Theorem dprdsn 19935

Description: A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)

Assertion

Ref	Expression
dprdsn	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐺dom DProd {⟨𝐴, 𝑆⟩} ∧ (𝐺 DProd {⟨𝐴, 𝑆⟩}) = 𝑆))

Proof of Theorem dprdsn

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

Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
2		eqid 2729	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		eqid 2729	. . 3 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		subgrcl 19028	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5	4	adantl 481	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
6		snex 5378	. . . 4 ⊢ {𝐴} ∈ V
7	6	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → {𝐴} ∈ V)
8		f1osng 6809	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}–1-1-onto→{𝑆})
9		f1of 6768	. . . . 5 ⊢ ({⟨𝐴, 𝑆⟩}:{𝐴}–1-1-onto→{𝑆} → {⟨𝐴, 𝑆⟩}:{𝐴}⟶{𝑆})
10	8, 9	syl 17	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}⟶{𝑆})
11		simpr 484	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
12	11	snssd 4763	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → {𝑆} ⊆ (SubGrp‘𝐺))
13	10, 12	fssd 6673	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}⟶(SubGrp‘𝐺))
14		simpr1 1195	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ {𝐴})
15		elsni 4596	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
16	14, 15	syl 17	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → 𝑥 = 𝐴)
17		simpr2 1196	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ {𝐴})
18		elsni 4596	. . . . . 6 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
19	17, 18	syl 17	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → 𝑦 = 𝐴)
20	16, 19	eqtr4d 2767	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → 𝑥 = 𝑦)
21		simpr3 1197	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
22	20, 21	pm2.21ddne 3009	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → ({⟨𝐴, 𝑆⟩}‘𝑥) ⊆ ((Cntz‘𝐺)‘({⟨𝐴, 𝑆⟩}‘𝑦)))
23	5	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → 𝐺 ∈ Grp)
24		eqid 2729	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
25	24	subgacs 19058	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
26		acsmre 17576	. . . . . . . 8 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
27	23, 25, 26	3syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
28	15	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → 𝑥 = 𝐴)
29	28	sneqd 4591	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {𝑥} = {𝐴})
30	29	difeq2d 4079	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({𝐴} ∖ {𝑥}) = ({𝐴} ∖ {𝐴}))
31		difid 4329	. . . . . . . . . . . . 13 ⊢ ({𝐴} ∖ {𝐴}) = ∅
32	30, 31	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({𝐴} ∖ {𝑥}) = ∅)
33	32	imaeq2d 6015	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ({⟨𝐴, 𝑆⟩} “ ∅))
34		ima0 6032	. . . . . . . . . . 11 ⊢ ({⟨𝐴, 𝑆⟩} “ ∅) = ∅
35	33, 34	eqtrdi 2780	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
36	35	unieqd 4874	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∪ ∅)
37		uni0 4889	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
38	36, 37	eqtrdi 2780	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
39		0ss 4353	. . . . . . . . 9 ⊢ ∅ ⊆ {(0g‘𝐺)}
40	39	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∅ ⊆ {(0g‘𝐺)})
41	38, 40	eqsstrd 3972	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) ⊆ {(0g‘𝐺)})
42	2	0subg 19048	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → {(0g‘𝐺)} ∈ (SubGrp‘𝐺))
43	23, 42	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {(0g‘𝐺)} ∈ (SubGrp‘𝐺))
44	3	mrcsscl 17544	. . . . . . 7 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) ⊆ {(0g‘𝐺)} ∧ {(0g‘𝐺)} ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ {(0g‘𝐺)})
45	27, 41, 43, 44	syl3anc 1373	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ((mrCls‘(SubGrp‘𝐺))‘∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ {(0g‘𝐺)})
46	2	subg0cl 19031	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑆)
47	46	ad2antlr 727	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (0g‘𝐺) ∈ 𝑆)
48	15	fveq2d 6830	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝐴} → ({⟨𝐴, 𝑆⟩}‘𝑥) = ({⟨𝐴, 𝑆⟩}‘𝐴))
49		fvsng 7120	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ({⟨𝐴, 𝑆⟩}‘𝐴) = 𝑆)
50	48, 49	sylan9eqr 2786	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩}‘𝑥) = 𝑆)
51	47, 50	eleqtrrd 2831	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (0g‘𝐺) ∈ ({⟨𝐴, 𝑆⟩}‘𝑥))
52	51	snssd 4763	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {(0g‘𝐺)} ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥))
53	45, 52	sstrd 3948	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ((mrCls‘(SubGrp‘𝐺))‘∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥))
54		sseqin2 4176	. . . . 5 ⊢ (((mrCls‘(SubGrp‘𝐺))‘∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥) ↔ (({⟨𝐴, 𝑆⟩}‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})))) = ((mrCls‘(SubGrp‘𝐺))‘∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))))
55	53, 54	sylib 218	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (({⟨𝐴, 𝑆⟩}‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})))) = ((mrCls‘(SubGrp‘𝐺))‘∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))))
56	55, 45	eqsstrd 3972	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (({⟨𝐴, 𝑆⟩}‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})))) ⊆ {(0g‘𝐺)})
57	1, 2, 3, 5, 7, 13, 22, 56	dmdprdd 19898	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺dom DProd {⟨𝐴, 𝑆⟩})
58	3	dprdspan 19926	. . . 4 ⊢ (𝐺dom DProd {⟨𝐴, 𝑆⟩} → (𝐺 DProd {⟨𝐴, 𝑆⟩}) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran {⟨𝐴, 𝑆⟩}))
59	57, 58	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐺 DProd {⟨𝐴, 𝑆⟩}) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran {⟨𝐴, 𝑆⟩}))
60		rnsnopg 6174	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝐴, 𝑆⟩} = {𝑆})
61	60	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = {𝑆})
62	61	unieqd 4874	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∪ ran {⟨𝐴, 𝑆⟩} = ∪ {𝑆})
63		unisng 4879	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ∪ {𝑆} = 𝑆)
64	63	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∪ {𝑆} = 𝑆)
65	62, 64	eqtrd 2764	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∪ ran {⟨𝐴, 𝑆⟩} = 𝑆)
66	65	fveq2d 6830	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ ran {⟨𝐴, 𝑆⟩}) = ((mrCls‘(SubGrp‘𝐺))‘𝑆))
67	5, 25, 26	3syl 18	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
68	3	mrcid 17537	. . . . 5 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘𝑆) = 𝑆)
69	67, 68	sylancom 588	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘𝑆) = 𝑆)
70	66, 69	eqtrd 2764	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ ran {⟨𝐴, 𝑆⟩}) = 𝑆)
71	59, 70	eqtrd 2764	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐺 DProd {⟨𝐴, 𝑆⟩}) = 𝑆)
72	57, 71	jca 511	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐺dom DProd {⟨𝐴, 𝑆⟩} ∧ (𝐺 DProd {⟨𝐴, 𝑆⟩}) = 𝑆))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3438   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  {csn 4579  ⟨cop 4585  ∪ cuni 4861   class class class wbr 5095  dom cdm 5623  ran crn 5624   “ cima 5626  ⟶wf 6482  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7353  Basecbs 17138  0_gc0g 17361  Moorecmre 17502  mrClscmrc 17503  ACScacs 17505  Grpcgrp 18830  SubGrpcsubg 19017  Cntzccntz 19212   DProd cdprd 19892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-of 7617  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8632  df-map 8762  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-fsupp 9271  df-oi 9421  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-n0 12403  df-z 12490  df-uz 12754  df-fz 13429  df-fzo 13576  df-seq 13927  df-hash 14256  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-0g 17363  df-gsum 17364  df-mre 17506  df-mrc 17507  df-acs 17509  df-mgm 18532  df-sgrp 18611  df-mnd 18627  df-mhm 18675  df-submnd 18676  df-grp 18833  df-minusg 18834  df-sbg 18835  df-mulg 18965  df-subg 19020  df-ghm 19110  df-gim 19156  df-cntz 19214  df-oppg 19243  df-cmn 19679  df-dprd 19894

This theorem is referenced by:  dprd2da  19941  dmdprdpr  19948  dprdpr  19949  dpjlem  19950  pgpfaclem1  19980