Theorem dprdres 19927

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 19904	. . . 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 19906	. . . 4 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
6		dprdres.3	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐼)
7	5, 6	fssresd 6695	. . 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 3938	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥 ∈ 𝐼)
13		eldifi 4084	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦 ∈ 𝐴)
14	13	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦 ∈ 𝐴)
15	10, 14	sseldd 3938	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦 ∈ 𝐼)
16		eldifsni 4744	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦 ≠ 𝑥)
17	16	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦 ≠ 𝑥)
18	17	necomd 2980	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥 ≠ 𝑦)
19		eqid 2729	. . . . . . . 8 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
20	8, 9, 12, 15, 18, 19	dprdcntz 19907	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → (𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)))
21	11	fvresd 6846	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆 ↾ 𝐴)‘𝑥) = (𝑆‘𝑥))
22	14	fvresd 6846	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆 ↾ 𝐴)‘𝑦) = (𝑆‘𝑦))
23	22	fveq2d 6830	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)) = ((Cntz‘𝐺)‘(𝑆‘𝑦)))
24	20, 21, 23	3sstr4d 3993	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆 ↾ 𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)))
25	24	ralrimiva 3121	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆 ↾ 𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)))
26		fvres 6845	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → ((𝑆 ↾ 𝐴)‘𝑥) = (𝑆‘𝑥))
27	26	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆 ↾ 𝐴)‘𝑥) = (𝑆‘𝑥))
28	27	ineq1d 4172	. . . . . 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 2729	. . . . . . . . . . . . 13 ⊢ (Base‘𝐺) = (Base‘𝐺)
30	29	subgacs 19058	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
31		acsmre 17576	. . . . . . . . . . . 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 2729	. . . . . . . . . 10 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
35		resss 5956	. . . . . . . . . . . . 13 ⊢ (𝑆 ↾ 𝐴) ⊆ 𝑆
36		imass1 6056	. . . . . . . . . . . . 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 4097	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝐼 → (𝐴 ∖ {𝑥}) ⊆ (𝐼 ∖ {𝑥}))
40		imass2 6057	. . . . . . . . . . . . 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 3949	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
43	42	unissd 4871	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
44		imassrn 6026	. . . . . . . . . . . 12 ⊢ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
45	5	frnd 6664	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
46	29	subgss 19024	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ (Base‘𝐺))
47		velpw 4558	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝒫 (Base‘𝐺) ↔ 𝑥 ⊆ (Base‘𝐺))
48	46, 47	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ∈ 𝒫 (Base‘𝐺))
49	48	ssriv 3941	. . . . . . . . . . . . . 14 ⊢ (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺)
50	45, 49	sstrdi 3950	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
51	50	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
52	44, 51	sstrid 3949	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
53		sspwuni 5052	. . . . . . . . . . 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 17548	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
56		sslin 4196	. . . . . . . . 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 3937	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐼)
61		eqid 2729	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
62	58, 59, 60, 61, 34	dprddisj 19908	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g‘𝐺)})
63	57, 62	sseqtrd 3974	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))) ⊆ {(0g‘𝐺)})
64	5	ffvelcdmda 7022	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ∈ (SubGrp‘𝐺))
65	60, 64	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑆‘𝑥) ∈ (SubGrp‘𝐺))
66	61	subg0cl 19031	. . . . . . . . . 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 3948	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (Base‘𝐺))
69	34	mrccl 17535	. . . . . . . . . . 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 19031	. . . . . . . . . 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 4153	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0g‘𝐺) ∈ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))))
74	73	snssd 4763	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {(0g‘𝐺)} ⊆ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))))
75	63, 74	eqssd 3955	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g‘𝐺)})
76	28, 75	eqtrd 2764	. . . . 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 3121	. . 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 19900	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
80	79, 6	ssexd 5266	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
81	7	fdmd 6666	. . . 4 ⊢ (𝜑 → dom (𝑆 ↾ 𝐴) = 𝐴)
82	19, 61, 34	dmdprd 19897	. . . 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 5885	. . . . . 6 ⊢ ((𝑆 ↾ 𝐴) ⊆ 𝑆 → ran (𝑆 ↾ 𝐴) ⊆ ran 𝑆)
86		uniss 4869	. . . . . 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 5052	. . . . 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 17548	. . 3 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran (𝑆 ↾ 𝐴)) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
92	34	dprdspan 19926	. . . 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 19926	. . . 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 3993	. 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 2925  ∀wral 3044  Vcvv 3438   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4553  {csn 4579  ∪ cuni 4861   class class class wbr 5095  dom cdm 5623  ran crn 5624   ↾ cres 5625   “ cima 5626  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353  Basecbs 17138  0_gc0g 17361  Moorecmre 17502  mrClscmrc 17503  ACScacs 17505  Grpcgrp 18830  SubGrpcsubg 19017  Cntzccntz 19212   DProd cdprd 19892

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 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-of 7617  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8632  df-map 8762  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-fsupp 9271  df-oi 9421  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-n0 12403  df-z 12490  df-uz 12754  df-fz 13429  df-fzo 13576  df-seq 13927  df-hash 14256  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-0g 17363  df-gsum 17364  df-mre 17506  df-mrc 17507  df-acs 17509  df-mgm 18532  df-sgrp 18611  df-mnd 18627  df-mhm 18675  df-submnd 18676  df-grp 18833  df-minusg 18834  df-sbg 18835  df-mulg 18965  df-subg 19020  df-ghm 19110  df-gim 19156  df-cntz 19214  df-oppg 19243  df-cmn 19679  df-dprd 19894

This theorem is referenced by:  dprdf1  19932  dprdcntz2  19937  dprddisj2  19938  dprd2dlem1  19940  dprd2da  19941  dmdprdsplit  19946  dprdsplit  19947  dpjf  19956  dpjidcl  19957  dpjlid  19960  dpjghm  19962  ablfac1eulem  19971  ablfac1eu  19972