Theorem xpccofval 16822

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 )

Assertion

Ref	Expression
xpccofval	|- 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 ) ) >. ) )

Distinct variable groups:   f, g, x, y, B   C, f, g, x, y   D, f, g, x, y   .x. , f, g, x, y   .xb , f, g, x, y   f, K, g, x, y   x, O, y

Allowed substitution hints:   T( x, y, f, g)   O( f, g)

Proof of Theorem xpccofval

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpccofval.t	. . . 4 |- T = ( C X.c D )
2		eqid 2622	. . . 4 |- ( Base ` C ) = ( Base ` C )
3		eqid 2622	. . . 4 |- ( Base ` D ) = ( Base ` D )
4		eqid 2622	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
5		eqid 2622	. . . 4 |- ( Hom ` D ) = ( Hom ` D )
6		xpccofval.o1	. . . 4 |- .x. = (comp ` C )
7		xpccofval.o2	. . . 4 |- .xb = (comp ` D )
8		simpl 473	. . . 4 |- ( ( C e. _V /\ D e. _V ) -> C e. _V )
9		simpr 477	. . . 4 |- ( ( C e. _V /\ D e. _V ) -> D e. _V )
10		xpccofval.b	. . . . . 6 |- B = ( Base ` T )
11	1, 2, 3	xpcbas 16818	. . . . . 6 |- ( ( Base ` C ) X. ( Base ` D ) ) = ( Base ` T )
12	10, 11	eqtr4i 2647	. . . . 5 |- B = ( ( Base ` C ) X. ( Base ` D ) )
13	12	a1i 11	. . . 4 |- ( ( C e. _V /\ D e. _V ) -> B = ( ( Base ` C ) X. ( Base ` D ) ) )
14		xpccofval.k	. . . . . 6 |- K = ( Hom ` T )
15	1, 10, 4, 5, 14	xpchomfval 16819	. . . . 5 |- K = ( u e. B , v e. B |-> ( ( ( 1st ` u ) ( Hom ` C ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` D ) ( 2nd ` v ) ) ) )
16	15	a1i 11	. . . 4 |- ( ( C e. _V /\ D e. _V ) -> K = ( u e. B , v e. B |-> ( ( ( 1st ` u ) ( Hom ` C ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` D ) ( 2nd ` v ) ) ) ) )
17		eqidd 2623	. . . 4 |- ( ( C e. _V /\ D e. _V ) -> ( 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 ) ) >. ) ) = ( 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 ) ) >. ) ) )
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 16, 17	xpcval 16817	. . 3 |- ( ( C e. _V /\ D e. _V ) -> T = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , K >. , <. (comp ` ndx ) , ( 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 ) ) >. ) ) >. } )
19		catstr 16617	. . 3 |- { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , K >. , <. (comp ` ndx ) , ( 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 ) ) >. ) ) >. } Struct <. 1 , ; 1 5 >.
20		ccoid 16077	. . 3 |- comp = Slot (comp ` ndx )
21		snsstp3 4349	. . 3 |- { <. (comp ` ndx ) , ( 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 ) ) >. ) ) >. } C_ { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , K >. , <. (comp ` ndx ) , ( 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 ) ) >. ) ) >. }
22		fvex 6201	. . . . . . 7 |- ( Base ` T ) e. _V
23	10, 22	eqeltri 2697	. . . . . 6 |- B e. _V
24	23, 23	xpex 6962	. . . . 5 |- ( B X. B ) e. _V
25	24, 23	mpt2ex 7247	. . . 4 |- ( 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 ) ) >. ) ) e. _V
26	25	a1i 11	. . 3 |- ( ( C e. _V /\ D e. _V ) -> ( 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 ) ) >. ) ) e. _V )
27		xpccofval.o	. . 3 |- O = (comp ` T )
28	18, 19, 20, 21, 26, 27	strfv3 15908	. 2 |- ( ( C e. _V /\ D e. _V ) -> 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 ) ) >. ) ) )
29		mpt20 6725	. . . 4 |- ( x e. (/) , y e. (/) |-> ( 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 ) ) >. ) ) = (/)
30	29	eqcomi 2631	. . 3 |- (/) = ( x e. (/) , y e. (/) |-> ( 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 ) ) >. ) )
31		fnxpc 16816	. . . . . . . 8 |- X.c Fn ( _V X. _V )
32		fndm 5990	. . . . . . . 8 |- ( X.c Fn ( _V X. _V ) -> dom X.c = ( _V X. _V ) )
33	31, 32	ax-mp 5	. . . . . . 7 |- dom X.c = ( _V X. _V )
34	33	ndmov 6818	. . . . . 6 |- ( -. ( C e. _V /\ D e. _V ) -> ( C X.c D ) = (/) )
35	1, 34	syl5eq 2668	. . . . 5 |- ( -. ( C e. _V /\ D e. _V ) -> T = (/) )
36	35	fveq2d 6195	. . . 4 |- ( -. ( C e. _V /\ D e. _V ) -> (comp ` T ) = (comp ` (/) ) )
37	20	str0 15911	. . . 4 |- (/) = (comp ` (/) )
38	36, 27, 37	3eqtr4g 2681	. . 3 |- ( -. ( C e. _V /\ D e. _V ) -> O = (/) )
39	35	fveq2d 6195	. . . . . . 7 |- ( -. ( C e. _V /\ D e. _V ) -> ( Base ` T ) = ( Base ` (/) ) )
40		base0 15912	. . . . . . 7 |- (/) = ( Base ` (/) )
41	39, 10, 40	3eqtr4g 2681	. . . . . 6 |- ( -. ( C e. _V /\ D e. _V ) -> B = (/) )
42	41	xpeq2d 5139	. . . . 5 |- ( -. ( C e. _V /\ D e. _V ) -> ( B X. B ) = ( B X. (/) ) )
43		xp0 5552	. . . . 5 |- ( B X. (/) ) = (/)
44	42, 43	syl6eq 2672	. . . 4 |- ( -. ( C e. _V /\ D e. _V ) -> ( B X. B ) = (/) )
45		eqidd 2623	. . . 4 |- ( -. ( C e. _V /\ D e. _V ) -> ( 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 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 ) ) >. ) )
46	44, 41, 45	mpt2eq123dv 6717	. . 3 |- ( -. ( C e. _V /\ D e. _V ) -> ( 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 ) ) >. ) ) = ( x e. (/) , y e. (/) |-> ( 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 ) ) >. ) ) )
47	30, 38, 46	3eqtr4a 2682	. 2 |- ( -. ( C e. _V /\ D e. _V ) -> 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 ) ) >. ) ) )
48	28, 47	pm2.61i 176	1 |- 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 ) ) >. ) )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   {ctp 4181   <.cop 4183   X. cxp 5112   dom cdm 5114   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   1c1 9937   5c5 11073  ;cdc 11493   ndxcnx 15854   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:  xpcco  16823