Theorem dprdsn 19979

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 2737	. . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
2		eqid 2737	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		eqid 2737	. . 3 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		subgrcl 19073	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5	4	adantl 481	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
6		snex 5385	. . . 4 ⊢ {𝐴} ∈ V
7	6	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → {𝐴} ∈ V)
8		f1osng 6824	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}–1-1-onto→{𝑆})
9		f1of 6782	. . . . 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 4767	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → {𝑆} ⊆ (SubGrp‘𝐺))
13	10, 12	fssd 6687	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}⟶(SubGrp‘𝐺))
14		simpr1 1196	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ {𝐴})
15		elsni 4599	. . . . . 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 1197	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ {𝐴})
18		elsni 4599	. . . . . 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 2775	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → 𝑥 = 𝑦)
21		simpr3 1198	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
22	20, 21	pm2.21ddne 3017	. . 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 2737	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
25	24	subgacs 19102	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
26		acsmre 17587	. . . . . . . 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 4594	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {𝑥} = {𝐴})
30	29	difeq2d 4080	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({𝐴} ∖ {𝑥}) = ({𝐴} ∖ {𝐴}))
31		difid 4330	. . . . . . . . . . . . 13 ⊢ ({𝐴} ∖ {𝐴}) = ∅
32	30, 31	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({𝐴} ∖ {𝑥}) = ∅)
33	32	imaeq2d 6027	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ({⟨𝐴, 𝑆⟩} “ ∅))
34		ima0 6044	. . . . . . . . . . 11 ⊢ ({⟨𝐴, 𝑆⟩} “ ∅) = ∅
35	33, 34	eqtrdi 2788	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
36	35	unieqd 4878	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∪ ∅)
37		uni0 4893	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
38	36, 37	eqtrdi 2788	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
39		0ss 4354	. . . . . . . . 9 ⊢ ∅ ⊆ {(0g‘𝐺)}
40	39	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∅ ⊆ {(0g‘𝐺)})
41	38, 40	eqsstrd 3970	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) ⊆ {(0g‘𝐺)})
42	2	0subg 19093	. . . . . . . 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 17555	. . . . . . 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 1374	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ((mrCls‘(SubGrp‘𝐺))‘∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ {(0g‘𝐺)})
46	2	subg0cl 19076	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑆)
47	46	ad2antlr 728	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (0g‘𝐺) ∈ 𝑆)
48	15	fveq2d 6846	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝐴} → ({⟨𝐴, 𝑆⟩}‘𝑥) = ({⟨𝐴, 𝑆⟩}‘𝐴))
49		fvsng 7136	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ({⟨𝐴, 𝑆⟩}‘𝐴) = 𝑆)
50	48, 49	sylan9eqr 2794	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩}‘𝑥) = 𝑆)
51	47, 50	eleqtrrd 2840	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (0g‘𝐺) ∈ ({⟨𝐴, 𝑆⟩}‘𝑥))
52	51	snssd 4767	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {(0g‘𝐺)} ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥))
53	45, 52	sstrd 3946	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ((mrCls‘(SubGrp‘𝐺))‘∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥))
54		sseqin2 4177	. . . . 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 3970	. . 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 19942	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺dom DProd {⟨𝐴, 𝑆⟩})
58	3	dprdspan 19970	. . . 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 6187	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝐴, 𝑆⟩} = {𝑆})
61	60	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = {𝑆})
62	61	unieqd 4878	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∪ ran {⟨𝐴, 𝑆⟩} = ∪ {𝑆})
63		unisng 4883	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ∪ {𝑆} = 𝑆)
64	63	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∪ {𝑆} = 𝑆)
65	62, 64	eqtrd 2772	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∪ ran {⟨𝐴, 𝑆⟩} = 𝑆)
66	65	fveq2d 6846	. . . 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 17548	. . . . 5 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘𝑆) = 𝑆)
69	67, 68	sylancom 589	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘𝑆) = 𝑆)
70	66, 69	eqtrd 2772	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ ran {⟨𝐴, 𝑆⟩}) = 𝑆)
71	59, 70	eqtrd 2772	. 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 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  {csn 4582  ⟨cop 4588  ∪ cuni 4865   class class class wbr 5100  dom cdm 5632  ran crn 5633   “ cima 5635  ⟶wf 6496  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  0_gc0g 17371  Moorecmre 17513  mrClscmrc 17514  ACScacs 17516  Grpcgrp 18875  SubGrpcsubg 19062  Cntzccntz 19256   DProd cdprd 19936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-of 7632  df-om 7819  df-1st 7943  df-2nd 7944  df-supp 8113  df-tpos 8178  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-map 8777  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9277  df-oi 9427  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-n0 12414  df-z 12501  df-uz 12764  df-fz 13436  df-fzo 13583  df-seq 13937  df-hash 14266  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-0g 17373  df-gsum 17374  df-mre 17517  df-mrc 17518  df-acs 17520  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-mhm 18720  df-submnd 18721  df-grp 18878  df-minusg 18879  df-sbg 18880  df-mulg 19010  df-subg 19065  df-ghm 19154  df-gim 19200  df-cntz 19258  df-oppg 19287  df-cmn 19723  df-dprd 19938

This theorem is referenced by:  dprd2da  19985  dmdprdpr  19992  dprdpr  19993  dpjlem  19994  pgpfaclem1  20024