Theorem dprdres 19967

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 19944	. . . 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 19946	. . . 4 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
6		dprdres.3	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐼)
7	5, 6	fssresd 6730	. . 3 ⊢ (𝜑 → (𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺))
8	1	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝐺dom DProd 𝑆)
9	4	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → dom 𝑆 = 𝐼)
10	6	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝐴 ⊆ 𝐼)
11		simplr 768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥 ∈ 𝐴)
12	10, 11	sseldd 3950	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥 ∈ 𝐼)
13		eldifi 4097	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦 ∈ 𝐴)
14	13	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦 ∈ 𝐴)
15	10, 14	sseldd 3950	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦 ∈ 𝐼)
16		eldifsni 4757	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦 ≠ 𝑥)
17	16	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦 ≠ 𝑥)
18	17	necomd 2981	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥 ≠ 𝑦)
19		eqid 2730	. . . . . . . 8 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
20	8, 9, 12, 15, 18, 19	dprdcntz 19947	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → (𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)))
21	11	fvresd 6881	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆 ↾ 𝐴)‘𝑥) = (𝑆‘𝑥))
22	14	fvresd 6881	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆 ↾ 𝐴)‘𝑦) = (𝑆‘𝑦))
23	22	fveq2d 6865	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)) = ((Cntz‘𝐺)‘(𝑆‘𝑦)))
24	20, 21, 23	3sstr4d 4005	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆 ↾ 𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)))
25	24	ralrimiva 3126	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆 ↾ 𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)))
26		fvres 6880	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → ((𝑆 ↾ 𝐴)‘𝑥) = (𝑆‘𝑥))
27	26	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆 ↾ 𝐴)‘𝑥) = (𝑆‘𝑥))
28	27	ineq1d 4185	. . . . . 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 2730	. . . . . . . . . . . . 13 ⊢ (Base‘𝐺) = (Base‘𝐺)
30	29	subgacs 19100	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
31		acsmre 17620	. . . . . . . . . . . 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 2730	. . . . . . . . . 10 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
35		resss 5975	. . . . . . . . . . . . 13 ⊢ (𝑆 ↾ 𝐴) ⊆ 𝑆
36		imass1 6075	. . . . . . . . . . . . 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 4110	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝐼 → (𝐴 ∖ {𝑥}) ⊆ (𝐼 ∖ {𝑥}))
40		imass2 6076	. . . . . . . . . . . . 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 3961	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
43	42	unissd 4884	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
44		imassrn 6045	. . . . . . . . . . . 12 ⊢ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
45	5	frnd 6699	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
46	29	subgss 19066	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ (Base‘𝐺))
47		velpw 4571	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝒫 (Base‘𝐺) ↔ 𝑥 ⊆ (Base‘𝐺))
48	46, 47	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ∈ 𝒫 (Base‘𝐺))
49	48	ssriv 3953	. . . . . . . . . . . . . 14 ⊢ (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺)
50	45, 49	sstrdi 3962	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
51	50	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
52	44, 51	sstrid 3961	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
53		sspwuni 5067	. . . . . . . . . . 11 ⊢ ((𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
54	52, 53	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
55	33, 34, 43, 54	mrcssd 17592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
56		sslin 4209	. . . . . . . . 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 3949	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐼)
61		eqid 2730	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
62	58, 59, 60, 61, 34	dprddisj 19948	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g‘𝐺)})
63	57, 62	sseqtrd 3986	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))) ⊆ {(0g‘𝐺)})
64	5	ffvelcdmda 7059	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ∈ (SubGrp‘𝐺))
65	60, 64	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑆‘𝑥) ∈ (SubGrp‘𝐺))
66	61	subg0cl 19073	. . . . . . . . . 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 3960	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (Base‘𝐺))
69	34	mrccl 17579	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
71	61	subg0cl 19073	. . . . . . . . . 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 4166	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0g‘𝐺) ∈ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))))
74	73	snssd 4776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {(0g‘𝐺)} ⊆ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))))
75	63, 74	eqssd 3967	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g‘𝐺)})
76	28, 75	eqtrd 2765	. . . . 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 3126	. . 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 19940	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
80	79, 6	ssexd 5282	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
81	7	fdmd 6701	. . . 4 ⊢ (𝜑 → dom (𝑆 ↾ 𝐴) = 𝐴)
82	19, 61, 34	dmdprd 19937	. . . 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 584	. . 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 1343	. 2 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐴))
85		rnss 5906	. . . . . 6 ⊢ ((𝑆 ↾ 𝐴) ⊆ 𝑆 → ran (𝑆 ↾ 𝐴) ⊆ ran 𝑆)
86		uniss 4882	. . . . . 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 5067	. . . . 5 ⊢ (ran 𝑆 ⊆ 𝒫 (Base‘𝐺) ↔ ∪ ran 𝑆 ⊆ (Base‘𝐺))
90	50, 89	sylib 218	. . . 4 ⊢ (𝜑 → ∪ ran 𝑆 ⊆ (Base‘𝐺))
91	32, 34, 88, 90	mrcssd 17592	. . 3 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran (𝑆 ↾ 𝐴)) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
92	34	dprdspan 19966	. . . 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 19966	. . . 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 4005	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  Vcvv 3450   ∖ cdif 3914   ∩ cin 3916   ⊆ wss 3917  𝒫 cpw 4566  {csn 4592  ∪ cuni 4874   class class class wbr 5110  dom cdm 5641  ran crn 5642   ↾ cres 5643   “ cima 5644  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  0_gc0g 17409  Moorecmre 17550  mrClscmrc 17551  ACScacs 17553  Grpcgrp 18872  SubGrpcsubg 19059  Cntzccntz 19254   DProd cdprd 19932

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-tpos 8208  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9320  df-oi 9470  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-n0 12450  df-z 12537  df-uz 12801  df-fz 13476  df-fzo 13623  df-seq 13974  df-hash 14303  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-0g 17411  df-gsum 17412  df-mre 17554  df-mrc 17555  df-acs 17557  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-mhm 18717  df-submnd 18718  df-grp 18875  df-minusg 18876  df-sbg 18877  df-mulg 19007  df-subg 19062  df-ghm 19152  df-gim 19198  df-cntz 19256  df-oppg 19285  df-cmn 19719  df-dprd 19934

This theorem is referenced by:  dprdf1  19972  dprdcntz2  19977  dprddisj2  19978  dprd2dlem1  19980  dprd2da  19981  dmdprdsplit  19986  dprdsplit  19987  dpjf  19996  dpjidcl  19997  dpjlid  20000  dpjghm  20002  ablfac1eulem  20011  ablfac1eu  20012