Theorem xpcco 16823

Description: Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
xpccofval.t	|- T = ( C X.c D )
xpccofval.b	|- B = ( Base ` T )
xpccofval.k	|- K = ( Hom ` T )
xpccofval.o1	|- .x. = (comp ` C )
xpccofval.o2	|- .xb = (comp ` D )
xpccofval.o	|- O = (comp ` T )
xpcco.x	|- ( ph -> X e. B )
xpcco.y	|- ( ph -> Y e. B )
xpcco.z	|- ( ph -> Z e. B )
xpcco.f	|- ( ph -> F e. ( X K Y ) )
xpcco.g	|- ( ph -> G e. ( Y K Z ) )

Assertion

Ref	Expression
xpcco	|- ( ph -> ( G ( <. X , Y >. O Z ) F ) = <. ( ( 1st ` G ) ( <. ( 1st ` X ) , ( 1st ` Y ) >. .x. ( 1st ` Z ) ) ( 1st ` F ) ) , ( ( 2nd ` G ) ( <. ( 2nd ` X ) , ( 2nd ` Y ) >. .xb ( 2nd ` Z ) ) ( 2nd ` F ) ) >. )

Proof of Theorem xpcco

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

Step	Hyp	Ref	Expression
1		xpccofval.t	. . 3 |- T = ( C X.c D )
2		xpccofval.b	. . 3 |- B = ( Base ` T )
3		xpccofval.k	. . 3 |- K = ( Hom ` T )
4		xpccofval.o1	. . 3 |- .x. = (comp ` C )
5		xpccofval.o2	. . 3 |- .xb = (comp ` D )
6		xpccofval.o	. . 3 |- O = (comp ` T )
7	1, 2, 3, 4, 5, 6	xpccofval 16822	. 2 |- O = ( x e. ( B X. B ) , y e. B |-> ( g e. ( ( 2nd ` x ) K y ) , f e. ( K ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. .x. ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. .xb ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) )
8		xpcco.x	. . . 4 |- ( ph -> X e. B )
9		xpcco.y	. . . 4 |- ( ph -> Y e. B )
10		opelxpi 5148	. . . 4 |- ( ( X e. B /\ Y e. B ) -> <. X , Y >. e. ( B X. B ) )
11	8, 9, 10	syl2anc 693	. . 3 |- ( ph -> <. X , Y >. e. ( B X. B ) )
12		xpcco.z	. . . 4 |- ( ph -> Z e. B )
13	12	adantr 481	. . 3 |- ( ( ph /\ x = <. X , Y >. ) -> Z e. B )
14		ovex 6678	. . . . 5 |- ( ( 2nd ` x ) K y ) e. _V
15		fvex 6201	. . . . 5 |- ( K ` x ) e. _V
16	14, 15	mpt2ex 7247	. . . 4 |- ( g e. ( ( 2nd ` x ) K y ) , f e. ( K ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. .x. ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. .xb ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) e. _V
17	16	a1i 11	. . 3 |- ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) -> ( g e. ( ( 2nd ` x ) K y ) , f e. ( K ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. .x. ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. .xb ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) e. _V )
18		xpcco.g	. . . . . 6 |- ( ph -> G e. ( Y K Z ) )
19	18	adantr 481	. . . . 5 |- ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) -> G e. ( Y K Z ) )
20		simprl 794	. . . . . . . 8 |- ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) -> x = <. X , Y >. )
21	20	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) -> ( 2nd ` x ) = ( 2nd ` <. X , Y >. ) )
22		op2ndg 7181	. . . . . . . . 9 |- ( ( X e. B /\ Y e. B ) -> ( 2nd ` <. X , Y >. ) = Y )
23	8, 9, 22	syl2anc 693	. . . . . . . 8 |- ( ph -> ( 2nd ` <. X , Y >. ) = Y )
24	23	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) -> ( 2nd ` <. X , Y >. ) = Y )
25	21, 24	eqtrd 2656	. . . . . 6 |- ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) -> ( 2nd ` x ) = Y )
26		simprr 796	. . . . . 6 |- ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) -> y = Z )
27	25, 26	oveq12d 6668	. . . . 5 |- ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) -> ( ( 2nd ` x ) K y ) = ( Y K Z ) )
28	19, 27	eleqtrrd 2704	. . . 4 |- ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) -> G e. ( ( 2nd ` x ) K y ) )
29		xpcco.f	. . . . . . 7 |- ( ph -> F e. ( X K Y ) )
30	29	adantr 481	. . . . . 6 |- ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) -> F e. ( X K Y ) )
31	20	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) -> ( K ` x ) = ( K ` <. X , Y >. ) )
32		df-ov 6653	. . . . . . 7 |- ( X K Y ) = ( K ` <. X , Y >. )
33	31, 32	syl6eqr 2674	. . . . . 6 |- ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) -> ( K ` x ) = ( X K Y ) )
34	30, 33	eleqtrrd 2704	. . . . 5 |- ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) -> F e. ( K ` x ) )
35	34	adantr 481	. . . 4 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ g = G ) -> F e. ( K ` x ) )
36		opex 4932	. . . . 5 |- <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. .x. ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. .xb ( 2nd ` y ) ) ( 2nd ` f ) ) >. e. _V
37	36	a1i 11	. . . 4 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. .x. ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. .xb ( 2nd ` y ) ) ( 2nd ` f ) ) >. e. _V )
38	20	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) -> ( 1st ` x ) = ( 1st ` <. X , Y >. ) )
39		op1stg 7180	. . . . . . . . . . . . 13 |- ( ( X e. B /\ Y e. B ) -> ( 1st ` <. X , Y >. ) = X )
40	8, 9, 39	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> ( 1st ` <. X , Y >. ) = X )
41	40	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) -> ( 1st ` <. X , Y >. ) = X )
42	38, 41	eqtrd 2656	. . . . . . . . . 10 |- ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) -> ( 1st ` x ) = X )
43	42	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> ( 1st ` x ) = X )
44	43	fveq2d 6195	. . . . . . . 8 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> ( 1st ` ( 1st ` x ) ) = ( 1st ` X ) )
45	25	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> ( 2nd ` x ) = Y )
46	45	fveq2d 6195	. . . . . . . 8 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> ( 1st ` ( 2nd ` x ) ) = ( 1st ` Y ) )
47	44, 46	opeq12d 4410	. . . . . . 7 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. = <. ( 1st ` X ) , ( 1st ` Y ) >. )
48		simplrr 801	. . . . . . . 8 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> y = Z )
49	48	fveq2d 6195	. . . . . . 7 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> ( 1st ` y ) = ( 1st ` Z ) )
50	47, 49	oveq12d 6668	. . . . . 6 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. .x. ( 1st ` y ) ) = ( <. ( 1st ` X ) , ( 1st ` Y ) >. .x. ( 1st ` Z ) ) )
51		simprl 794	. . . . . . 7 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> g = G )
52	51	fveq2d 6195	. . . . . 6 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> ( 1st ` g ) = ( 1st ` G ) )
53		simprr 796	. . . . . . 7 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> f = F )
54	53	fveq2d 6195	. . . . . 6 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> ( 1st ` f ) = ( 1st ` F ) )
55	50, 52, 54	oveq123d 6671	. . . . 5 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. .x. ( 1st ` y ) ) ( 1st ` f ) ) = ( ( 1st ` G ) ( <. ( 1st ` X ) , ( 1st ` Y ) >. .x. ( 1st ` Z ) ) ( 1st ` F ) ) )
56	43	fveq2d 6195	. . . . . . . 8 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> ( 2nd ` ( 1st ` x ) ) = ( 2nd ` X ) )
57	45	fveq2d 6195	. . . . . . . 8 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> ( 2nd ` ( 2nd ` x ) ) = ( 2nd ` Y ) )
58	56, 57	opeq12d 4410	. . . . . . 7 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. = <. ( 2nd ` X ) , ( 2nd ` Y ) >. )
59	48	fveq2d 6195	. . . . . . 7 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> ( 2nd ` y ) = ( 2nd ` Z ) )
60	58, 59	oveq12d 6668	. . . . . 6 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. .xb ( 2nd ` y ) ) = ( <. ( 2nd ` X ) , ( 2nd ` Y ) >. .xb ( 2nd ` Z ) ) )
61	51	fveq2d 6195	. . . . . 6 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> ( 2nd ` g ) = ( 2nd ` G ) )
62	53	fveq2d 6195	. . . . . 6 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> ( 2nd ` f ) = ( 2nd ` F ) )
63	60, 61, 62	oveq123d 6671	. . . . 5 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. .xb ( 2nd ` y ) ) ( 2nd ` f ) ) = ( ( 2nd ` G ) ( <. ( 2nd ` X ) , ( 2nd ` Y ) >. .xb ( 2nd ` Z ) ) ( 2nd ` F ) ) )
64	55, 63	opeq12d 4410	. . . 4 |- ( ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) /\ ( g = G /\ f = F ) ) -> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. .x. ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. .xb ( 2nd ` y ) ) ( 2nd ` f ) ) >. = <. ( ( 1st ` G ) ( <. ( 1st ` X ) , ( 1st ` Y ) >. .x. ( 1st ` Z ) ) ( 1st ` F ) ) , ( ( 2nd ` G ) ( <. ( 2nd ` X ) , ( 2nd ` Y ) >. .xb ( 2nd ` Z ) ) ( 2nd ` F ) ) >. )
65	28, 35, 37, 64	ovmpt2dv2 6794	. . 3 |- ( ( ph /\ ( x = <. X , Y >. /\ y = Z ) ) -> ( ( <. X , Y >. O Z ) = ( g e. ( ( 2nd ` x ) K y ) , f e. ( K ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. .x. ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. .xb ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) -> ( G ( <. X , Y >. O Z ) F ) = <. ( ( 1st ` G ) ( <. ( 1st ` X ) , ( 1st ` Y ) >. .x. ( 1st ` Z ) ) ( 1st ` F ) ) , ( ( 2nd ` G ) ( <. ( 2nd ` X ) , ( 2nd ` Y ) >. .xb ( 2nd ` Z ) ) ( 2nd ` F ) ) >. ) )
66	11, 13, 17, 65	ovmpt2dv 6793	. 2 |- ( ph -> ( O = ( x e. ( B X. B ) , y e. B |-> ( g e. ( ( 2nd ` x ) K y ) , f e. ( K ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. .x. ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. .xb ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) -> ( G ( <. X , Y >. O Z ) F ) = <. ( ( 1st ` G ) ( <. ( 1st ` X ) , ( 1st ` Y ) >. .x. ( 1st ` Z ) ) ( 1st ` F ) ) , ( ( 2nd ` G ) ( <. ( 2nd ` X ) , ( 2nd ` Y ) >. .xb ( 2nd ` Z ) ) ( 2nd ` F ) ) >. ) )
67	7, 66	mpi 20	1 |- ( ph -> ( G ( <. X , Y >. O Z ) F ) = <. ( ( 1st ` G ) ( <. ( 1st ` X ) , ( 1st ` Y ) >. .x. ( 1st ` Z ) ) ( 1st ` F ) ) , ( ( 2nd ` G ) ( <. ( 2nd ` X ) , ( 2nd ` Y ) >. .xb ( 2nd ` Z ) ) ( 2nd ` F ) ) >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952  compcco 15953   X._c cxpc 16808

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-xpc 16812

This theorem is referenced by:  xpcco1st  16824  xpcco2nd  16825  xpcco2  16827  xpccatid  16828