Theorem dprdsn 19554

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 2738	. . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
2		eqid 2738	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		eqid 2738	. . 3 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		subgrcl 18675	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5	4	adantl 481	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
6		snex 5349	. . . 4 ⊢ {𝐴} ∈ V
7	6	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → {𝐴} ∈ V)
8		f1osng 6740	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}–1-1-onto→{𝑆})
9		f1of 6700	. . . . 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 4739	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → {𝑆} ⊆ (SubGrp‘𝐺))
13	10, 12	fssd 6602	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}⟶(SubGrp‘𝐺))
14		simpr1 1192	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ {𝐴})
15		elsni 4575	. . . . . 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 1193	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ {𝐴})
18		elsni 4575	. . . . . 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 2781	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → 𝑥 = 𝑦)
21		simpr3 1194	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
22	20, 21	pm2.21ddne 3028	. . 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 2738	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
25	24	subgacs 18704	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
26		acsmre 17278	. . . . . . . 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 4570	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {𝑥} = {𝐴})
30	29	difeq2d 4053	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({𝐴} ∖ {𝑥}) = ({𝐴} ∖ {𝐴}))
31		difid 4301	. . . . . . . . . . . . 13 ⊢ ({𝐴} ∖ {𝐴}) = ∅
32	30, 31	eqtrdi 2795	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({𝐴} ∖ {𝑥}) = ∅)
33	32	imaeq2d 5958	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ({⟨𝐴, 𝑆⟩} “ ∅))
34		ima0 5974	. . . . . . . . . . 11 ⊢ ({⟨𝐴, 𝑆⟩} “ ∅) = ∅
35	33, 34	eqtrdi 2795	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
36	35	unieqd 4850	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∪ ∅)
37		uni0 4866	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
38	36, 37	eqtrdi 2795	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
39		0ss 4327	. . . . . . . . 9 ⊢ ∅ ⊆ {(0g‘𝐺)}
40	39	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∅ ⊆ {(0g‘𝐺)})
41	38, 40	eqsstrd 3955	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) ⊆ {(0g‘𝐺)})
42	2	0subg 18695	. . . . . . . 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 17246	. . . . . . 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 1369	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ((mrCls‘(SubGrp‘𝐺))‘∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ {(0g‘𝐺)})
46	2	subg0cl 18678	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑆)
47	46	ad2antlr 723	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (0g‘𝐺) ∈ 𝑆)
48	15	fveq2d 6760	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝐴} → ({⟨𝐴, 𝑆⟩}‘𝑥) = ({⟨𝐴, 𝑆⟩}‘𝐴))
49		fvsng 7034	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ({⟨𝐴, 𝑆⟩}‘𝐴) = 𝑆)
50	48, 49	sylan9eqr 2801	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩}‘𝑥) = 𝑆)
51	47, 50	eleqtrrd 2842	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (0g‘𝐺) ∈ ({⟨𝐴, 𝑆⟩}‘𝑥))
52	51	snssd 4739	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {(0g‘𝐺)} ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥))
53	45, 52	sstrd 3927	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ((mrCls‘(SubGrp‘𝐺))‘∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥))
54		sseqin2 4146	. . . . 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 217	. . . 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 3955	. . 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 19517	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺dom DProd {⟨𝐴, 𝑆⟩})
58	3	dprdspan 19545	. . . 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 6113	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝐴, 𝑆⟩} = {𝑆})
61	60	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = {𝑆})
62	61	unieqd 4850	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∪ ran {⟨𝐴, 𝑆⟩} = ∪ {𝑆})
63		unisng 4857	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ∪ {𝑆} = 𝑆)
64	63	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∪ {𝑆} = 𝑆)
65	62, 64	eqtrd 2778	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∪ ran {⟨𝐴, 𝑆⟩} = 𝑆)
66	65	fveq2d 6760	. . . 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 17239	. . . . 5 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘𝑆) = 𝑆)
69	67, 68	sylancom 587	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘𝑆) = 𝑆)
70	66, 69	eqtrd 2778	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ ran {⟨𝐴, 𝑆⟩}) = 𝑆)
71	59, 70	eqtrd 2778	. 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 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  Vcvv 3422   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  {csn 4558  ⟨cop 4564  ∪ cuni 4836   class class class wbr 5070  dom cdm 5580  ran crn 5581   “ cima 5583  ⟶wf 6414  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  0_gc0g 17067  Moorecmre 17208  mrClscmrc 17209  ACScacs 17211  Grpcgrp 18492  SubGrpcsubg 18664  Cntzccntz 18836   DProd cdprd 19511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-tpos 8013  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-seq 13650  df-hash 13973  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-0g 17069  df-gsum 17070  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-mhm 18345  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-mulg 18616  df-subg 18667  df-ghm 18747  df-gim 18790  df-cntz 18838  df-oppg 18865  df-cmn 19303  df-dprd 19513

This theorem is referenced by:  dprd2da  19560  dmdprdpr  19567  dprdpr  19568  dpjlem  19569  pgpfaclem1  19599