Theorem dprdres 16642

Description: Restriction of a direct product (dropping factors). (Contributed by Mario Carneiro, 25-Apr-2016.)

Hypotheses

Ref	Expression
dprdres.1	|- ( ph -> G dom DProd S )
dprdres.2	|- ( ph -> dom S = I )
dprdres.3	|- ( ph -> A C_ I )

Assertion

Ref	Expression
dprdres	|- ( ph -> ( G dom DProd ( S |` A ) /\ ( G DProd ( S |` A ) ) C_ ( G DProd S ) ) )

Proof of Theorem dprdres

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

Step	Hyp	Ref	Expression
1		dprdres.1	. . . 4 |- ( ph -> G dom DProd S )
2		dprdgrp 16606	. . . 4 |- ( G dom DProd S -> G e. Grp )
3	1, 2	syl 16	. . 3 |- ( ph -> G e. Grp )
4		dprdres.2	. . . . 5 |- ( ph -> dom S = I )
5	1, 4	dprdf2 16608	. . . 4 |- ( ph -> S : I --> (SubGrp ` G ) )
6		dprdres.3	. . . 4 |- ( ph -> A C_ I )
7		fssres 5681	. . . 4 |- ( ( S : I --> (SubGrp ` G ) /\ A C_ I ) -> ( S |` A ) : A --> (SubGrp ` G ) )
8	5, 6, 7	syl2anc 661	. . 3 |- ( ph -> ( S |` A ) : A --> (SubGrp ` G ) )
9	1	ad2antrr 725	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> G dom DProd S )
10	4	ad2antrr 725	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> dom S = I )
11	6	ad2antrr 725	. . . . . . . . 9 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> A C_ I )
12		simplr 754	. . . . . . . . 9 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> x e. A )
13	11, 12	sseldd 3460	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> x e. I )
14		eldifi 3581	. . . . . . . . . 10 |- ( y e. ( A \ { x } ) -> y e. A )
15	14	adantl 466	. . . . . . . . 9 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> y e. A )
16	11, 15	sseldd 3460	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> y e. I )
17		eldifsni 4104	. . . . . . . . . 10 |- ( y e. ( A \ { x } ) -> y =/= x )
18	17	adantl 466	. . . . . . . . 9 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> y =/= x )
19	18	necomd 2720	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> x =/= y )
20		eqid 2452	. . . . . . . 8 |- (Cntz ` G ) = (Cntz ` G )
21	9, 10, 13, 16, 19, 20	dprdcntz 16609	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) )
22		fvres 5808	. . . . . . . 8 |- ( x e. A -> ( ( S |` A ) ` x ) = ( S ` x ) )
23	12, 22	syl 16	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> ( ( S |` A ) ` x ) = ( S ` x ) )
24		fvres 5808	. . . . . . . . 9 |- ( y e. A -> ( ( S |` A ) ` y ) = ( S ` y ) )
25	15, 24	syl 16	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> ( ( S |` A ) ` y ) = ( S ` y ) )
26	25	fveq2d 5798	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> ( (Cntz ` G ) ` ( ( S |` A ) ` y ) ) = ( (Cntz ` G ) ` ( S ` y ) ) )
27	21, 23, 26	3sstr4d 3502	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> ( ( S |` A ) ` x ) C_ ( (Cntz ` G ) ` ( ( S |` A ) ` y ) ) )
28	27	ralrimiva 2827	. . . . 5 |- ( ( ph /\ x e. A ) -> A. y e. ( A \ { x } ) ( ( S |` A ) ` x ) C_ ( (Cntz ` G ) ` ( ( S |` A ) ` y ) ) )
29	22	adantl 466	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( ( S |` A ) ` x ) = ( S ` x ) )
30	29	ineq1d 3654	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( ( ( S |` A ) ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) ) = ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) ) )
31		eqid 2452	. . . . . . . . . . . . 13 |- ( Base ` G ) = ( Base ` G )
32	31	subgacs 15830	. . . . . . . . . . . 12 |- ( G e. Grp -> (SubGrp ` G ) e. (ACS ` ( Base ` G ) ) )
33		acsmre 14704	. . . . . . . . . . . 12 |- ( (SubGrp ` G ) e. (ACS ` ( Base ` G ) ) -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
34	3, 32, 33	3syl 20	. . . . . . . . . . 11 |- ( ph -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
35	34	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
36		eqid 2452	. . . . . . . . . 10 |- (mrCls ` (SubGrp ` G ) ) = (mrCls ` (SubGrp ` G ) )
37		resss 5237	. . . . . . . . . . . . 13 |- ( S |` A ) C_ S
38		imass1 5306	. . . . . . . . . . . . 13 |- ( ( S |` A ) C_ S -> ( ( S |` A ) " ( A \ { x } ) ) C_ ( S " ( A \ { x } ) ) )
39	37, 38	ax-mp 5	. . . . . . . . . . . 12 |- ( ( S |` A ) " ( A \ { x } ) ) C_ ( S " ( A \ { x } ) )
40	6	adantr 465	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. A ) -> A C_ I )
41		ssdif 3594	. . . . . . . . . . . . 13 |- ( A C_ I -> ( A \ { x } ) C_ ( I \ { x } ) )
42		imass2 5307	. . . . . . . . . . . . 13 |- ( ( A \ { x } ) C_ ( I \ { x } ) -> ( S " ( A \ { x } ) ) C_ ( S " ( I \ { x } ) ) )
43	40, 41, 42	3syl 20	. . . . . . . . . . . 12 |- ( ( ph /\ x e. A ) -> ( S " ( A \ { x } ) ) C_ ( S " ( I \ { x } ) ) )
44	39, 43	syl5ss 3470	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( ( S |` A ) " ( A \ { x } ) ) C_ ( S " ( I \ { x } ) ) )
45		uniss 4215	. . . . . . . . . . 11 |- ( ( ( S |` A ) " ( A \ { x } ) ) C_ ( S " ( I \ { x } ) ) -> U. ( ( S |` A ) " ( A \ { x } ) ) C_ U. ( S " ( I \ { x } ) ) )
46	44, 45	syl 16	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> U. ( ( S |` A ) " ( A \ { x } ) ) C_ U. ( S " ( I \ { x } ) ) )
47		imassrn 5283	. . . . . . . . . . . 12 |- ( S " ( I \ { x } ) ) C_ ran S
48		frn 5668	. . . . . . . . . . . . . . 15 |- ( S : I --> (SubGrp ` G ) -> ran S C_ (SubGrp ` G ) )
49	5, 48	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> ran S C_ (SubGrp ` G ) )
50	31	subgss 15796	. . . . . . . . . . . . . . . . 17 |- ( x e. (SubGrp ` G ) -> x C_ ( Base ` G ) )
51		selpw 3970	. . . . . . . . . . . . . . . . 17 |- ( x e. ~P ( Base ` G ) <-> x C_ ( Base ` G ) )
52	50, 51	sylibr 212	. . . . . . . . . . . . . . . 16 |- ( x e. (SubGrp ` G ) -> x e. ~P ( Base ` G ) )
53	52	ssriv 3463	. . . . . . . . . . . . . . 15 |- (SubGrp ` G ) C_ ~P ( Base ` G )
54	53	a1i 11	. . . . . . . . . . . . . 14 |- ( ph -> (SubGrp ` G ) C_ ~P ( Base ` G ) )
55	49, 54	sstrd 3469	. . . . . . . . . . . . 13 |- ( ph -> ran S C_ ~P ( Base ` G ) )
56	55	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ x e. A ) -> ran S C_ ~P ( Base ` G ) )
57	47, 56	syl5ss 3470	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( S " ( I \ { x } ) ) C_ ~P ( Base ` G ) )
58		sspwuni 4359	. . . . . . . . . . 11 |- ( ( S " ( I \ { x } ) ) C_ ~P ( Base ` G ) <-> U. ( S " ( I \ { x } ) ) C_ ( Base ` G ) )
59	57, 58	sylib 196	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> U. ( S " ( I \ { x } ) ) C_ ( Base ` G ) )
60	35, 36, 46, 59	mrcssd 14676	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) C_ ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) )
61		sslin 3679	. . . . . . . . 9 |- ( ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) C_ ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) -> ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) ) C_ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) )
62	60, 61	syl 16	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) ) C_ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) )
63	1	adantr 465	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> G dom DProd S )
64	4	adantr 465	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> dom S = I )
65	6	sselda 3459	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> x e. I )
66		eqid 2452	. . . . . . . . 9 |- ( 0g ` G ) = ( 0g ` G )
67	63, 64, 65, 66, 36	dprddisj 16610	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) = { ( 0g ` G ) } )
68	62, 67	sseqtrd 3495	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) ) C_ { ( 0g ` G ) } )
69	5	ffvelrnda 5947	. . . . . . . . . . 11 |- ( ( ph /\ x e. I ) -> ( S ` x ) e. (SubGrp ` G ) )
70	65, 69	syldan 470	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( S ` x ) e. (SubGrp ` G ) )
71	66	subg0cl 15803	. . . . . . . . . 10 |- ( ( S ` x ) e. (SubGrp ` G ) -> ( 0g ` G ) e. ( S ` x ) )
72	70, 71	syl 16	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( 0g ` G ) e. ( S ` x ) )
73	46, 59	sstrd 3469	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> U. ( ( S |` A ) " ( A \ { x } ) ) C_ ( Base ` G ) )
74	36	mrccl 14663	. . . . . . . . . . 11 |- ( ( (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) /\ U. ( ( S |` A ) " ( A \ { x } ) ) C_ ( Base ` G ) ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) e. (SubGrp ` G ) )
75	35, 73, 74	syl2anc 661	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) e. (SubGrp ` G ) )
76	66	subg0cl 15803	. . . . . . . . . 10 |- ( ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) e. (SubGrp ` G ) -> ( 0g ` G ) e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) )
77	75, 76	syl 16	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( 0g ` G ) e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) )
78	72, 77	elind 3643	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( 0g ` G ) e. ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) ) )
79	78	snssd 4121	. . . . . . 7 |- ( ( ph /\ x e. A ) -> { ( 0g ` G ) } C_ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) ) )
80	68, 79	eqssd 3476	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) ) = { ( 0g ` G ) } )
81	30, 80	eqtrd 2493	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ( ( S |` A ) ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) ) = { ( 0g ` G ) } )
82	28, 81	jca 532	. . . 4 |- ( ( ph /\ x e. A ) -> ( A. y e. ( A \ { x } ) ( ( S |` A ) ` x ) C_ ( (Cntz ` G ) ` ( ( S |` A ) ` y ) ) /\ ( ( ( S |` A ) ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) ) = { ( 0g ` G ) } ) )
83	82	ralrimiva 2827	. . 3 |- ( ph -> A. x e. A ( A. y e. ( A \ { x } ) ( ( S |` A ) ` x ) C_ ( (Cntz ` G ) ` ( ( S |` A ) ` y ) ) /\ ( ( ( S |` A ) ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) ) = { ( 0g ` G ) } ) )
84		reldmdprd 16596	. . . . . . . 8 |- Rel dom DProd
85	84	brrelex2i 4983	. . . . . . 7 |- ( G dom DProd S -> S e. _V )
86		dmexg 6614	. . . . . . 7 |- ( S e. _V -> dom S e. _V )
87	1, 85, 86	3syl 20	. . . . . 6 |- ( ph -> dom S e. _V )
88	4, 87	eqeltrrd 2541	. . . . 5 |- ( ph -> I e. _V )
89	88, 6	ssexd 4542	. . . 4 |- ( ph -> A e. _V )
90		fdm 5666	. . . . 5 |- ( ( S |` A ) : A --> (SubGrp ` G ) -> dom ( S |` A ) = A )
91	8, 90	syl 16	. . . 4 |- ( ph -> dom ( S |` A ) = A )
92	20, 66, 36	dmdprd 16597	. . . 4 |- ( ( A e. _V /\ dom ( S |` A ) = A ) -> ( G dom DProd ( S |` A ) <-> ( G e. Grp /\ ( S |` A ) : A --> (SubGrp ` G ) /\ A. x e. A ( A. y e. ( A \ { x } ) ( ( S |` A ) ` x ) C_ ( (Cntz ` G ) ` ( ( S |` A ) ` y ) ) /\ ( ( ( S |` A ) ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) ) )
93	89, 91, 92	syl2anc 661	. . 3 |- ( ph -> ( G dom DProd ( S |` A ) <-> ( G e. Grp /\ ( S |` A ) : A --> (SubGrp ` G ) /\ A. x e. A ( A. y e. ( A \ { x } ) ( ( S |` A ) ` x ) C_ ( (Cntz ` G ) ` ( ( S |` A ) ` y ) ) /\ ( ( ( S |` A ) ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) ) )
94	3, 8, 83, 93	mpbir3and 1171	. 2 |- ( ph -> G dom DProd ( S |` A ) )
95		rnss 5171	. . . . . 6 |- ( ( S |` A ) C_ S -> ran ( S |` A ) C_ ran S )
96		uniss 4215	. . . . . 6 |- ( ran ( S |` A ) C_ ran S -> U. ran ( S |` A ) C_ U. ran S )
97	37, 95, 96	mp2b 10	. . . . 5 |- U. ran ( S |` A ) C_ U. ran S
98	97	a1i 11	. . . 4 |- ( ph -> U. ran ( S |` A ) C_ U. ran S )
99		sspwuni 4359	. . . . 5 |- ( ran S C_ ~P ( Base ` G ) <-> U. ran S C_ ( Base ` G ) )
100	55, 99	sylib 196	. . . 4 |- ( ph -> U. ran S C_ ( Base ` G ) )
101	34, 36, 98, 100	mrcssd 14676	. . 3 |- ( ph -> ( (mrCls ` (SubGrp ` G ) ) ` U. ran ( S |` A ) ) C_ ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) )
102	36	dprdspan 16641	. . . 4 |- ( G dom DProd ( S |` A ) -> ( G DProd ( S |` A ) ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ran ( S |` A ) ) )
103	94, 102	syl 16	. . 3 |- ( ph -> ( G DProd ( S |` A ) ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ran ( S |` A ) ) )
104	36	dprdspan 16641	. . . 4 |- ( G dom DProd S -> ( G DProd S ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) )
105	1, 104	syl 16	. . 3 |- ( ph -> ( G DProd S ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) )
106	101, 103, 105	3sstr4d 3502	. 2 |- ( ph -> ( G DProd ( S |` A ) ) C_ ( G DProd S ) )
107	94, 106	jca 532	1 |- ( ph -> ( G dom DProd ( S |` A ) /\ ( G DProd ( S |` A ) ) C_ ( G DProd S ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2645   A.wral 2796   _Vcvv 3072   \ cdif 3428   i^i cin 3430   C_ wss 3431   ~Pcpw 3963   {csn 3980   U.cuni 4194   class class class wbr 4395   dom cdm 4943   ran crn 4944   |` cres 4945   "cima 4946   -->wf 5517   ` cfv 5521  (class class class)co 6195   Basecbs 14287   0gc0g 14492  Moorecmre 14634  mrClscmrc 14635  ACScacs 14637   Grpcgrp 15524  SubGrpcsubg 15789  Cntzccntz 15947   DProd cdprd 16592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-of 6425  df-om 6582  df-1st 6682  df-2nd 6683  df-supp 6796  df-tpos 6850  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-map 7321  df-ixp 7369  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-fsupp 7727  df-oi 7830  df-card 8215  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-n0 10686  df-z 10753  df-uz 10968  df-fz 11550  df-fzo 11661  df-seq 11919  df-hash 12216  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-0g 14494  df-gsum 14495  df-mre 14638  df-mrc 14639  df-acs 14641  df-mnd 15529  df-mhm 15578  df-submnd 15579  df-grp 15659  df-minusg 15660  df-sbg 15661  df-mulg 15662  df-subg 15792  df-ghm 15859  df-gim 15901  df-cntz 15949  df-oppg 15975  df-cmn 16395  df-dprd 16594

This theorem is referenced by:  dprdf1  16647  dprdcntz2  16653  dprddisj2  16654  dprddisj2OLD  16655  dprd2dlem1  16657  dprd2da  16658  dmdprdsplit  16663  dprdsplit  16664  dpjf  16673  dpjidcl  16674  dpjlid  16677  dpjghm  16679  dpjidclOLD  16681  ablfac1eulem  16690  ablfac1eu  16691