co 6650   2ndc2nd 7167   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325  Hom_achoma 16673  Id_acida 16703  comp_accoa 16704

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-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-ot 4186  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-1st 7168  df-2nd 7169  df-cat 16329  df-doma 16674  df-coda 16675  df-homa 16676  df-arw 16677  df-coa 16706

This theorem is referenced by: (None)