Theorem dprdres 19546

Description: Restriction of a direct product (dropping factors). (Contributed by Mario Carneiro, 25-Apr-2016.)

Hypotheses

Ref	Expression
dprdres.1	⊢ (𝜑 → 𝐺dom DProd 𝑆)
dprdres.2	⊢ (𝜑 → dom 𝑆 = 𝐼)
dprdres.3	⊢ (𝜑 → 𝐴 ⊆ 𝐼)

Assertion

Ref	Expression
dprdres	⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ 𝐴) ∧ (𝐺 DProd (𝑆 ↾ 𝐴)) ⊆ (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 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
2		dprdgrp 19523	. . . 4 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
4		dprdres.2	. . . . 5 ⊢ (𝜑 → dom 𝑆 = 𝐼)
5	1, 4	dprdf2 19525	. . . 4 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
6		dprdres.3	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐼)
7	5, 6	fssresd 6625	. . 3 ⊢ (𝜑 → (𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺))
8	1	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝐺dom DProd 𝑆)
9	4	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → dom 𝑆 = 𝐼)
10	6	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝐴 ⊆ 𝐼)
11		simplr 765	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥 ∈ 𝐴)
12	10, 11	sseldd 3918	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥 ∈ 𝐼)
13		eldifi 4057	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦 ∈ 𝐴)
14	13	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦 ∈ 𝐴)
15	10, 14	sseldd 3918	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦 ∈ 𝐼)
16		eldifsni 4720	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦 ≠ 𝑥)
17	16	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦 ≠ 𝑥)
18	17	necomd 2998	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥 ≠ 𝑦)
19		eqid 2738	. . . . . . . 8 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
20	8, 9, 12, 15, 18, 19	dprdcntz 19526	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → (𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)))
21	11	fvresd 6776	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆 ↾ 𝐴)‘𝑥) = (𝑆‘𝑥))
22	14	fvresd 6776	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆 ↾ 𝐴)‘𝑦) = (𝑆‘𝑦))
23	22	fveq2d 6760	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)) = ((Cntz‘𝐺)‘(𝑆‘𝑦)))
24	20, 21, 23	3sstr4d 3964	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆 ↾ 𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)))
25	24	ralrimiva 3107	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆 ↾ 𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)))
26		fvres 6775	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → ((𝑆 ↾ 𝐴)‘𝑥) = (𝑆‘𝑥))
27	26	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆 ↾ 𝐴)‘𝑥) = (𝑆‘𝑥))
28	27	ineq1d 4142	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑆 ↾ 𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))) = ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))))
29		eqid 2738	. . . . . . . . . . . . 13 ⊢ (Base‘𝐺) = (Base‘𝐺)
30	29	subgacs 18704	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
31		acsmre 17278	. . . . . . . . . . . 12 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
32	3, 30, 31	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
33	32	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
34		eqid 2738	. . . . . . . . . 10 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
35		resss 5905	. . . . . . . . . . . . 13 ⊢ (𝑆 ↾ 𝐴) ⊆ 𝑆
36		imass1 5998	. . . . . . . . . . . . 13 ⊢ ((𝑆 ↾ 𝐴) ⊆ 𝑆 → ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐴 ∖ {𝑥})))
37	35, 36	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐴 ∖ {𝑥}))
38	6	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ 𝐼)
39		ssdif 4070	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝐼 → (𝐴 ∖ {𝑥}) ⊆ (𝐼 ∖ {𝑥}))
40		imass2 5999	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ {𝑥}) ⊆ (𝐼 ∖ {𝑥}) → (𝑆 “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
41	38, 39, 40	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
42	37, 41	sstrid 3928	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
43	42	unissd 4846	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
44		imassrn 5969	. . . . . . . . . . . 12 ⊢ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
45	5	frnd 6592	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
46	29	subgss 18671	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ (Base‘𝐺))
47		velpw 4535	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝒫 (Base‘𝐺) ↔ 𝑥 ⊆ (Base‘𝐺))
48	46, 47	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ∈ 𝒫 (Base‘𝐺))
49	48	ssriv 3921	. . . . . . . . . . . . . 14 ⊢ (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺)
50	45, 49	sstrdi 3929	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
51	50	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
52	44, 51	sstrid 3928	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
53		sspwuni 5025	. . . . . . . . . . 11 ⊢ ((𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
54	52, 53	sylib 217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
55	33, 34, 43, 54	mrcssd 17250	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
56		sslin 4165	. . . . . . . . 9 ⊢ (((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))) ⊆ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))))
57	55, 56	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))) ⊆ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))))
58	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺dom DProd 𝑆)
59	4	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom 𝑆 = 𝐼)
60	6	sselda 3917	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐼)
61		eqid 2738	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
62	58, 59, 60, 61, 34	dprddisj 19527	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g‘𝐺)})
63	57, 62	sseqtrd 3957	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))) ⊆ {(0g‘𝐺)})
64	5	ffvelrnda 6943	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ∈ (SubGrp‘𝐺))
65	60, 64	syldan 590	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑆‘𝑥) ∈ (SubGrp‘𝐺))
66	61	subg0cl 18678	. . . . . . . . . 10 ⊢ ((𝑆‘𝑥) ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ (𝑆‘𝑥))
67	65, 66	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0g‘𝐺) ∈ (𝑆‘𝑥))
68	43, 54	sstrd 3927	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (Base‘𝐺))
69	34	mrccl 17237	. . . . . . . . . . 11 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
70	33, 68, 69	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
71	61	subg0cl 18678	. . . . . . . . . 10 ⊢ (((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥}))))
72	70, 71	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0g‘𝐺) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥}))))
73	67, 72	elind 4124	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0g‘𝐺) ∈ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))))
74	73	snssd 4739	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {(0g‘𝐺)} ⊆ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))))
75	63, 74	eqssd 3934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g‘𝐺)})
76	28, 75	eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑆 ↾ 𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g‘𝐺)})
77	25, 76	jca 511	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆 ↾ 𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)) ∧ (((𝑆 ↾ 𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g‘𝐺)}))
78	77	ralrimiva 3107	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆 ↾ 𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)) ∧ (((𝑆 ↾ 𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g‘𝐺)}))
79	1, 4	dprddomcld 19519	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
80	79, 6	ssexd 5243	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
81	7	fdmd 6595	. . . 4 ⊢ (𝜑 → dom (𝑆 ↾ 𝐴) = 𝐴)
82	19, 61, 34	dmdprd 19516	. . . 4 ⊢ ((𝐴 ∈ V ∧ dom (𝑆 ↾ 𝐴) = 𝐴) → (𝐺dom DProd (𝑆 ↾ 𝐴) ↔ (𝐺 ∈ Grp ∧ (𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆 ↾ 𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)) ∧ (((𝑆 ↾ 𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g‘𝐺)}))))
83	80, 81, 82	syl2anc 583	. . 3 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ 𝐴) ↔ (𝐺 ∈ Grp ∧ (𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆 ↾ 𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)) ∧ (((𝑆 ↾ 𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g‘𝐺)}))))
84	3, 7, 78, 83	mpbir3and 1340	. 2 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐴))
85		rnss 5837	. . . . . 6 ⊢ ((𝑆 ↾ 𝐴) ⊆ 𝑆 → ran (𝑆 ↾ 𝐴) ⊆ ran 𝑆)
86		uniss 4844	. . . . . 6 ⊢ (ran (𝑆 ↾ 𝐴) ⊆ ran 𝑆 → ∪ ran (𝑆 ↾ 𝐴) ⊆ ∪ ran 𝑆)
87	35, 85, 86	mp2b 10	. . . . 5 ⊢ ∪ ran (𝑆 ↾ 𝐴) ⊆ ∪ ran 𝑆
88	87	a1i 11	. . . 4 ⊢ (𝜑 → ∪ ran (𝑆 ↾ 𝐴) ⊆ ∪ ran 𝑆)
89		sspwuni 5025	. . . . 5 ⊢ (ran 𝑆 ⊆ 𝒫 (Base‘𝐺) ↔ ∪ ran 𝑆 ⊆ (Base‘𝐺))
90	50, 89	sylib 217	. . . 4 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ (Base‘𝐺))
91	32, 34, 88, 90	mrcssd 17250	. . 3 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran (𝑆 ↾ 𝐴)) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
92	34	dprdspan 19545	. . . 4 ⊢ (𝐺dom DProd (𝑆 ↾ 𝐴) → (𝐺 DProd (𝑆 ↾ 𝐴)) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran (𝑆 ↾ 𝐴)))
93	84, 92	syl 17	. . 3 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐴)) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran (𝑆 ↾ 𝐴)))
94	34	dprdspan 19545	. . . 4 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
95	1, 94	syl 17	. . 3 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
96	91, 93, 95	3sstr4d 3964	. 2 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐴)) ⊆ (𝐺 DProd 𝑆))
97	84, 96	jca 511	1 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ 𝐴) ∧ (𝐺 DProd (𝑆 ↾ 𝐴)) ⊆ (𝐺 DProd 𝑆)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  Vcvv 3422   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  {csn 4558  ∪ cuni 4836   class class class wbr 5070  dom cdm 5580  ran crn 5581   ↾ cres 5582   “ cima 5583  ⟶wf 6414  ‘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:  dprdf1  19551  dprdcntz2  19556  dprddisj2  19557  dprd2dlem1  19559  dprd2da  19560  dmdprdsplit  19565  dprdsplit  19566  dpjf  19575  dpjidcl  19576  dpjlid  19579  dpjghm  19581  ablfac1eulem  19590  ablfac1eu  19591