Theorem dprdres 19971

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 19948	. . . 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 19950	. . . 4 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
6		dprdres.3	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐼)
7	5, 6	fssresd 6709	. . 3 ⊢ (𝜑 → (𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺))
8	1	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝐺dom DProd 𝑆)
9	4	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → dom 𝑆 = 𝐼)
10	6	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝐴 ⊆ 𝐼)
11		simplr 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥 ∈ 𝐴)
12	10, 11	sseldd 3936	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥 ∈ 𝐼)
13		eldifi 4085	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦 ∈ 𝐴)
14	13	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦 ∈ 𝐴)
15	10, 14	sseldd 3936	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦 ∈ 𝐼)
16		eldifsni 4748	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦 ≠ 𝑥)
17	16	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦 ≠ 𝑥)
18	17	necomd 2988	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥 ≠ 𝑦)
19		eqid 2737	. . . . . . . 8 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
20	8, 9, 12, 15, 18, 19	dprdcntz 19951	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → (𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)))
21	11	fvresd 6862	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆 ↾ 𝐴)‘𝑥) = (𝑆‘𝑥))
22	14	fvresd 6862	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆 ↾ 𝐴)‘𝑦) = (𝑆‘𝑦))
23	22	fveq2d 6846	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)) = ((Cntz‘𝐺)‘(𝑆‘𝑦)))
24	20, 21, 23	3sstr4d 3991	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆 ↾ 𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)))
25	24	ralrimiva 3130	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆 ↾ 𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)))
26		fvres 6861	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → ((𝑆 ↾ 𝐴)‘𝑥) = (𝑆‘𝑥))
27	26	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆 ↾ 𝐴)‘𝑥) = (𝑆‘𝑥))
28	27	ineq1d 4173	. . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (Base‘𝐺) = (Base‘𝐺)
30	29	subgacs 19102	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
31		acsmre 17587	. . . . . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
35		resss 5968	. . . . . . . . . . . . 13 ⊢ (𝑆 ↾ 𝐴) ⊆ 𝑆
36		imass1 6068	. . . . . . . . . . . . 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 4098	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝐼 → (𝐴 ∖ {𝑥}) ⊆ (𝐼 ∖ {𝑥}))
40		imass2 6069	. . . . . . . . . . . . 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 3947	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
43	42	unissd 4875	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
44		imassrn 6038	. . . . . . . . . . . 12 ⊢ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
45	5	frnd 6678	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
46	29	subgss 19069	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ (Base‘𝐺))
47		velpw 4561	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝒫 (Base‘𝐺) ↔ 𝑥 ⊆ (Base‘𝐺))
48	46, 47	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ∈ 𝒫 (Base‘𝐺))
49	48	ssriv 3939	. . . . . . . . . . . . . 14 ⊢ (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺)
50	45, 49	sstrdi 3948	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
51	50	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
52	44, 51	sstrid 3947	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
53		sspwuni 5057	. . . . . . . . . . 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 17559	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
56		sslin 4197	. . . . . . . . 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 3935	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐼)
61		eqid 2737	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
62	58, 59, 60, 61, 34	dprddisj 19952	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g‘𝐺)})
63	57, 62	sseqtrd 3972	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))) ⊆ {(0g‘𝐺)})
64	5	ffvelcdmda 7038	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ∈ (SubGrp‘𝐺))
65	60, 64	syldan 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑆‘𝑥) ∈ (SubGrp‘𝐺))
66	61	subg0cl 19076	. . . . . . . . . 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 3946	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (Base‘𝐺))
69	34	mrccl 17546	. . . . . . . . . . 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 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
71	61	subg0cl 19076	. . . . . . . . . 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 4154	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0g‘𝐺) ∈ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))))
74	73	snssd 4767	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {(0g‘𝐺)} ⊆ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))))
75	63, 74	eqssd 3953	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g‘𝐺)})
76	28, 75	eqtrd 2772	. . . . 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 3130	. . 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 19944	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
80	79, 6	ssexd 5271	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
81	7	fdmd 6680	. . . 4 ⊢ (𝜑 → dom (𝑆 ↾ 𝐴) = 𝐴)
82	19, 61, 34	dmdprd 19941	. . . 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 585	. . 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 1344	. 2 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐴))
85		rnss 5896	. . . . . 6 ⊢ ((𝑆 ↾ 𝐴) ⊆ 𝑆 → ran (𝑆 ↾ 𝐴) ⊆ ran 𝑆)
86		uniss 4873	. . . . . 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 5057	. . . . 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 17559	. . 3 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran (𝑆 ↾ 𝐴)) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
92	34	dprdspan 19970	. . . 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 19970	. . . 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 3991	. 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 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3442   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4556  {csn 4582  ∪ cuni 4865   class class class wbr 5100  dom cdm 5632  ran crn 5633   ↾ cres 5634   “ cima 5635  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  0_gc0g 17371  Moorecmre 17513  mrClscmrc 17514  ACScacs 17516  Grpcgrp 18875  SubGrpcsubg 19062  Cntzccntz 19256   DProd cdprd 19936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-of 7632  df-om 7819  df-1st 7943  df-2nd 7944  df-supp 8113  df-tpos 8178  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-map 8777  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9277  df-oi 9427  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-n0 12414  df-z 12501  df-uz 12764  df-fz 13436  df-fzo 13583  df-seq 13937  df-hash 14266  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-0g 17373  df-gsum 17374  df-mre 17517  df-mrc 17518  df-acs 17520  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-mhm 18720  df-submnd 18721  df-grp 18878  df-minusg 18879  df-sbg 18880  df-mulg 19010  df-subg 19065  df-ghm 19154  df-gim 19200  df-cntz 19258  df-oppg 19287  df-cmn 19723  df-dprd 19938

This theorem is referenced by:  dprdf1  19976  dprdcntz2  19981  dprddisj2  19982  dprd2dlem1  19984  dprd2da  19985  dmdprdsplit  19990  dprdsplit  19991  dpjf  20000  dpjidcl  20001  dpjlid  20004  dpjghm  20006  ablfac1eulem  20015  ablfac1eu  20016