Theorem dprdsn 20080

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 2740	. . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
2		eqid 2740	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		eqid 2740	. . 3 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		subgrcl 19171	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5	4	adantl 481	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
6		snex 5451	. . . 4 ⊢ {𝐴} ∈ V
7	6	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → {𝐴} ∈ V)
8		f1osng 6903	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}–1-1-onto→{𝑆})
9		f1of 6862	. . . . 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 4834	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → {𝑆} ⊆ (SubGrp‘𝐺))
13	10, 12	fssd 6764	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → {⟨𝐴, 𝑆⟩}:{𝐴}⟶(SubGrp‘𝐺))
14		simpr1 1194	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ {𝐴})
15		elsni 4665	. . . . . 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 1195	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ {𝐴})
18		elsni 4665	. . . . . 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 2783	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → 𝑥 = 𝑦)
21		simpr3 1196	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
22	20, 21	pm2.21ddne 3032	. . 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 2740	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
25	24	subgacs 19201	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
26		acsmre 17710	. . . . . . . 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 4660	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {𝑥} = {𝐴})
30	29	difeq2d 4149	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({𝐴} ∖ {𝑥}) = ({𝐴} ∖ {𝐴}))
31		difid 4398	. . . . . . . . . . . . 13 ⊢ ({𝐴} ∖ {𝐴}) = ∅
32	30, 31	eqtrdi 2796	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({𝐴} ∖ {𝑥}) = ∅)
33	32	imaeq2d 6089	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ({⟨𝐴, 𝑆⟩} “ ∅))
34		ima0 6106	. . . . . . . . . . 11 ⊢ ({⟨𝐴, 𝑆⟩} “ ∅) = ∅
35	33, 34	eqtrdi 2796	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
36	35	unieqd 4944	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∪ ∅)
37		uni0 4959	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
38	36, 37	eqtrdi 2796	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) = ∅)
39		0ss 4423	. . . . . . . . 9 ⊢ ∅ ⊆ {(0g‘𝐺)}
40	39	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∅ ⊆ {(0g‘𝐺)})
41	38, 40	eqsstrd 4047	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥})) ⊆ {(0g‘𝐺)})
42	2	0subg 19191	. . . . . . . 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 17678	. . . . . . 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 1371	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ((mrCls‘(SubGrp‘𝐺))‘∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ {(0g‘𝐺)})
46	2	subg0cl 19174	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑆)
47	46	ad2antlr 726	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (0g‘𝐺) ∈ 𝑆)
48	15	fveq2d 6924	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝐴} → ({⟨𝐴, 𝑆⟩}‘𝑥) = ({⟨𝐴, 𝑆⟩}‘𝐴))
49		fvsng 7214	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ({⟨𝐴, 𝑆⟩}‘𝐴) = 𝑆)
50	48, 49	sylan9eqr 2802	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ({⟨𝐴, 𝑆⟩}‘𝑥) = 𝑆)
51	47, 50	eleqtrrd 2847	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → (0g‘𝐺) ∈ ({⟨𝐴, 𝑆⟩}‘𝑥))
52	51	snssd 4834	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → {(0g‘𝐺)} ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥))
53	45, 52	sstrd 4019	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ {𝐴}) → ((mrCls‘(SubGrp‘𝐺))‘∪ ({⟨𝐴, 𝑆⟩} “ ({𝐴} ∖ {𝑥}))) ⊆ ({⟨𝐴, 𝑆⟩}‘𝑥))
54		sseqin2 4244	. . . . 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 4047	. . 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 20043	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺dom DProd {⟨𝐴, 𝑆⟩})
58	3	dprdspan 20071	. . . 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 6252	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝐴, 𝑆⟩} = {𝑆})
61	60	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ran {⟨𝐴, 𝑆⟩} = {𝑆})
62	61	unieqd 4944	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∪ ran {⟨𝐴, 𝑆⟩} = ∪ {𝑆})
63		unisng 4949	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ∪ {𝑆} = 𝑆)
64	63	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∪ {𝑆} = 𝑆)
65	62, 64	eqtrd 2780	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∪ ran {⟨𝐴, 𝑆⟩} = 𝑆)
66	65	fveq2d 6924	. . . 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 17671	. . . . 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 2780	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ ran {⟨𝐴, 𝑆⟩}) = 𝑆)
71	59, 70	eqtrd 2780	. 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 1537   ∈ wcel 2108   ≠ wne 2946  Vcvv 3488   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  {csn 4648  ⟨cop 4654  ∪ cuni 4931   class class class wbr 5166  dom cdm 5700  ran crn 5701   “ cima 5703  ⟶wf 6569  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  0_gc0g 17499  Moorecmre 17640  mrClscmrc 17641  ACScacs 17643  Grpcgrp 18973  SubGrpcsubg 19160  Cntzccntz 19355   DProd cdprd 20037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-tpos 8267  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-seq 14053  df-hash 14380  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-0g 17501  df-gsum 17502  df-mre 17644  df-mrc 17645  df-acs 17647  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-mulg 19108  df-subg 19163  df-ghm 19253  df-gim 19299  df-cntz 19357  df-oppg 19386  df-cmn 19824  df-dprd 20039

This theorem is referenced by:  dprd2da  20086  dmdprdpr  20093  dprdpr  20094  dpjlem  20095  pgpfaclem1  20125