Theorem xpcco2 16827

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

Hypotheses

Ref	Expression
xpcco2.t	|- T = ( C X.c D )
xpcco2.x	|- X = ( Base ` C )
xpcco2.y	|- Y = ( Base ` D )
xpcco2.h	|- H = ( Hom ` C )
xpcco2.j	|- J = ( Hom ` D )
xpcco2.m	|- ( ph -> M e. X )
xpcco2.n	|- ( ph -> N e. Y )
xpcco2.p	|- ( ph -> P e. X )
xpcco2.q	|- ( ph -> Q e. Y )
xpcco2.o1	|- .x. = (comp ` C )
xpcco2.o2	|- .xb = (comp ` D )
xpcco2.o	|- O = (comp ` T )
xpcco2.r	|- ( ph -> R e. X )
xpcco2.s	|- ( ph -> S e. Y )
xpcco2.f	|- ( ph -> F e. ( M H P ) )
xpcco2.g	|- ( ph -> G e. ( N J Q ) )
xpcco2.k	|- ( ph -> K e. ( P H R ) )
xpcco2.l	|- ( ph -> L e. ( Q J S ) )

Assertion

Ref	Expression
xpcco2	|- ( ph -> ( <. K , L >. ( <. <. M , N >. , <. P , Q >. >. O <. R , S >. ) <. F , G >. ) = <. ( K ( <. M , P >. .x. R ) F ) , ( L ( <. N , Q >. .xb S ) G ) >. )

Proof of Theorem xpcco2

Step	Hyp	Ref	Expression
1		xpcco2.t	. . 3 |- T = ( C X.c D )
2		xpcco2.x	. . . 4 |- X = ( Base ` C )
3		xpcco2.y	. . . 4 |- Y = ( Base ` D )
4	1, 2, 3	xpcbas 16818	. . 3 |- ( X X. Y ) = ( Base ` T )
5		eqid 2622	. . 3 |- ( Hom ` T ) = ( Hom ` T )
6		xpcco2.o1	. . 3 |- .x. = (comp ` C )
7		xpcco2.o2	. . 3 |- .xb = (comp ` D )
8		xpcco2.o	. . 3 |- O = (comp ` T )
9		xpcco2.m	. . . 4 |- ( ph -> M e. X )
10		xpcco2.n	. . . 4 |- ( ph -> N e. Y )
11		opelxpi 5148	. . . 4 |- ( ( M e. X /\ N e. Y ) -> <. M , N >. e. ( X X. Y ) )
12	9, 10, 11	syl2anc 693	. . 3 |- ( ph -> <. M , N >. e. ( X X. Y ) )
13		xpcco2.p	. . . 4 |- ( ph -> P e. X )
14		xpcco2.q	. . . 4 |- ( ph -> Q e. Y )
15		opelxpi 5148	. . . 4 |- ( ( P e. X /\ Q e. Y ) -> <. P , Q >. e. ( X X. Y ) )
16	13, 14, 15	syl2anc 693	. . 3 |- ( ph -> <. P , Q >. e. ( X X. Y ) )
17		xpcco2.r	. . . 4 |- ( ph -> R e. X )
18		xpcco2.s	. . . 4 |- ( ph -> S e. Y )
19		opelxpi 5148	. . . 4 |- ( ( R e. X /\ S e. Y ) -> <. R , S >. e. ( X X. Y ) )
20	17, 18, 19	syl2anc 693	. . 3 |- ( ph -> <. R , S >. e. ( X X. Y ) )
21		xpcco2.f	. . . . 5 |- ( ph -> F e. ( M H P ) )
22		xpcco2.g	. . . . 5 |- ( ph -> G e. ( N J Q ) )
23		opelxpi 5148	. . . . 5 |- ( ( F e. ( M H P ) /\ G e. ( N J Q ) ) -> <. F , G >. e. ( ( M H P ) X. ( N J Q ) ) )
24	21, 22, 23	syl2anc 693	. . . 4 |- ( ph -> <. F , G >. e. ( ( M H P ) X. ( N J Q ) ) )
25		xpcco2.h	. . . . 5 |- H = ( Hom ` C )
26		xpcco2.j	. . . . 5 |- J = ( Hom ` D )
27	1, 2, 3, 25, 26, 9, 10, 13, 14, 5	xpchom2 16826	. . . 4 |- ( ph -> ( <. M , N >. ( Hom ` T ) <. P , Q >. ) = ( ( M H P ) X. ( N J Q ) ) )
28	24, 27	eleqtrrd 2704	. . 3 |- ( ph -> <. F , G >. e. ( <. M , N >. ( Hom ` T ) <. P , Q >. ) )
29		xpcco2.k	. . . . 5 |- ( ph -> K e. ( P H R ) )
30		xpcco2.l	. . . . 5 |- ( ph -> L e. ( Q J S ) )
31		opelxpi 5148	. . . . 5 |- ( ( K e. ( P H R ) /\ L e. ( Q J S ) ) -> <. K , L >. e. ( ( P H R ) X. ( Q J S ) ) )
32	29, 30, 31	syl2anc 693	. . . 4 |- ( ph -> <. K , L >. e. ( ( P H R ) X. ( Q J S ) ) )
33	1, 2, 3, 25, 26, 13, 14, 17, 18, 5	xpchom2 16826	. . . 4 |- ( ph -> ( <. P , Q >. ( Hom ` T ) <. R , S >. ) = ( ( P H R ) X. ( Q J S ) ) )
34	32, 33	eleqtrrd 2704	. . 3 |- ( ph -> <. K , L >. e. ( <. P , Q >. ( Hom ` T ) <. R , S >. ) )
35	1, 4, 5, 6, 7, 8, 12, 16, 20, 28, 34	xpcco 16823	. 2 |- ( ph -> ( <. K , L >. ( <. <. M , N >. , <. P , Q >. >. O <. R , S >. ) <. F , G >. ) = <. ( ( 1st ` <. K , L >. ) ( <. ( 1st ` <. M , N >. ) , ( 1st ` <. P , Q >. ) >. .x. ( 1st ` <. R , S >. ) ) ( 1st ` <. F , G >. ) ) , ( ( 2nd ` <. K , L >. ) ( <. ( 2nd ` <. M , N >. ) , ( 2nd ` <. P , Q >. ) >. .xb ( 2nd ` <. R , S >. ) ) ( 2nd ` <. F , G >. ) ) >. )
36		op1stg 7180	. . . . . . 7 |- ( ( M e. X /\ N e. Y ) -> ( 1st ` <. M , N >. ) = M )
37	9, 10, 36	syl2anc 693	. . . . . 6 |- ( ph -> ( 1st ` <. M , N >. ) = M )
38		op1stg 7180	. . . . . . 7 |- ( ( P e. X /\ Q e. Y ) -> ( 1st ` <. P , Q >. ) = P )
39	13, 14, 38	syl2anc 693	. . . . . 6 |- ( ph -> ( 1st ` <. P , Q >. ) = P )
40	37, 39	opeq12d 4410	. . . . 5 |- ( ph -> <. ( 1st ` <. M , N >. ) , ( 1st ` <. P , Q >. ) >. = <. M , P >. )
41		op1stg 7180	. . . . . 6 |- ( ( R e. X /\ S e. Y ) -> ( 1st ` <. R , S >. ) = R )
42	17, 18, 41	syl2anc 693	. . . . 5 |- ( ph -> ( 1st ` <. R , S >. ) = R )
43	40, 42	oveq12d 6668	. . . 4 |- ( ph -> ( <. ( 1st ` <. M , N >. ) , ( 1st ` <. P , Q >. ) >. .x. ( 1st ` <. R , S >. ) ) = ( <. M , P >. .x. R ) )
44		op1stg 7180	. . . . 5 |- ( ( K e. ( P H R ) /\ L e. ( Q J S ) ) -> ( 1st ` <. K , L >. ) = K )
45	29, 30, 44	syl2anc 693	. . . 4 |- ( ph -> ( 1st ` <. K , L >. ) = K )
46		op1stg 7180	. . . . 5 |- ( ( F e. ( M H P ) /\ G e. ( N J Q ) ) -> ( 1st ` <. F , G >. ) = F )
47	21, 22, 46	syl2anc 693	. . . 4 |- ( ph -> ( 1st ` <. F , G >. ) = F )
48	43, 45, 47	oveq123d 6671	. . 3 |- ( ph -> ( ( 1st ` <. K , L >. ) ( <. ( 1st ` <. M , N >. ) , ( 1st ` <. P , Q >. ) >. .x. ( 1st ` <. R , S >. ) ) ( 1st ` <. F , G >. ) ) = ( K ( <. M , P >. .x. R ) F ) )
49		op2ndg 7181	. . . . . . 7 |- ( ( M e. X /\ N e. Y ) -> ( 2nd ` <. M , N >. ) = N )
50	9, 10, 49	syl2anc 693	. . . . . 6 |- ( ph -> ( 2nd ` <. M , N >. ) = N )
51		op2ndg 7181	. . . . . . 7 |- ( ( P e. X /\ Q e. Y ) -> ( 2nd ` <. P , Q >. ) = Q )
52	13, 14, 51	syl2anc 693	. . . . . 6 |- ( ph -> ( 2nd ` <. P , Q >. ) = Q )
53	50, 52	opeq12d 4410	. . . . 5 |- ( ph -> <. ( 2nd ` <. M , N >. ) , ( 2nd ` <. P , Q >. ) >. = <. N , Q >. )
54		op2ndg 7181	. . . . . 6 |- ( ( R e. X /\ S e. Y ) -> ( 2nd ` <. R , S >. ) = S )
55	17, 18, 54	syl2anc 693	. . . . 5 |- ( ph -> ( 2nd ` <. R , S >. ) = S )
56	53, 55	oveq12d 6668	. . . 4 |- ( ph -> ( <. ( 2nd ` <. M , N >. ) , ( 2nd ` <. P , Q >. ) >. .xb ( 2nd ` <. R , S >. ) ) = ( <. N , Q >. .xb S ) )
57		op2ndg 7181	. . . . 5 |- ( ( K e. ( P H R ) /\ L e. ( Q J S ) ) -> ( 2nd ` <. K , L >. ) = L )
58	29, 30, 57	syl2anc 693	. . . 4 |- ( ph -> ( 2nd ` <. K , L >. ) = L )
59		op2ndg 7181	. . . . 5 |- ( ( F e. ( M H P ) /\ G e. ( N J Q ) ) -> ( 2nd ` <. F , G >. ) = G )
60	21, 22, 59	syl2anc 693	. . . 4 |- ( ph -> ( 2nd ` <. F , G >. ) = G )
61	56, 58, 60	oveq123d 6671	. . 3 |- ( ph -> ( ( 2nd ` <. K , L >. ) ( <. ( 2nd ` <. M , N >. ) , ( 2nd ` <. P , Q >. ) >. .xb ( 2nd ` <. R , S >. ) ) ( 2nd ` <. F , G >. ) ) = ( L ( <. N , Q >. .xb S ) G ) )
62	48, 61	opeq12d 4410	. 2 |- ( ph -> <. ( ( 1st ` <. K , L >. ) ( <. ( 1st ` <. M , N >. ) , ( 1st ` <. P , Q >. ) >. .x. ( 1st ` <. R , S >. ) ) ( 1st ` <. F , G >. ) ) , ( ( 2nd ` <. K , L >. ) ( <. ( 2nd ` <. M , N >. ) , ( 2nd ` <. P , Q >. ) >. .xb ( 2nd ` <. R , S >. ) ) ( 2nd ` <. F , G >. ) ) >. = <. ( K ( <. M , P >. .x. R ) F ) , ( L ( <. N , Q >. .xb S ) G ) >. )
63	35, 62	eqtrd 2656	1 |- ( ph -> ( <. K , L >. ( <. <. M , N >. , <. P , Q >. >. O <. R , S >. ) <. F , G >. ) = <. ( K ( <. M , P >. .x. R ) F ) , ( L ( <. N , Q >. .xb S ) G ) >. )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   <.cop 4183   X. cxp 5112   ` cfv 5888  (class class class)co 6650   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:  prfcl  16843  evlfcllem  16861  curf1cl  16868  curf2cl  16871  curfcl  16872  uncfcurf  16879  hofcl  16899