Theorem hofcllem 16898

Description: Lemma for hofcl 16899. (Contributed by Mario Carneiro, 15-Jan-2017.)

Hypotheses

Ref	Expression
hofcl.m	|- M = (HomF ` C )
hofcl.o	|- O = (oppCat ` C )
hofcl.d	|- D = ( SetCat ` U )
hofcl.c	|- ( ph -> C e. Cat )
hofcl.u	|- ( ph -> U e. V )
hofcl.h	|- ( ph -> ran ( Hom f ` C ) C_ U )
hofcllem.b	|- B = ( Base ` C )
hofcllem.h	|- H = ( Hom ` C )
hofcllem.x	|- ( ph -> X e. B )
hofcllem.y	|- ( ph -> Y e. B )
hofcllem.z	|- ( ph -> Z e. B )
hofcllem.w	|- ( ph -> W e. B )
hofcllem.s	|- ( ph -> S e. B )
hofcllem.t	|- ( ph -> T e. B )
hofcllem.m	|- ( ph -> K e. ( Z H X ) )
hofcllem.n	|- ( ph -> L e. ( Y H W ) )
hofcllem.p	|- ( ph -> P e. ( S H Z ) )
hofcllem.q	|- ( ph -> Q e. ( W H T ) )

Assertion

Ref	Expression
hofcllem	|- ( ph -> ( ( K ( <. S , Z >. (comp ` C ) X ) P ) ( <. X , Y >. ( 2nd ` M ) <. S , T >. ) ( Q ( <. Y , W >. (comp ` C ) T ) L ) ) = ( ( P ( <. Z , W >. ( 2nd ` M ) <. S , T >. ) Q ) ( <. ( X H Y ) , ( Z H W ) >. (comp ` D ) ( S H T ) ) ( K ( <. X , Y >. ( 2nd ` M ) <. Z , W >. ) L ) ) )

Proof of Theorem hofcllem

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

Step	Hyp	Ref	Expression
1		hofcllem.b	. . . . 5 |- B = ( Base ` C )
2		hofcllem.h	. . . . 5 |- H = ( Hom ` C )
3		eqid 2622	. . . . 5 |- (comp ` C ) = (comp ` C )
4		hofcl.c	. . . . . 6 |- ( ph -> C e. Cat )
5	4	adantr 481	. . . . 5 |- ( ( ph /\ f e. ( X H Y ) ) -> C e. Cat )
6		hofcllem.s	. . . . . 6 |- ( ph -> S e. B )
7	6	adantr 481	. . . . 5 |- ( ( ph /\ f e. ( X H Y ) ) -> S e. B )
8		hofcllem.z	. . . . . 6 |- ( ph -> Z e. B )
9	8	adantr 481	. . . . 5 |- ( ( ph /\ f e. ( X H Y ) ) -> Z e. B )
10		hofcllem.x	. . . . . 6 |- ( ph -> X e. B )
11	10	adantr 481	. . . . 5 |- ( ( ph /\ f e. ( X H Y ) ) -> X e. B )
12		hofcllem.p	. . . . . 6 |- ( ph -> P e. ( S H Z ) )
13	12	adantr 481	. . . . 5 |- ( ( ph /\ f e. ( X H Y ) ) -> P e. ( S H Z ) )
14		hofcllem.m	. . . . . 6 |- ( ph -> K e. ( Z H X ) )
15	14	adantr 481	. . . . 5 |- ( ( ph /\ f e. ( X H Y ) ) -> K e. ( Z H X ) )
16		hofcllem.t	. . . . . 6 |- ( ph -> T e. B )
17	16	adantr 481	. . . . 5 |- ( ( ph /\ f e. ( X H Y ) ) -> T e. B )
18		hofcllem.y	. . . . . . 7 |- ( ph -> Y e. B )
19	18	adantr 481	. . . . . 6 |- ( ( ph /\ f e. ( X H Y ) ) -> Y e. B )
20		simpr 477	. . . . . 6 |- ( ( ph /\ f e. ( X H Y ) ) -> f e. ( X H Y ) )
21		hofcllem.w	. . . . . . . 8 |- ( ph -> W e. B )
22		hofcllem.n	. . . . . . . 8 |- ( ph -> L e. ( Y H W ) )
23		hofcllem.q	. . . . . . . 8 |- ( ph -> Q e. ( W H T ) )
24	1, 2, 3, 4, 18, 21, 16, 22, 23	catcocl 16346	. . . . . . 7 |- ( ph -> ( Q ( <. Y , W >. (comp ` C ) T ) L ) e. ( Y H T ) )
25	24	adantr 481	. . . . . 6 |- ( ( ph /\ f e. ( X H Y ) ) -> ( Q ( <. Y , W >. (comp ` C ) T ) L ) e. ( Y H T ) )
26	1, 2, 3, 5, 11, 19, 17, 20, 25	catcocl 16346	. . . . 5 |- ( ( ph /\ f e. ( X H Y ) ) -> ( ( Q ( <. Y , W >. (comp ` C ) T ) L ) ( <. X , Y >. (comp ` C ) T ) f ) e. ( X H T ) )
27	1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 26	catass 16347	. . . 4 |- ( ( ph /\ f e. ( X H Y ) ) -> ( ( ( ( Q ( <. Y , W >. (comp ` C ) T ) L ) ( <. X , Y >. (comp ` C ) T ) f ) ( <. Z , X >. (comp ` C ) T ) K ) ( <. S , Z >. (comp ` C ) T ) P ) = ( ( ( Q ( <. Y , W >. (comp ` C ) T ) L ) ( <. X , Y >. (comp ` C ) T ) f ) ( <. S , X >. (comp ` C ) T ) ( K ( <. S , Z >. (comp ` C ) X ) P ) ) )
28	21	adantr 481	. . . . . . . 8 |- ( ( ph /\ f e. ( X H Y ) ) -> W e. B )
29	22	adantr 481	. . . . . . . 8 |- ( ( ph /\ f e. ( X H Y ) ) -> L e. ( Y H W ) )
30	23	adantr 481	. . . . . . . 8 |- ( ( ph /\ f e. ( X H Y ) ) -> Q e. ( W H T ) )
31	1, 2, 3, 5, 11, 19, 28, 20, 29, 17, 30	catass 16347	. . . . . . 7 |- ( ( ph /\ f e. ( X H Y ) ) -> ( ( Q ( <. Y , W >. (comp ` C ) T ) L ) ( <. X , Y >. (comp ` C ) T ) f ) = ( Q ( <. X , W >. (comp ` C ) T ) ( L ( <. X , Y >. (comp ` C ) W ) f ) ) )
32	31	oveq1d 6665	. . . . . 6 |- ( ( ph /\ f e. ( X H Y ) ) -> ( ( ( Q ( <. Y , W >. (comp ` C ) T ) L ) ( <. X , Y >. (comp ` C ) T ) f ) ( <. Z , X >. (comp ` C ) T ) K ) = ( ( Q ( <. X , W >. (comp ` C ) T ) ( L ( <. X , Y >. (comp ` C ) W ) f ) ) ( <. Z , X >. (comp ` C ) T ) K ) )
33	1, 2, 3, 5, 11, 19, 28, 20, 29	catcocl 16346	. . . . . . 7 |- ( ( ph /\ f e. ( X H Y ) ) -> ( L ( <. X , Y >. (comp ` C ) W ) f ) e. ( X H W ) )
34	1, 2, 3, 5, 9, 11, 28, 15, 33, 17, 30	catass 16347	. . . . . 6 |- ( ( ph /\ f e. ( X H Y ) ) -> ( ( Q ( <. X , W >. (comp ` C ) T ) ( L ( <. X , Y >. (comp ` C ) W ) f ) ) ( <. Z , X >. (comp ` C ) T ) K ) = ( Q ( <. Z , W >. (comp ` C ) T ) ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) ) )
35	32, 34	eqtrd 2656	. . . . 5 |- ( ( ph /\ f e. ( X H Y ) ) -> ( ( ( Q ( <. Y , W >. (comp ` C ) T ) L ) ( <. X , Y >. (comp ` C ) T ) f ) ( <. Z , X >. (comp ` C ) T ) K ) = ( Q ( <. Z , W >. (comp ` C ) T ) ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) ) )
36	35	oveq1d 6665	. . . 4 |- ( ( ph /\ f e. ( X H Y ) ) -> ( ( ( ( Q ( <. Y , W >. (comp ` C ) T ) L ) ( <. X , Y >. (comp ` C ) T ) f ) ( <. Z , X >. (comp ` C ) T ) K ) ( <. S , Z >. (comp ` C ) T ) P ) = ( ( Q ( <. Z , W >. (comp ` C ) T ) ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) ) ( <. S , Z >. (comp ` C ) T ) P ) )
37	27, 36	eqtr3d 2658	. . 3 |- ( ( ph /\ f e. ( X H Y ) ) -> ( ( ( Q ( <. Y , W >. (comp ` C ) T ) L ) ( <. X , Y >. (comp ` C ) T ) f ) ( <. S , X >. (comp ` C ) T ) ( K ( <. S , Z >. (comp ` C ) X ) P ) ) = ( ( Q ( <. Z , W >. (comp ` C ) T ) ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) ) ( <. S , Z >. (comp ` C ) T ) P ) )
38	37	mpteq2dva 4744	. 2 |- ( ph -> ( f e. ( X H Y ) |-> ( ( ( Q ( <. Y , W >. (comp ` C ) T ) L ) ( <. X , Y >. (comp ` C ) T ) f ) ( <. S , X >. (comp ` C ) T ) ( K ( <. S , Z >. (comp ` C ) X ) P ) ) ) = ( f e. ( X H Y ) |-> ( ( Q ( <. Z , W >. (comp ` C ) T ) ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) ) ( <. S , Z >. (comp ` C ) T ) P ) ) )
39		hofcl.m	. . 3 |- M = (HomF ` C )
40	1, 2, 3, 4, 6, 8, 10, 12, 14	catcocl 16346	. . 3 |- ( ph -> ( K ( <. S , Z >. (comp ` C ) X ) P ) e. ( S H X ) )
41	39, 4, 1, 2, 10, 18, 6, 16, 3, 40, 24	hof2val 16896	. 2 |- ( ph -> ( ( K ( <. S , Z >. (comp ` C ) X ) P ) ( <. X , Y >. ( 2nd ` M ) <. S , T >. ) ( Q ( <. Y , W >. (comp ` C ) T ) L ) ) = ( f e. ( X H Y ) |-> ( ( ( Q ( <. Y , W >. (comp ` C ) T ) L ) ( <. X , Y >. (comp ` C ) T ) f ) ( <. S , X >. (comp ` C ) T ) ( K ( <. S , Z >. (comp ` C ) X ) P ) ) ) )
42	39, 4, 1, 2, 8, 21, 6, 16, 3, 12, 23	hof2val 16896	. . . 4 |- ( ph -> ( P ( <. Z , W >. ( 2nd ` M ) <. S , T >. ) Q ) = ( g e. ( Z H W ) |-> ( ( Q ( <. Z , W >. (comp ` C ) T ) g ) ( <. S , Z >. (comp ` C ) T ) P ) ) )
43	39, 4, 1, 2, 10, 18, 8, 21, 3, 14, 22	hof2val 16896	. . . 4 |- ( ph -> ( K ( <. X , Y >. ( 2nd ` M ) <. Z , W >. ) L ) = ( f e. ( X H Y ) |-> ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) ) )
44	42, 43	oveq12d 6668	. . 3 |- ( ph -> ( ( P ( <. Z , W >. ( 2nd ` M ) <. S , T >. ) Q ) ( <. ( X H Y ) , ( Z H W ) >. (comp ` D ) ( S H T ) ) ( K ( <. X , Y >. ( 2nd ` M ) <. Z , W >. ) L ) ) = ( ( g e. ( Z H W ) |-> ( ( Q ( <. Z , W >. (comp ` C ) T ) g ) ( <. S , Z >. (comp ` C ) T ) P ) ) ( <. ( X H Y ) , ( Z H W ) >. (comp ` D ) ( S H T ) ) ( f e. ( X H Y ) |-> ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) ) ) )
45		hofcl.d	. . . 4 |- D = ( SetCat ` U )
46		hofcl.u	. . . 4 |- ( ph -> U e. V )
47		eqid 2622	. . . 4 |- (comp ` D ) = (comp ` D )
48		eqid 2622	. . . . . 6 |- ( Hom f ` C ) = ( Hom f ` C )
49	48, 1, 2, 10, 18	homfval 16352	. . . . 5 |- ( ph -> ( X ( Hom f ` C ) Y ) = ( X H Y ) )
50	48, 1	homffn 16353	. . . . . . . 8 |- ( Hom f ` C ) Fn ( B X. B )
51	50	a1i 11	. . . . . . 7 |- ( ph -> ( Hom f ` C ) Fn ( B X. B ) )
52		hofcl.h	. . . . . . 7 |- ( ph -> ran ( Hom f ` C ) C_ U )
53		df-f 5892	. . . . . . 7 |- ( ( Hom f ` C ) : ( B X. B ) --> U <-> ( ( Hom f ` C ) Fn ( B X. B ) /\ ran ( Hom f ` C ) C_ U ) )
54	51, 52, 53	sylanbrc 698	. . . . . 6 |- ( ph -> ( Hom f ` C ) : ( B X. B ) --> U )
55	54, 10, 18	fovrnd 6806	. . . . 5 |- ( ph -> ( X ( Hom f ` C ) Y ) e. U )
56	49, 55	eqeltrrd 2702	. . . 4 |- ( ph -> ( X H Y ) e. U )
57	48, 1, 2, 8, 21	homfval 16352	. . . . 5 |- ( ph -> ( Z ( Hom f ` C ) W ) = ( Z H W ) )
58	54, 8, 21	fovrnd 6806	. . . . 5 |- ( ph -> ( Z ( Hom f ` C ) W ) e. U )
59	57, 58	eqeltrrd 2702	. . . 4 |- ( ph -> ( Z H W ) e. U )
60	48, 1, 2, 6, 16	homfval 16352	. . . . 5 |- ( ph -> ( S ( Hom f ` C ) T ) = ( S H T ) )
61	54, 6, 16	fovrnd 6806	. . . . 5 |- ( ph -> ( S ( Hom f ` C ) T ) e. U )
62	60, 61	eqeltrrd 2702	. . . 4 |- ( ph -> ( S H T ) e. U )
63	1, 2, 3, 5, 9, 11, 28, 15, 33	catcocl 16346	. . . . 5 |- ( ( ph /\ f e. ( X H Y ) ) -> ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) e. ( Z H W ) )
64		eqid 2622	. . . . 5 |- ( f e. ( X H Y ) |-> ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) ) = ( f e. ( X H Y ) |-> ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) )
65	63, 64	fmptd 6385	. . . 4 |- ( ph -> ( f e. ( X H Y ) |-> ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) ) : ( X H Y ) --> ( Z H W ) )
66	4	adantr 481	. . . . . 6 |- ( ( ph /\ g e. ( Z H W ) ) -> C e. Cat )
67	6	adantr 481	. . . . . 6 |- ( ( ph /\ g e. ( Z H W ) ) -> S e. B )
68	8	adantr 481	. . . . . 6 |- ( ( ph /\ g e. ( Z H W ) ) -> Z e. B )
69	16	adantr 481	. . . . . 6 |- ( ( ph /\ g e. ( Z H W ) ) -> T e. B )
70	12	adantr 481	. . . . . 6 |- ( ( ph /\ g e. ( Z H W ) ) -> P e. ( S H Z ) )
71	21	adantr 481	. . . . . . 7 |- ( ( ph /\ g e. ( Z H W ) ) -> W e. B )
72		simpr 477	. . . . . . 7 |- ( ( ph /\ g e. ( Z H W ) ) -> g e. ( Z H W ) )
73	23	adantr 481	. . . . . . 7 |- ( ( ph /\ g e. ( Z H W ) ) -> Q e. ( W H T ) )
74	1, 2, 3, 66, 68, 71, 69, 72, 73	catcocl 16346	. . . . . 6 |- ( ( ph /\ g e. ( Z H W ) ) -> ( Q ( <. Z , W >. (comp ` C ) T ) g ) e. ( Z H T ) )
75	1, 2, 3, 66, 67, 68, 69, 70, 74	catcocl 16346	. . . . 5 |- ( ( ph /\ g e. ( Z H W ) ) -> ( ( Q ( <. Z , W >. (comp ` C ) T ) g ) ( <. S , Z >. (comp ` C ) T ) P ) e. ( S H T ) )
76		eqid 2622	. . . . 5 |- ( g e. ( Z H W ) |-> ( ( Q ( <. Z , W >. (comp ` C ) T ) g ) ( <. S , Z >. (comp ` C ) T ) P ) ) = ( g e. ( Z H W ) |-> ( ( Q ( <. Z , W >. (comp ` C ) T ) g ) ( <. S , Z >. (comp ` C ) T ) P ) )
77	75, 76	fmptd 6385	. . . 4 |- ( ph -> ( g e. ( Z H W ) |-> ( ( Q ( <. Z , W >. (comp ` C ) T ) g ) ( <. S , Z >. (comp ` C ) T ) P ) ) : ( Z H W ) --> ( S H T ) )
78	45, 46, 47, 56, 59, 62, 65, 77	setcco 16733	. . 3 |- ( ph -> ( ( g e. ( Z H W ) |-> ( ( Q ( <. Z , W >. (comp ` C ) T ) g ) ( <. S , Z >. (comp ` C ) T ) P ) ) ( <. ( X H Y ) , ( Z H W ) >. (comp ` D ) ( S H T ) ) ( f e. ( X H Y ) |-> ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) ) ) = ( ( g e. ( Z H W ) |-> ( ( Q ( <. Z , W >. (comp ` C ) T ) g ) ( <. S , Z >. (comp ` C ) T ) P ) ) o. ( f e. ( X H Y ) |-> ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) ) ) )
79		eqidd 2623	. . . 4 |- ( ph -> ( f e. ( X H Y ) |-> ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) ) = ( f e. ( X H Y ) |-> ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) ) )
80		eqidd 2623	. . . 4 |- ( ph -> ( g e. ( Z H W ) |-> ( ( Q ( <. Z , W >. (comp ` C ) T ) g ) ( <. S , Z >. (comp ` C ) T ) P ) ) = ( g e. ( Z H W ) |-> ( ( Q ( <. Z , W >. (comp ` C ) T ) g ) ( <. S , Z >. (comp ` C ) T ) P ) ) )
81		oveq2 6658	. . . . 5 |- ( g = ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) -> ( Q ( <. Z , W >. (comp ` C ) T ) g ) = ( Q ( <. Z , W >. (comp ` C ) T ) ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) ) )
82	81	oveq1d 6665	. . . 4 |- ( g = ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) -> ( ( Q ( <. Z , W >. (comp ` C ) T ) g ) ( <. S , Z >. (comp ` C ) T ) P ) = ( ( Q ( <. Z , W >. (comp ` C ) T ) ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) ) ( <. S , Z >. (comp ` C ) T ) P ) )
83	63, 79, 80, 82	fmptco 6396	. . 3 |- ( ph -> ( ( g e. ( Z H W ) |-> ( ( Q ( <. Z , W >. (comp ` C ) T ) g ) ( <. S , Z >. (comp ` C ) T ) P ) ) o. ( f e. ( X H Y ) |-> ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) ) ) = ( f e. ( X H Y ) |-> ( ( Q ( <. Z , W >. (comp ` C ) T ) ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) ) ( <. S , Z >. (comp ` C ) T ) P ) ) )
84	44, 78, 83	3eqtrd 2660	. 2 |- ( ph -> ( ( P ( <. Z , W >. ( 2nd ` M ) <. S , T >. ) Q ) ( <. ( X H Y ) , ( Z H W ) >. (comp ` D ) ( S H T ) ) ( K ( <. X , Y >. ( 2nd ` M ) <. Z , W >. ) L ) ) = ( f e. ( X H Y ) |-> ( ( Q ( <. Z , W >. (comp ` C ) T ) ( ( L ( <. X , Y >. (comp ` C ) W ) f ) ( <. Z , X >. (comp ` C ) W ) K ) ) ( <. S , Z >. (comp ` C ) T ) P ) ) )
85	38, 41, 84	3eqtr4d 2666	1 |- ( ph -> ( ( K ( <. S , Z >. (comp ` C ) X ) P ) ( <. X , Y >. ( 2nd ` M ) <. S , T >. ) ( Q ( <. Y , W >. (comp ` C ) T ) L ) ) = ( ( P ( <. Z , W >. ( 2nd ` M ) <. S , T >. ) Q ) ( <. ( X H Y ) , ( Z H W ) >. (comp ` D ) ( S H T ) ) ( K ( <. X , Y >. ( 2nd ` M ) <. Z , W >. ) L ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   <.cop 4183   |-> cmpt 4729   X. cxp 5112   ran crn 5115   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   2ndc2nd 7167   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Hom _f chomf 16327  oppCatcoppc 16371   SetCatcsetc 16725  Hom_Fchof 16888

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-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-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-homf 16331  df-setc 16726  df-hof 16890

This theorem is referenced by:  hofcl  16899