Theorem dprdres 17596

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 17572	. . . 4 |- ( G dom DProd S -> G e. Grp )
3	1, 2	syl 17	. . 3 |- ( ph -> G e. Grp )
4		dprdres.2	. . . . 5 |- ( ph -> dom S = I )
5	1, 4	dprdf2 17574	. . . 4 |- ( ph -> S : I --> (SubGrp ` G ) )
6		dprdres.3	. . . 4 |- ( ph -> A C_ I )
7	5, 6	fssresd 5767	. . 3 |- ( ph -> ( S |` A ) : A --> (SubGrp ` G ) )
8	1	ad2antrr 730	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> G dom DProd S )
9	4	ad2antrr 730	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> dom S = I )
10	6	ad2antrr 730	. . . . . . . . 9 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> A C_ I )
11		simplr 760	. . . . . . . . 9 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> x e. A )
12	10, 11	sseldd 3471	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> x e. I )
13		eldifi 3593	. . . . . . . . . 10 |- ( y e. ( A \ { x } ) -> y e. A )
14	13	adantl 467	. . . . . . . . 9 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> y e. A )
15	10, 14	sseldd 3471	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> y e. I )
16		eldifsni 4129	. . . . . . . . . 10 |- ( y e. ( A \ { x } ) -> y =/= x )
17	16	adantl 467	. . . . . . . . 9 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> y =/= x )
18	17	necomd 2702	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> x =/= y )
19		eqid 2429	. . . . . . . 8 |- (Cntz ` G ) = (Cntz ` G )
20	8, 9, 12, 15, 18, 19	dprdcntz 17575	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) )
21		fvres 5895	. . . . . . . 8 |- ( x e. A -> ( ( S |` A ) ` x ) = ( S ` x ) )
22	11, 21	syl 17	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> ( ( S |` A ) ` x ) = ( S ` x ) )
23		fvres 5895	. . . . . . . . 9 |- ( y e. A -> ( ( S |` A ) ` y ) = ( S ` y ) )
24	14, 23	syl 17	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> ( ( S |` A ) ` y ) = ( S ` y ) )
25	24	fveq2d 5885	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> ( (Cntz ` G ) ` ( ( S |` A ) ` y ) ) = ( (Cntz ` G ) ` ( S ` y ) ) )
26	20, 22, 25	3sstr4d 3513	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ y e. ( A \ { x } ) ) -> ( ( S |` A ) ` x ) C_ ( (Cntz ` G ) ` ( ( S |` A ) ` y ) ) )
27	26	ralrimiva 2846	. . . . 5 |- ( ( ph /\ x e. A ) -> A. y e. ( A \ { x } ) ( ( S |` A ) ` x ) C_ ( (Cntz ` G ) ` ( ( S |` A ) ` y ) ) )
28	21	adantl 467	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( ( S |` A ) ` x ) = ( S ` x ) )
29	28	ineq1d 3669	. . . . . 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 } ) ) ) ) )
30		eqid 2429	. . . . . . . . . . . . 13 |- ( Base ` G ) = ( Base ` G )
31	30	subgacs 16803	. . . . . . . . . . . 12 |- ( G e. Grp -> (SubGrp ` G ) e. (ACS ` ( Base ` G ) ) )
32		acsmre 15509	. . . . . . . . . . . 12 |- ( (SubGrp ` G ) e. (ACS ` ( Base ` G ) ) -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
33	3, 31, 32	3syl 18	. . . . . . . . . . 11 |- ( ph -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
34	33	adantr 466	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> (SubGrp ` G ) e. (Moore ` ( Base ` G ) ) )
35		eqid 2429	. . . . . . . . . 10 |- (mrCls ` (SubGrp ` G ) ) = (mrCls ` (SubGrp ` G ) )
36		resss 5148	. . . . . . . . . . . . 13 |- ( S |` A ) C_ S
37		imass1 5223	. . . . . . . . . . . . 13 |- ( ( S |` A ) C_ S -> ( ( S |` A ) " ( A \ { x } ) ) C_ ( S " ( A \ { x } ) ) )
38	36, 37	ax-mp 5	. . . . . . . . . . . 12 |- ( ( S |` A ) " ( A \ { x } ) ) C_ ( S " ( A \ { x } ) )
39	6	adantr 466	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. A ) -> A C_ I )
40		ssdif 3606	. . . . . . . . . . . . 13 |- ( A C_ I -> ( A \ { x } ) C_ ( I \ { x } ) )
41		imass2 5224	. . . . . . . . . . . . 13 |- ( ( A \ { x } ) C_ ( I \ { x } ) -> ( S " ( A \ { x } ) ) C_ ( S " ( I \ { x } ) ) )
42	39, 40, 41	3syl 18	. . . . . . . . . . . 12 |- ( ( ph /\ x e. A ) -> ( S " ( A \ { x } ) ) C_ ( S " ( I \ { x } ) ) )
43	38, 42	syl5ss 3481	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( ( S |` A ) " ( A \ { x } ) ) C_ ( S " ( I \ { x } ) ) )
44	43	unissd 4246	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> U. ( ( S |` A ) " ( A \ { x } ) ) C_ U. ( S " ( I \ { x } ) ) )
45		imassrn 5199	. . . . . . . . . . . 12 |- ( S " ( I \ { x } ) ) C_ ran S
46		frn 5752	. . . . . . . . . . . . . . 15 |- ( S : I --> (SubGrp ` G ) -> ran S C_ (SubGrp ` G ) )
47	5, 46	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> ran S C_ (SubGrp ` G ) )
48	30	subgss 16769	. . . . . . . . . . . . . . . 16 |- ( x e. (SubGrp ` G ) -> x C_ ( Base ` G ) )
49		selpw 3992	. . . . . . . . . . . . . . . 16 |- ( x e. ~P ( Base ` G ) <-> x C_ ( Base ` G ) )
50	48, 49	sylibr 215	. . . . . . . . . . . . . . 15 |- ( x e. (SubGrp ` G ) -> x e. ~P ( Base ` G ) )
51	50	ssriv 3474	. . . . . . . . . . . . . 14 |- (SubGrp ` G ) C_ ~P ( Base ` G )
52	47, 51	syl6ss 3482	. . . . . . . . . . . . 13 |- ( ph -> ran S C_ ~P ( Base ` G ) )
53	52	adantr 466	. . . . . . . . . . . 12 |- ( ( ph /\ x e. A ) -> ran S C_ ~P ( Base ` G ) )
54	45, 53	syl5ss 3481	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( S " ( I \ { x } ) ) C_ ~P ( Base ` G ) )
55		sspwuni 4391	. . . . . . . . . . 11 |- ( ( S " ( I \ { x } ) ) C_ ~P ( Base ` G ) <-> U. ( S " ( I \ { x } ) ) C_ ( Base ` G ) )
56	54, 55	sylib 199	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> U. ( S " ( I \ { x } ) ) C_ ( Base ` G ) )
57	34, 35, 44, 56	mrcssd 15481	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) C_ ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) )
58		sslin 3694	. . . . . . . . 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 } ) ) ) ) )
59	57, 58	syl 17	. . . . . . . 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 } ) ) ) ) )
60	1	adantr 466	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> G dom DProd S )
61	4	adantr 466	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> dom S = I )
62	6	sselda 3470	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> x e. I )
63		eqid 2429	. . . . . . . . 9 |- ( 0g ` G ) = ( 0g ` G )
64	60, 61, 62, 63, 35	dprddisj 17576	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) = { ( 0g ` G ) } )
65	59, 64	sseqtrd 3506	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) ) C_ { ( 0g ` G ) } )
66	5	ffvelrnda 6037	. . . . . . . . . . 11 |- ( ( ph /\ x e. I ) -> ( S ` x ) e. (SubGrp ` G ) )
67	62, 66	syldan 472	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( S ` x ) e. (SubGrp ` G ) )
68	63	subg0cl 16776	. . . . . . . . . 10 |- ( ( S ` x ) e. (SubGrp ` G ) -> ( 0g ` G ) e. ( S ` x ) )
69	67, 68	syl 17	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( 0g ` G ) e. ( S ` x ) )
70	44, 56	sstrd 3480	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> U. ( ( S |` A ) " ( A \ { x } ) ) C_ ( Base ` G ) )
71	35	mrccl 15468	. . . . . . . . . . 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 ) )
72	34, 70, 71	syl2anc 665	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) e. (SubGrp ` G ) )
73	63	subg0cl 16776	. . . . . . . . . 10 |- ( ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) e. (SubGrp ` G ) -> ( 0g ` G ) e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) )
74	72, 73	syl 17	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( 0g ` G ) e. ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) )
75	69, 74	elind 3656	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( 0g ` G ) e. ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) ) )
76	75	snssd 4148	. . . . . . 7 |- ( ( ph /\ x e. A ) -> { ( 0g ` G ) } C_ ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) ) )
77	65, 76	eqssd 3487	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( ( S ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) ) = { ( 0g ` G ) } )
78	29, 77	eqtrd 2470	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ( ( S |` A ) ` x ) i^i ( (mrCls ` (SubGrp ` G ) ) ` U. ( ( S |` A ) " ( A \ { x } ) ) ) ) = { ( 0g ` G ) } )
79	27, 78	jca 534	. . . 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 ) } ) )
80	79	ralrimiva 2846	. . 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 ) } ) )
81	1, 4	dprddomcld 17568	. . . . 5 |- ( ph -> I e. _V )
82	81, 6	ssexd 4572	. . . 4 |- ( ph -> A e. _V )
83		fdm 5750	. . . . 5 |- ( ( S |` A ) : A --> (SubGrp ` G ) -> dom ( S |` A ) = A )
84	7, 83	syl 17	. . . 4 |- ( ph -> dom ( S |` A ) = A )
85	19, 63, 35	dmdprd 17565	. . . 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 ) } ) ) ) )
86	82, 84, 85	syl2anc 665	. . 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 ) } ) ) ) )
87	3, 7, 80, 86	mpbir3and 1188	. 2 |- ( ph -> G dom DProd ( S |` A ) )
88		rnss 5083	. . . . . 6 |- ( ( S |` A ) C_ S -> ran ( S |` A ) C_ ran S )
89		uniss 4243	. . . . . 6 |- ( ran ( S |` A ) C_ ran S -> U. ran ( S |` A ) C_ U. ran S )
90	36, 88, 89	mp2b 10	. . . . 5 |- U. ran ( S |` A ) C_ U. ran S
91	90	a1i 11	. . . 4 |- ( ph -> U. ran ( S |` A ) C_ U. ran S )
92		sspwuni 4391	. . . . 5 |- ( ran S C_ ~P ( Base ` G ) <-> U. ran S C_ ( Base ` G ) )
93	52, 92	sylib 199	. . . 4 |- ( ph -> U. ran S C_ ( Base ` G ) )
94	33, 35, 91, 93	mrcssd 15481	. . 3 |- ( ph -> ( (mrCls ` (SubGrp ` G ) ) ` U. ran ( S |` A ) ) C_ ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) )
95	35	dprdspan 17595	. . . 4 |- ( G dom DProd ( S |` A ) -> ( G DProd ( S |` A ) ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ran ( S |` A ) ) )
96	87, 95	syl 17	. . 3 |- ( ph -> ( G DProd ( S |` A ) ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ran ( S |` A ) ) )
97	35	dprdspan 17595	. . . 4 |- ( G dom DProd S -> ( G DProd S ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) )
98	1, 97	syl 17	. . 3 |- ( ph -> ( G DProd S ) = ( (mrCls ` (SubGrp ` G ) ) ` U. ran S ) )
99	94, 96, 98	3sstr4d 3513	. 2 |- ( ph -> ( G DProd ( S |` A ) ) C_ ( G DProd S ) )
100	87, 99	jca 534	1 |- ( ph -> ( G dom DProd ( S |` A ) /\ ( G DProd ( S |` A ) ) C_ ( G DProd S ) ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1870   =/= wne 2625   A.wral 2782   _Vcvv 3087   \ cdif 3439   i^i cin 3441   C_ wss 3442   ~Pcpw 3985   {csn 4002   U.cuni 4222   class class class wbr 4426   dom cdm 4854   ran crn 4855   |` cres 4856   "cima 4857   -->wf 5597   ` cfv 5601  (class class class)co 6305   Basecbs 15084   0gc0g 15297  Moorecmre 15439  mrClscmrc 15440  ACScacs 15442   Grpcgrp 16620  SubGrpcsubg 16762  Cntzccntz 16920   DProd cdprd 17560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-seq 12211  df-hash 12513  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-0g 15299  df-gsum 15300  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-mhm 16533  df-submnd 16534  df-grp 16624  df-minusg 16625  df-sbg 16626  df-mulg 16627  df-subg 16765  df-ghm 16832  df-gim 16874  df-cntz 16922  df-oppg 16948  df-cmn 17367  df-dprd 17562

This theorem is referenced by:  dprdf1  17601  dprdcntz2  17606  dprddisj2  17607  dprd2dlem1  17609  dprd2da  17610  dmdprdsplit  17615  dprdsplit  17616  dpjf  17625  dpjidcl  17626  dpjlid  17629  dpjghm  17631  ablfac1eulem  17640  ablfac1eu  17641