Theorem coaval 16718

Description: Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
homdmcoa.o	|- .x. = (compa ` C )
homdmcoa.h	|- H = (Homa ` C )
homdmcoa.f	|- ( ph -> F e. ( X H Y ) )
homdmcoa.g	|- ( ph -> G e. ( Y H Z ) )
coaval.x	|- .xb = (comp ` C )

Assertion

Ref	Expression
coaval	|- ( ph -> ( G .x. F ) = <. X , Z , ( ( 2nd ` G ) ( <. X , Y >. .xb Z ) ( 2nd ` F ) ) >. )

Proof of Theorem coaval

Dummy variables f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		homdmcoa.o	. . 3 |- .x. = (compa ` C )
2		eqid 2622	. . 3 |- (Nat ` C ) = (Nat ` C )
3		coaval.x	. . 3 |- .xb = (comp ` C )
4	1, 2, 3	coafval 16714	. 2 |- .x. = ( g e. (Nat ` C ) , f e. { h e. (Nat ` C ) | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. )
5		homdmcoa.h	. . . . 5 |- H = (Homa ` C )
6	2, 5	homarw 16696	. . . 4 |- ( Y H Z ) C_ (Nat ` C )
7		homdmcoa.g	. . . 4 |- ( ph -> G e. ( Y H Z ) )
8	6, 7	sseldi 3601	. . 3 |- ( ph -> G e. (Nat ` C ) )
9	2, 5	homarw 16696	. . . . 5 |- ( X H Y ) C_ (Nat ` C )
10		homdmcoa.f	. . . . . 6 |- ( ph -> F e. ( X H Y ) )
11	10	adantr 481	. . . . 5 |- ( ( ph /\ g = G ) -> F e. ( X H Y ) )
12	9, 11	sseldi 3601	. . . 4 |- ( ( ph /\ g = G ) -> F e. (Nat ` C ) )
13	5	homacd 16691	. . . . . 6 |- ( F e. ( X H Y ) -> (coda ` F ) = Y )
14	11, 13	syl 17	. . . . 5 |- ( ( ph /\ g = G ) -> (coda ` F ) = Y )
15		simpr 477	. . . . . . 7 |- ( ( ph /\ g = G ) -> g = G )
16	15	fveq2d 6195	. . . . . 6 |- ( ( ph /\ g = G ) -> (domA ` g ) = (domA ` G ) )
17	7	adantr 481	. . . . . . 7 |- ( ( ph /\ g = G ) -> G e. ( Y H Z ) )
18	5	homadm 16690	. . . . . . 7 |- ( G e. ( Y H Z ) -> (domA ` G ) = Y )
19	17, 18	syl 17	. . . . . 6 |- ( ( ph /\ g = G ) -> (domA ` G ) = Y )
20	16, 19	eqtrd 2656	. . . . 5 |- ( ( ph /\ g = G ) -> (domA ` g ) = Y )
21	14, 20	eqtr4d 2659	. . . 4 |- ( ( ph /\ g = G ) -> (coda ` F ) = (domA ` g ) )
22		fveq2 6191	. . . . . 6 |- ( h = F -> (coda ` h ) = (coda ` F ) )
23	22	eqeq1d 2624	. . . . 5 |- ( h = F -> ( (coda ` h ) = (domA ` g ) <-> (coda ` F ) = (domA ` g ) ) )
24	23	elrab 3363	. . . 4 |- ( F e. { h e. (Nat ` C ) | (coda ` h ) = (domA ` g ) } <-> ( F e. (Nat ` C ) /\ (coda ` F ) = (domA ` g ) ) )
25	12, 21, 24	sylanbrc 698	. . 3 |- ( ( ph /\ g = G ) -> F e. { h e. (Nat ` C ) | (coda ` h ) = (domA ` g ) } )
26		otex 4933	. . . 4 |- <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. e. _V
27	26	a1i 11	. . 3 |- ( ( ph /\ ( g = G /\ f = F ) ) -> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. e. _V )
28		simprr 796	. . . . . 6 |- ( ( ph /\ ( g = G /\ f = F ) ) -> f = F )
29	28	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( g = G /\ f = F ) ) -> (domA ` f ) = (domA ` F ) )
30	5	homadm 16690	. . . . . . 7 |- ( F e. ( X H Y ) -> (domA ` F ) = X )
31	11, 30	syl 17	. . . . . 6 |- ( ( ph /\ g = G ) -> (domA ` F ) = X )
32	31	adantrr 753	. . . . 5 |- ( ( ph /\ ( g = G /\ f = F ) ) -> (domA ` F ) = X )
33	29, 32	eqtrd 2656	. . . 4 |- ( ( ph /\ ( g = G /\ f = F ) ) -> (domA ` f ) = X )
34	15	fveq2d 6195	. . . . . 6 |- ( ( ph /\ g = G ) -> (coda ` g ) = (coda ` G ) )
35	5	homacd 16691	. . . . . . 7 |- ( G e. ( Y H Z ) -> (coda ` G ) = Z )
36	17, 35	syl 17	. . . . . 6 |- ( ( ph /\ g = G ) -> (coda ` G ) = Z )
37	34, 36	eqtrd 2656	. . . . 5 |- ( ( ph /\ g = G ) -> (coda ` g ) = Z )
38	37	adantrr 753	. . . 4 |- ( ( ph /\ ( g = G /\ f = F ) ) -> (coda ` g ) = Z )
39	20	adantrr 753	. . . . . . 7 |- ( ( ph /\ ( g = G /\ f = F ) ) -> (domA ` g ) = Y )
40	33, 39	opeq12d 4410	. . . . . 6 |- ( ( ph /\ ( g = G /\ f = F ) ) -> <. (domA ` f ) , (domA ` g ) >. = <. X , Y >. )
41	40, 38	oveq12d 6668	. . . . 5 |- ( ( ph /\ ( g = G /\ f = F ) ) -> ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) = ( <. X , Y >. .xb Z ) )
42		simprl 794	. . . . . 6 |- ( ( ph /\ ( g = G /\ f = F ) ) -> g = G )
43	42	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( g = G /\ f = F ) ) -> ( 2nd ` g ) = ( 2nd ` G ) )
44	28	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( g = G /\ f = F ) ) -> ( 2nd ` f ) = ( 2nd ` F ) )
45	41, 43, 44	oveq123d 6671	. . . 4 |- ( ( ph /\ ( g = G /\ f = F ) ) -> ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) = ( ( 2nd ` G ) ( <. X , Y >. .xb Z ) ( 2nd ` F ) ) )
46	33, 38, 45	oteq123d 4417	. . 3 |- ( ( ph /\ ( g = G /\ f = F ) ) -> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. = <. X , Z , ( ( 2nd ` G ) ( <. X , Y >. .xb Z ) ( 2nd ` F ) ) >. )
47	8, 25, 27, 46	ovmpt2dv2 6794	. 2 |- ( ph -> ( .x. = ( g e. (Nat ` C ) , f e. { h e. (Nat ` C ) | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. ) -> ( G .x. F ) = <. X , Z , ( ( 2nd ` G ) ( <. X , Y >. .xb Z ) ( 2nd ` F ) ) >. ) )
48	4, 47	mpi 20	1 |- ( ph -> ( G .x. F ) = <. X , Z , ( ( 2nd ` G ) ( <. X , Y >. .xb Z ) ( 2nd ` F ) ) >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   <.cop 4183   <.cotp 4185   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   2ndc2nd 7167  compcco 15953  domAcdoma 16670  cod_accoda 16671  Natcarw 16672  Hom_achoma 16673  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-doma 16674  df-coda 16675  df-homa 16676  df-arw 16677  df-coa 16706

This theorem is referenced by:  coa2  16719  coahom  16720  arwlid  16722  arwrid  16723  arwass  16724