Theorem evlfcllem 16861

Description: Lemma for evlfcl 16862. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
evlfcl.e	|- E = ( C evalF D )
evlfcl.q	|- Q = ( C FuncCat D )
evlfcl.c	|- ( ph -> C e. Cat )
evlfcl.d	|- ( ph -> D e. Cat )
evlfcl.n	|- N = ( C Nat D )
evlfcl.f	|- ( ph -> ( F e. ( C Func D ) /\ X e. ( Base ` C ) ) )
evlfcl.g	|- ( ph -> ( G e. ( C Func D ) /\ Y e. ( Base ` C ) ) )
evlfcl.h	|- ( ph -> ( H e. ( C Func D ) /\ Z e. ( Base ` C ) ) )
evlfcl.a	|- ( ph -> ( A e. ( F N G ) /\ K e. ( X ( Hom ` C ) Y ) ) )
evlfcl.b	|- ( ph -> ( B e. ( G N H ) /\ L e. ( Y ( Hom ` C ) Z ) ) )

Assertion

Ref	Expression
evlfcllem	|- ( ph -> ( ( <. F , X >. ( 2nd ` E ) <. H , Z >. ) ` ( <. B , L >. ( <. <. F , X >. , <. G , Y >. >. (comp ` ( Q X.c C ) ) <. H , Z >. ) <. A , K >. ) ) = ( ( ( <. G , Y >. ( 2nd ` E ) <. H , Z >. ) ` <. B , L >. ) ( <. ( ( 1st ` E ) ` <. F , X >. ) , ( ( 1st ` E ) ` <. G , Y >. ) >. (comp ` D ) ( ( 1st ` E ) ` <. H , Z >. ) ) ( ( <. F , X >. ( 2nd ` E ) <. G , Y >. ) ` <. A , K >. ) ) )

Proof of Theorem evlfcllem

Step	Hyp	Ref	Expression
1		evlfcl.e	. . . 4 |- E = ( C evalF D )
2		evlfcl.c	. . . 4 |- ( ph -> C e. Cat )
3		evlfcl.d	. . . 4 |- ( ph -> D e. Cat )
4		eqid 2622	. . . 4 |- ( Base ` C ) = ( Base ` C )
5		eqid 2622	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
6		eqid 2622	. . . 4 |- (comp ` D ) = (comp ` D )
7		evlfcl.n	. . . 4 |- N = ( C Nat D )
8		evlfcl.f	. . . . 5 |- ( ph -> ( F e. ( C Func D ) /\ X e. ( Base ` C ) ) )
9	8	simpld 475	. . . 4 |- ( ph -> F e. ( C Func D ) )
10		evlfcl.h	. . . . 5 |- ( ph -> ( H e. ( C Func D ) /\ Z e. ( Base ` C ) ) )
11	10	simpld 475	. . . 4 |- ( ph -> H e. ( C Func D ) )
12	8	simprd 479	. . . 4 |- ( ph -> X e. ( Base ` C ) )
13	10	simprd 479	. . . 4 |- ( ph -> Z e. ( Base ` C ) )
14		eqid 2622	. . . 4 |- ( <. F , X >. ( 2nd ` E ) <. H , Z >. ) = ( <. F , X >. ( 2nd ` E ) <. H , Z >. )
15		evlfcl.q	. . . . 5 |- Q = ( C FuncCat D )
16		eqid 2622	. . . . 5 |- (comp ` Q ) = (comp ` Q )
17		evlfcl.a	. . . . . 6 |- ( ph -> ( A e. ( F N G ) /\ K e. ( X ( Hom ` C ) Y ) ) )
18	17	simpld 475	. . . . 5 |- ( ph -> A e. ( F N G ) )
19		evlfcl.b	. . . . . 6 |- ( ph -> ( B e. ( G N H ) /\ L e. ( Y ( Hom ` C ) Z ) ) )
20	19	simpld 475	. . . . 5 |- ( ph -> B e. ( G N H ) )
21	15, 7, 16, 18, 20	fuccocl 16624	. . . 4 |- ( ph -> ( B ( <. F , G >. (comp ` Q ) H ) A ) e. ( F N H ) )
22		eqid 2622	. . . . 5 |- (comp ` C ) = (comp ` C )
23		evlfcl.g	. . . . . 6 |- ( ph -> ( G e. ( C Func D ) /\ Y e. ( Base ` C ) ) )
24	23	simprd 479	. . . . 5 |- ( ph -> Y e. ( Base ` C ) )
25	17	simprd 479	. . . . 5 |- ( ph -> K e. ( X ( Hom ` C ) Y ) )
26	19	simprd 479	. . . . 5 |- ( ph -> L e. ( Y ( Hom ` C ) Z ) )
27	4, 5, 22, 2, 12, 24, 13, 25, 26	catcocl 16346	. . . 4 |- ( ph -> ( L ( <. X , Y >. (comp ` C ) Z ) K ) e. ( X ( Hom ` C ) Z ) )
28	1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 14, 21, 27	evlf2val 16859	. . 3 |- ( ph -> ( ( B ( <. F , G >. (comp ` Q ) H ) A ) ( <. F , X >. ( 2nd ` E ) <. H , Z >. ) ( L ( <. X , Y >. (comp ` C ) Z ) K ) ) = ( ( ( B ( <. F , G >. (comp ` Q ) H ) A ) ` Z ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( X ( 2nd ` F ) Z ) ` ( L ( <. X , Y >. (comp ` C ) Z ) K ) ) ) )
29	15, 7, 4, 6, 16, 18, 20, 13	fuccoval 16623	. . . 4 |- ( ph -> ( ( B ( <. F , G >. (comp ` Q ) H ) A ) ` Z ) = ( ( B ` Z ) ( <. ( ( 1st ` F ) ` Z ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( A ` Z ) ) )
30	29	oveq1d 6665	. . 3 |- ( ph -> ( ( ( B ( <. F , G >. (comp ` Q ) H ) A ) ` Z ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( X ( 2nd ` F ) Z ) ` ( L ( <. X , Y >. (comp ` C ) Z ) K ) ) ) = ( ( ( B ` Z ) ( <. ( ( 1st ` F ) ` Z ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( A ` Z ) ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( X ( 2nd ` F ) Z ) ` ( L ( <. X , Y >. (comp ` C ) Z ) K ) ) ) )
31		relfunc 16522	. . . . . . 7 |- Rel ( C Func D )
32		1st2ndbr 7217	. . . . . . 7 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
33	31, 9, 32	sylancr 695	. . . . . 6 |- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
34	4, 5, 22, 6, 33, 12, 24, 13, 25, 26	funcco 16531	. . . . 5 |- ( ph -> ( ( X ( 2nd ` F ) Z ) ` ( L ( <. X , Y >. (comp ` C ) Z ) K ) ) = ( ( ( Y ( 2nd ` F ) Z ) ` L ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. (comp ` D ) ( ( 1st ` F ) ` Z ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) )
35	34	oveq2d 6666	. . . 4 |- ( ph -> ( ( ( B ` Z ) ( <. ( ( 1st ` F ) ` Z ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( A ` Z ) ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( X ( 2nd ` F ) Z ) ` ( L ( <. X , Y >. (comp ` C ) Z ) K ) ) ) = ( ( ( B ` Z ) ( <. ( ( 1st ` F ) ` Z ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( A ` Z ) ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( ( Y ( 2nd ` F ) Z ) ` L ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. (comp ` D ) ( ( 1st ` F ) ` Z ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) ) )
36	7, 18	nat1st2nd 16611	. . . . . . . . 9 |- ( ph -> A e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )
37	7, 36, 4, 5, 6, 24, 13, 26	nati 16615	. . . . . . . 8 |- ( ph -> ( ( A ` Z ) ( <. ( ( 1st ` F ) ` Y ) , ( ( 1st ` F ) ` Z ) >. (comp ` D ) ( ( 1st ` G ) ` Z ) ) ( ( Y ( 2nd ` F ) Z ) ` L ) ) = ( ( ( Y ( 2nd ` G ) Z ) ` L ) ( <. ( ( 1st ` F ) ` Y ) , ( ( 1st ` G ) ` Y ) >. (comp ` D ) ( ( 1st ` G ) ` Z ) ) ( A ` Y ) ) )
38	37	oveq2d 6666	. . . . . . 7 |- ( ph -> ( ( B ` Z ) ( <. ( ( 1st ` F ) ` Y ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( A ` Z ) ( <. ( ( 1st ` F ) ` Y ) , ( ( 1st ` F ) ` Z ) >. (comp ` D ) ( ( 1st ` G ) ` Z ) ) ( ( Y ( 2nd ` F ) Z ) ` L ) ) ) = ( ( B ` Z ) ( <. ( ( 1st ` F ) ` Y ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( ( Y ( 2nd ` G ) Z ) ` L ) ( <. ( ( 1st ` F ) ` Y ) , ( ( 1st ` G ) ` Y ) >. (comp ` D ) ( ( 1st ` G ) ` Z ) ) ( A ` Y ) ) ) )
39		eqid 2622	. . . . . . . 8 |- ( Base ` D ) = ( Base ` D )
40		eqid 2622	. . . . . . . 8 |- ( Hom ` D ) = ( Hom ` D )
41	4, 39, 33	funcf1 16526	. . . . . . . . 9 |- ( ph -> ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) )
42	41, 24	ffvelrnd 6360	. . . . . . . 8 |- ( ph -> ( ( 1st ` F ) ` Y ) e. ( Base ` D ) )
43	41, 13	ffvelrnd 6360	. . . . . . . 8 |- ( ph -> ( ( 1st ` F ) ` Z ) e. ( Base ` D ) )
44	23	simpld 475	. . . . . . . . . . 11 |- ( ph -> G e. ( C Func D ) )
45		1st2ndbr 7217	. . . . . . . . . . 11 |- ( ( Rel ( C Func D ) /\ G e. ( C Func D ) ) -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) )
46	31, 44, 45	sylancr 695	. . . . . . . . . 10 |- ( ph -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) )
47	4, 39, 46	funcf1 16526	. . . . . . . . 9 |- ( ph -> ( 1st ` G ) : ( Base ` C ) --> ( Base ` D ) )
48	47, 13	ffvelrnd 6360	. . . . . . . 8 |- ( ph -> ( ( 1st ` G ) ` Z ) e. ( Base ` D ) )
49	4, 5, 40, 33, 24, 13	funcf2 16528	. . . . . . . . 9 |- ( ph -> ( Y ( 2nd ` F ) Z ) : ( Y ( Hom ` C ) Z ) --> ( ( ( 1st ` F ) ` Y ) ( Hom ` D ) ( ( 1st ` F ) ` Z ) ) )
50	49, 26	ffvelrnd 6360	. . . . . . . 8 |- ( ph -> ( ( Y ( 2nd ` F ) Z ) ` L ) e. ( ( ( 1st ` F ) ` Y ) ( Hom ` D ) ( ( 1st ` F ) ` Z ) ) )
51	7, 36, 4, 40, 13	natcl 16613	. . . . . . . 8 |- ( ph -> ( A ` Z ) e. ( ( ( 1st ` F ) ` Z ) ( Hom ` D ) ( ( 1st ` G ) ` Z ) ) )
52		1st2ndbr 7217	. . . . . . . . . . 11 |- ( ( Rel ( C Func D ) /\ H e. ( C Func D ) ) -> ( 1st ` H ) ( C Func D ) ( 2nd ` H ) )
53	31, 11, 52	sylancr 695	. . . . . . . . . 10 |- ( ph -> ( 1st ` H ) ( C Func D ) ( 2nd ` H ) )
54	4, 39, 53	funcf1 16526	. . . . . . . . 9 |- ( ph -> ( 1st ` H ) : ( Base ` C ) --> ( Base ` D ) )
55	54, 13	ffvelrnd 6360	. . . . . . . 8 |- ( ph -> ( ( 1st ` H ) ` Z ) e. ( Base ` D ) )
56	7, 20	nat1st2nd 16611	. . . . . . . . 9 |- ( ph -> B e. ( <. ( 1st ` G ) , ( 2nd ` G ) >. N <. ( 1st ` H ) , ( 2nd ` H ) >. ) )
57	7, 56, 4, 40, 13	natcl 16613	. . . . . . . 8 |- ( ph -> ( B ` Z ) e. ( ( ( 1st ` G ) ` Z ) ( Hom ` D ) ( ( 1st ` H ) ` Z ) ) )
58	39, 40, 6, 3, 42, 43, 48, 50, 51, 55, 57	catass 16347	. . . . . . 7 |- ( ph -> ( ( ( B ` Z ) ( <. ( ( 1st ` F ) ` Z ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( A ` Z ) ) ( <. ( ( 1st ` F ) ` Y ) , ( ( 1st ` F ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( Y ( 2nd ` F ) Z ) ` L ) ) = ( ( B ` Z ) ( <. ( ( 1st ` F ) ` Y ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( A ` Z ) ( <. ( ( 1st ` F ) ` Y ) , ( ( 1st ` F ) ` Z ) >. (comp ` D ) ( ( 1st ` G ) ` Z ) ) ( ( Y ( 2nd ` F ) Z ) ` L ) ) ) )
59	47, 24	ffvelrnd 6360	. . . . . . . 8 |- ( ph -> ( ( 1st ` G ) ` Y ) e. ( Base ` D ) )
60	7, 36, 4, 40, 24	natcl 16613	. . . . . . . 8 |- ( ph -> ( A ` Y ) e. ( ( ( 1st ` F ) ` Y ) ( Hom ` D ) ( ( 1st ` G ) ` Y ) ) )
61	4, 5, 40, 46, 24, 13	funcf2 16528	. . . . . . . . 9 |- ( ph -> ( Y ( 2nd ` G ) Z ) : ( Y ( Hom ` C ) Z ) --> ( ( ( 1st ` G ) ` Y ) ( Hom ` D ) ( ( 1st ` G ) ` Z ) ) )
62	61, 26	ffvelrnd 6360	. . . . . . . 8 |- ( ph -> ( ( Y ( 2nd ` G ) Z ) ` L ) e. ( ( ( 1st ` G ) ` Y ) ( Hom ` D ) ( ( 1st ` G ) ` Z ) ) )
63	39, 40, 6, 3, 42, 59, 48, 60, 62, 55, 57	catass 16347	. . . . . . 7 |- ( ph -> ( ( ( B ` Z ) ( <. ( ( 1st ` G ) ` Y ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( Y ( 2nd ` G ) Z ) ` L ) ) ( <. ( ( 1st ` F ) ` Y ) , ( ( 1st ` G ) ` Y ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( A ` Y ) ) = ( ( B ` Z ) ( <. ( ( 1st ` F ) ` Y ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( ( Y ( 2nd ` G ) Z ) ` L ) ( <. ( ( 1st ` F ) ` Y ) , ( ( 1st ` G ) ` Y ) >. (comp ` D ) ( ( 1st ` G ) ` Z ) ) ( A ` Y ) ) ) )
64	38, 58, 63	3eqtr4d 2666	. . . . . 6 |- ( ph -> ( ( ( B ` Z ) ( <. ( ( 1st ` F ) ` Z ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( A ` Z ) ) ( <. ( ( 1st ` F ) ` Y ) , ( ( 1st ` F ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( Y ( 2nd ` F ) Z ) ` L ) ) = ( ( ( B ` Z ) ( <. ( ( 1st ` G ) ` Y ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( Y ( 2nd ` G ) Z ) ` L ) ) ( <. ( ( 1st ` F ) ` Y ) , ( ( 1st ` G ) ` Y ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( A ` Y ) ) )
65	64	oveq1d 6665	. . . . 5 |- ( ph -> ( ( ( ( B ` Z ) ( <. ( ( 1st ` F ) ` Z ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( A ` Z ) ) ( <. ( ( 1st ` F ) ` Y ) , ( ( 1st ` F ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( Y ( 2nd ` F ) Z ) ` L ) ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) = ( ( ( ( B ` Z ) ( <. ( ( 1st ` G ) ` Y ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( Y ( 2nd ` G ) Z ) ` L ) ) ( <. ( ( 1st ` F ) ` Y ) , ( ( 1st ` G ) ` Y ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( A ` Y ) ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) )
66	41, 12	ffvelrnd 6360	. . . . . 6 |- ( ph -> ( ( 1st ` F ) ` X ) e. ( Base ` D ) )
67	4, 5, 40, 33, 12, 24	funcf2 16528	. . . . . . 7 |- ( ph -> ( X ( 2nd ` F ) Y ) : ( X ( Hom ` C ) Y ) --> ( ( ( 1st ` F ) ` X ) ( Hom ` D ) ( ( 1st ` F ) ` Y ) ) )
68	67, 25	ffvelrnd 6360	. . . . . 6 |- ( ph -> ( ( X ( 2nd ` F ) Y ) ` K ) e. ( ( ( 1st ` F ) ` X ) ( Hom ` D ) ( ( 1st ` F ) ` Y ) ) )
69	39, 40, 6, 3, 43, 48, 55, 51, 57	catcocl 16346	. . . . . 6 |- ( ph -> ( ( B ` Z ) ( <. ( ( 1st ` F ) ` Z ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( A ` Z ) ) e. ( ( ( 1st ` F ) ` Z ) ( Hom ` D ) ( ( 1st ` H ) ` Z ) ) )
70	39, 40, 6, 3, 66, 42, 43, 68, 50, 55, 69	catass 16347	. . . . 5 |- ( ph -> ( ( ( ( B ` Z ) ( <. ( ( 1st ` F ) ` Z ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( A ` Z ) ) ( <. ( ( 1st ` F ) ` Y ) , ( ( 1st ` F ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( Y ( 2nd ` F ) Z ) ` L ) ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) = ( ( ( B ` Z ) ( <. ( ( 1st ` F ) ` Z ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( A ` Z ) ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( ( Y ( 2nd ` F ) Z ) ` L ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. (comp ` D ) ( ( 1st ` F ) ` Z ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) ) )
71	39, 40, 6, 3, 59, 48, 55, 62, 57	catcocl 16346	. . . . . 6 |- ( ph -> ( ( B ` Z ) ( <. ( ( 1st ` G ) ` Y ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( Y ( 2nd ` G ) Z ) ` L ) ) e. ( ( ( 1st ` G ) ` Y ) ( Hom ` D ) ( ( 1st ` H ) ` Z ) ) )
72	39, 40, 6, 3, 66, 42, 59, 68, 60, 55, 71	catass 16347	. . . . 5 |- ( ph -> ( ( ( ( B ` Z ) ( <. ( ( 1st ` G ) ` Y ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( Y ( 2nd ` G ) Z ) ` L ) ) ( <. ( ( 1st ` F ) ` Y ) , ( ( 1st ` G ) ` Y ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( A ` Y ) ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) = ( ( ( B ` Z ) ( <. ( ( 1st ` G ) ` Y ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( Y ( 2nd ` G ) Z ) ` L ) ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` G ) ` Y ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( A ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. (comp ` D ) ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) ) )
73	65, 70, 72	3eqtr3d 2664	. . . 4 |- ( ph -> ( ( ( B ` Z ) ( <. ( ( 1st ` F ) ` Z ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( A ` Z ) ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( ( Y ( 2nd ` F ) Z ) ` L ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. (comp ` D ) ( ( 1st ` F ) ` Z ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) ) = ( ( ( B ` Z ) ( <. ( ( 1st ` G ) ` Y ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( Y ( 2nd ` G ) Z ) ` L ) ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` G ) ` Y ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( A ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. (comp ` D ) ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) ) )
74	35, 73	eqtrd 2656	. . 3 |- ( ph -> ( ( ( B ` Z ) ( <. ( ( 1st ` F ) ` Z ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( A ` Z ) ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( X ( 2nd ` F ) Z ) ` ( L ( <. X , Y >. (comp ` C ) Z ) K ) ) ) = ( ( ( B ` Z ) ( <. ( ( 1st ` G ) ` Y ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( Y ( 2nd ` G ) Z ) ` L ) ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` G ) ` Y ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( A ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. (comp ` D ) ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) ) )
75	28, 30, 74	3eqtrd 2660	. 2 |- ( ph -> ( ( B ( <. F , G >. (comp ` Q ) H ) A ) ( <. F , X >. ( 2nd ` E ) <. H , Z >. ) ( L ( <. X , Y >. (comp ` C ) Z ) K ) ) = ( ( ( B ` Z ) ( <. ( ( 1st ` G ) ` Y ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( Y ( 2nd ` G ) Z ) ` L ) ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` G ) ` Y ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( A ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. (comp ` D ) ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) ) )
76		eqid 2622	. . . . 5 |- ( Q X.c C ) = ( Q X.c C )
77	15	fucbas 16620	. . . . 5 |- ( C Func D ) = ( Base ` Q )
78	15, 7	fuchom 16621	. . . . 5 |- N = ( Hom ` Q )
79		eqid 2622	. . . . 5 |- (comp ` ( Q X.c C ) ) = (comp ` ( Q X.c C ) )
80	76, 77, 4, 78, 5, 9, 12, 44, 24, 16, 22, 79, 11, 13, 18, 25, 20, 26	xpcco2 16827	. . . 4 |- ( ph -> ( <. B , L >. ( <. <. F , X >. , <. G , Y >. >. (comp ` ( Q X.c C ) ) <. H , Z >. ) <. A , K >. ) = <. ( B ( <. F , G >. (comp ` Q ) H ) A ) , ( L ( <. X , Y >. (comp ` C ) Z ) K ) >. )
81	80	fveq2d 6195	. . 3 |- ( ph -> ( ( <. F , X >. ( 2nd ` E ) <. H , Z >. ) ` ( <. B , L >. ( <. <. F , X >. , <. G , Y >. >. (comp ` ( Q X.c C ) ) <. H , Z >. ) <. A , K >. ) ) = ( ( <. F , X >. ( 2nd ` E ) <. H , Z >. ) ` <. ( B ( <. F , G >. (comp ` Q ) H ) A ) , ( L ( <. X , Y >. (comp ` C ) Z ) K ) >. ) )
82		df-ov 6653	. . 3 |- ( ( B ( <. F , G >. (comp ` Q ) H ) A ) ( <. F , X >. ( 2nd ` E ) <. H , Z >. ) ( L ( <. X , Y >. (comp ` C ) Z ) K ) ) = ( ( <. F , X >. ( 2nd ` E ) <. H , Z >. ) ` <. ( B ( <. F , G >. (comp ` Q ) H ) A ) , ( L ( <. X , Y >. (comp ` C ) Z ) K ) >. )
83	81, 82	syl6eqr 2674	. 2 |- ( ph -> ( ( <. F , X >. ( 2nd ` E ) <. H , Z >. ) ` ( <. B , L >. ( <. <. F , X >. , <. G , Y >. >. (comp ` ( Q X.c C ) ) <. H , Z >. ) <. A , K >. ) ) = ( ( B ( <. F , G >. (comp ` Q ) H ) A ) ( <. F , X >. ( 2nd ` E ) <. H , Z >. ) ( L ( <. X , Y >. (comp ` C ) Z ) K ) ) )
84		df-ov 6653	. . . . . 6 |- ( F ( 1st ` E ) X ) = ( ( 1st ` E ) ` <. F , X >. )
85	1, 2, 3, 4, 9, 12	evlf1 16860	. . . . . 6 |- ( ph -> ( F ( 1st ` E ) X ) = ( ( 1st ` F ) ` X ) )
86	84, 85	syl5eqr 2670	. . . . 5 |- ( ph -> ( ( 1st ` E ) ` <. F , X >. ) = ( ( 1st ` F ) ` X ) )
87		df-ov 6653	. . . . . 6 |- ( G ( 1st ` E ) Y ) = ( ( 1st ` E ) ` <. G , Y >. )
88	1, 2, 3, 4, 44, 24	evlf1 16860	. . . . . 6 |- ( ph -> ( G ( 1st ` E ) Y ) = ( ( 1st ` G ) ` Y ) )
89	87, 88	syl5eqr 2670	. . . . 5 |- ( ph -> ( ( 1st ` E ) ` <. G , Y >. ) = ( ( 1st ` G ) ` Y ) )
90	86, 89	opeq12d 4410	. . . 4 |- ( ph -> <. ( ( 1st ` E ) ` <. F , X >. ) , ( ( 1st ` E ) ` <. G , Y >. ) >. = <. ( ( 1st ` F ) ` X ) , ( ( 1st ` G ) ` Y ) >. )
91		df-ov 6653	. . . . 5 |- ( H ( 1st ` E ) Z ) = ( ( 1st ` E ) ` <. H , Z >. )
92	1, 2, 3, 4, 11, 13	evlf1 16860	. . . . 5 |- ( ph -> ( H ( 1st ` E ) Z ) = ( ( 1st ` H ) ` Z ) )
93	91, 92	syl5eqr 2670	. . . 4 |- ( ph -> ( ( 1st ` E ) ` <. H , Z >. ) = ( ( 1st ` H ) ` Z ) )
94	90, 93	oveq12d 6668	. . 3 |- ( ph -> ( <. ( ( 1st ` E ) ` <. F , X >. ) , ( ( 1st ` E ) ` <. G , Y >. ) >. (comp ` D ) ( ( 1st ` E ) ` <. H , Z >. ) ) = ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` G ) ` Y ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) )
95		df-ov 6653	. . . 4 |- ( B ( <. G , Y >. ( 2nd ` E ) <. H , Z >. ) L ) = ( ( <. G , Y >. ( 2nd ` E ) <. H , Z >. ) ` <. B , L >. )
96		eqid 2622	. . . . 5 |- ( <. G , Y >. ( 2nd ` E ) <. H , Z >. ) = ( <. G , Y >. ( 2nd ` E ) <. H , Z >. )
97	1, 2, 3, 4, 5, 6, 7, 44, 11, 24, 13, 96, 20, 26	evlf2val 16859	. . . 4 |- ( ph -> ( B ( <. G , Y >. ( 2nd ` E ) <. H , Z >. ) L ) = ( ( B ` Z ) ( <. ( ( 1st ` G ) ` Y ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( Y ( 2nd ` G ) Z ) ` L ) ) )
98	95, 97	syl5eqr 2670	. . 3 |- ( ph -> ( ( <. G , Y >. ( 2nd ` E ) <. H , Z >. ) ` <. B , L >. ) = ( ( B ` Z ) ( <. ( ( 1st ` G ) ` Y ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( Y ( 2nd ` G ) Z ) ` L ) ) )
99		df-ov 6653	. . . 4 |- ( A ( <. F , X >. ( 2nd ` E ) <. G , Y >. ) K ) = ( ( <. F , X >. ( 2nd ` E ) <. G , Y >. ) ` <. A , K >. )
100		eqid 2622	. . . . 5 |- ( <. F , X >. ( 2nd ` E ) <. G , Y >. ) = ( <. F , X >. ( 2nd ` E ) <. G , Y >. )
101	1, 2, 3, 4, 5, 6, 7, 9, 44, 12, 24, 100, 18, 25	evlf2val 16859	. . . 4 |- ( ph -> ( A ( <. F , X >. ( 2nd ` E ) <. G , Y >. ) K ) = ( ( A ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. (comp ` D ) ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) )
102	99, 101	syl5eqr 2670	. . 3 |- ( ph -> ( ( <. F , X >. ( 2nd ` E ) <. G , Y >. ) ` <. A , K >. ) = ( ( A ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. (comp ` D ) ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) )
103	94, 98, 102	oveq123d 6671	. 2 |- ( ph -> ( ( ( <. G , Y >. ( 2nd ` E ) <. H , Z >. ) ` <. B , L >. ) ( <. ( ( 1st ` E ) ` <. F , X >. ) , ( ( 1st ` E ) ` <. G , Y >. ) >. (comp ` D ) ( ( 1st ` E ) ` <. H , Z >. ) ) ( ( <. F , X >. ( 2nd ` E ) <. G , Y >. ) ` <. A , K >. ) ) = ( ( ( B ` Z ) ( <. ( ( 1st ` G ) ` Y ) , ( ( 1st ` G ) ` Z ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( Y ( 2nd ` G ) Z ) ` L ) ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` G ) ` Y ) >. (comp ` D ) ( ( 1st ` H ) ` Z ) ) ( ( A ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. (comp ` D ) ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) ) )
104	75, 83, 103	3eqtr4d 2666	1 |- ( ph -> ( ( <. F , X >. ( 2nd ` E ) <. H , Z >. ) ` ( <. B , L >. ( <. <. F , X >. , <. G , Y >. >. (comp ` ( Q X.c C ) ) <. H , Z >. ) <. A , K >. ) ) = ( ( ( <. G , Y >. ( 2nd ` E ) <. H , Z >. ) ` <. B , L >. ) ( <. ( ( 1st ` E ) ` <. F , X >. ) , ( ( 1st ` E ) ` <. G , Y >. ) >. (comp ` D ) ( ( 1st ` E ) ` <. H , Z >. ) ) ( ( <. F , X >. ( 2nd ` E ) <. G , Y >. ) ` <. A , K >. ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Func cfunc 16514   Nat cnat 16601   FuncCat cfuc 16602   X._c cxpc 16808   eval_F cevlf 16849

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-map 7859  df-ixp 7909  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-cat 16329  df-func 16518  df-nat 16603  df-fuc 16604  df-xpc 16812  df-evlf 16853

This theorem is referenced by:  evlfcl  16862