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Theorem subgdmdprd 20002

Description: A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)

**Hypothesis**
Ref	Expression
subgdprd.1	⊢ 𝐻 = (𝐺 ↾_s 𝐴)

**Assertion**
Ref	Expression
subgdmdprd	⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴)))

Proof of Theorem subgdmdprd Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldmdprd 19965	. . . 4 ⊢ Rel dom DProd
2	1	brrelex2i 5681	. . 3 ⊢ (𝐻dom DProd 𝑆 → 𝑆 ∈ V)
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝐻dom DProd 𝑆 → 𝑆 ∈ V))
4	1	brrelex2i 5681	. . . 4 ⊢ (𝐺dom DProd 𝑆 → 𝑆 ∈ V)
5	4	adantr 480	. . 3 ⊢ ((𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴) → 𝑆 ∈ V)
6	5	a1i 11	. 2 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → ((𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴) → 𝑆 ∈ V))
7		ffvelcdm 7027	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ 𝑥 ∈ dom 𝑆) → (𝑆‘𝑥) ∈ (SubGrp‘𝐻))
8	7	ad2ant2lr 749	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆‘𝑥) ∈ (SubGrp‘𝐻))
9		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐻) = (Base‘𝐻)
10	9	subgss 19094	. . . . . . . . . . . . . . 15 ⊢ ((𝑆‘𝑥) ∈ (SubGrp‘𝐻) → (𝑆‘𝑥) ⊆ (Base‘𝐻))
11	8, 10	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆‘𝑥) ⊆ (Base‘𝐻))
12		subgdprd.1	. . . . . . . . . . . . . . . 16 ⊢ 𝐻 = (𝐺 ↾_s 𝐴)
13	12	subgbas 19097	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐴 = (Base‘𝐻))
14	13	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝐴 = (Base‘𝐻))
15	11, 14	sseqtrrd 3960	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆‘𝑥) ⊆ 𝐴)
16	15	biantrud 531	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → ((𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ↔ ((𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ (𝑆‘𝑥) ⊆ 𝐴)))
17		simpll 767	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝐴 ∈ (SubGrp‘𝐺))
18		simplr 769	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝑆:dom 𝑆⟶(SubGrp‘𝐻))
19		eldifi 4072	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (dom 𝑆 ∖ {𝑥}) → 𝑦 ∈ dom 𝑆)
20	19	ad2antll 730	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝑦 ∈ dom 𝑆)
21	18, 20	ffvelcdmd 7031	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆‘𝑦) ∈ (SubGrp‘𝐻))
22	9	subgss 19094	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆‘𝑦) ∈ (SubGrp‘𝐻) → (𝑆‘𝑦) ⊆ (Base‘𝐻))
23	21, 22	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆‘𝑦) ⊆ (Base‘𝐻))
24	23, 14	sseqtrrd 3960	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆‘𝑦) ⊆ 𝐴)
25		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
26		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (Cntz‘𝐻) = (Cntz‘𝐻)
27	12, 25, 26	resscntz 19299	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ (𝑆‘𝑦) ⊆ 𝐴) → ((Cntz‘𝐻)‘(𝑆‘𝑦)) = (((Cntz‘𝐺)‘(𝑆‘𝑦)) ∩ 𝐴))
28	17, 24, 27	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → ((Cntz‘𝐻)‘(𝑆‘𝑦)) = (((Cntz‘𝐺)‘(𝑆‘𝑦)) ∩ 𝐴))
29	28	sseq2d 3955	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → ((𝑆‘𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆‘𝑦)) ↔ (𝑆‘𝑥) ⊆ (((Cntz‘𝐺)‘(𝑆‘𝑦)) ∩ 𝐴)))
30		ssin 4180	. . . . . . . . . . . . 13 ⊢ (((𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ (𝑆‘𝑥) ⊆ 𝐴) ↔ (𝑆‘𝑥) ⊆ (((Cntz‘𝐺)‘(𝑆‘𝑦)) ∩ 𝐴))
31	29, 30	bitr4di 289	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → ((𝑆‘𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆‘𝑦)) ↔ ((𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ (𝑆‘𝑥) ⊆ 𝐴)))
32	16, 31	bitr4d 282	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → ((𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ↔ (𝑆‘𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆‘𝑦))))
33	32	anassrs 467	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥})) → ((𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ↔ (𝑆‘𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆‘𝑦))))
34	33	ralbidva 3159	. . . . . . . . 9 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ↔ ∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆‘𝑦))))
35		subgrcl 19098	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
36	35	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐺 ∈ Grp)
37		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐺) = (Base‘𝐺)
38	37	subgacs 19127	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
39		acsmre 17609	. . . . . . . . . . . . . 14 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
40	36, 38, 39	3syl 18	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
41	12	subggrp 19096	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
42	41	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐻 ∈ Grp)
43	9	subgacs 19127	. . . . . . . . . . . . . . 15 ⊢ (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
44		acsmre 17609	. . . . . . . . . . . . . . 15 ⊢ ((SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)) → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
45	42, 43, 44	3syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
46		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (mrCls‘(SubGrp‘𝐻)) = (mrCls‘(SubGrp‘𝐻))
47		imassrn 6030	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ ran 𝑆
48		frn 6669	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆:dom 𝑆⟶(SubGrp‘𝐻) → ran 𝑆 ⊆ (SubGrp‘𝐻))
49	48	ad2antlr 728	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ran 𝑆 ⊆ (SubGrp‘𝐻))
50	47, 49	sstrid 3934	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (SubGrp‘𝐻))
51		mresspw 17545	. . . . . . . . . . . . . . . . 17 ⊢ ((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
52	45, 51	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
53	50, 52	sstrd 3933	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐻))
54		sspwuni 5043	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐻) ↔ ∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐻))
55	53, 54	sylib 218	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐻))
56	45, 46, 55	mrcssidd 17582	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))))
57	46	mrccl 17568	. . . . . . . . . . . . . . . 16 ⊢ (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ ∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐻)) → ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻))
58	45, 55, 57	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻))
59	12	subsubg 19116	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻) ↔ (((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ⊆ 𝐴)))
60	59	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻) ↔ (((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ⊆ 𝐴)))
61	58, 60	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ⊆ 𝐴))
62	61	simpld 494	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
63		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
64	63	mrcsscl 17577	. . . . . . . . . . . . 13 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∧ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))))
65	40, 56, 62, 64	syl3anc 1374	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))))
66	13	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐴 = (Base‘𝐻))
67	55, 66	sseqtrrd 3960	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ 𝐴)
68	37	subgss 19094	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐴 ⊆ (Base‘𝐺))
69	68	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐴 ⊆ (Base‘𝐺))
70	67, 69	sstrd 3933	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐺))
71	40, 63, 70	mrcssidd 17582	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))))
72	63	mrccl 17568	. . . . . . . . . . . . . . 15 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
73	40, 70, 72	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
74		simpll 767	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐴 ∈ (SubGrp‘𝐺))
75	63	mrcsscl 17577	. . . . . . . . . . . . . . 15 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ 𝐴 ∧ 𝐴 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ⊆ 𝐴)
76	40, 67, 74, 75	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ⊆ 𝐴)
77	12	subsubg 19116	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻) ↔ (((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ⊆ 𝐴)))
78	77	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻) ↔ (((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) ∧ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ⊆ 𝐴)))
79	73, 76, 78	mpbir2and 714	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻))
80	46	mrcsscl 17577	. . . . . . . . . . . . 13 ⊢ (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ ∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∧ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻)) → ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))))
81	45, 71, 79, 80	syl3anc 1374	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))))
82	65, 81	eqssd 3940	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) = ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))))
83	82	ineq2d 4161	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))))
84		eqid 2737	. . . . . . . . . . . . 13 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
85	12, 84	subg0 19099	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (0_g‘𝐺) = (0_g‘𝐻))
86	85	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (0_g‘𝐺) = (0_g‘𝐻))
87	86	sneqd 4580	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → {(0_g‘𝐺)} = {(0_g‘𝐻)})
88	83, 87	eqeq12d 2753	. . . . . . . . 9 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐺)} ↔ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐻)}))
89	34, 88	anbi12d 633	. . . . . . . 8 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐺)}) ↔ (∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐻)})))
90	89	ralbidva 3159	. . . . . . 7 ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) → (∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐺)}) ↔ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐻)})))
91	90	pm5.32da 579	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → ((𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐺)})) ↔ (𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐻)}))))
92	12	subsubg 19116	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ⊆ 𝐴)))
93		elin 3906	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((SubGrp‘𝐺) ∩ 𝒫 𝐴) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝒫 𝐴))
94		velpw 4547	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
95	94	anbi2i 624	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝒫 𝐴) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ⊆ 𝐴))
96	93, 95	bitri 275	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ((SubGrp‘𝐺) ∩ 𝒫 𝐴) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ⊆ 𝐴))
97	92, 96	bitr4di 289	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝑥 ∈ (SubGrp‘𝐻) ↔ 𝑥 ∈ ((SubGrp‘𝐺) ∩ 𝒫 𝐴)))
98	97	eqrdv 2735	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (SubGrp‘𝐻) = ((SubGrp‘𝐺) ∩ 𝒫 𝐴))
99	98	sseq2d 3955	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (ran 𝑆 ⊆ (SubGrp‘𝐻) ↔ ran 𝑆 ⊆ ((SubGrp‘𝐺) ∩ 𝒫 𝐴)))
100		ssin 4180	. . . . . . . . . 10 ⊢ ((ran 𝑆 ⊆ (SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ↔ ran 𝑆 ⊆ ((SubGrp‘𝐺) ∩ 𝒫 𝐴))
101	99, 100	bitr4di 289	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (ran 𝑆 ⊆ (SubGrp‘𝐻) ↔ (ran 𝑆 ⊆ (SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
102	101	anbi2d 631	. . . . . . . 8 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → ((𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ (SubGrp‘𝐻)) ↔ (𝑆 Fn dom 𝑆 ∧ (ran 𝑆 ⊆ (SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴))))
103		df-f 6496	. . . . . . . 8 ⊢ (𝑆:dom 𝑆⟶(SubGrp‘𝐻) ↔ (𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ (SubGrp‘𝐻)))
104		df-f 6496	. . . . . . . . . 10 ⊢ (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ↔ (𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ (SubGrp‘𝐺)))
105	104	anbi1i 625	. . . . . . . . 9 ⊢ ((𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ↔ ((𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ (SubGrp‘𝐺)) ∧ ran 𝑆 ⊆ 𝒫 𝐴))
106		anass 468	. . . . . . . . 9 ⊢ (((𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ (SubGrp‘𝐺)) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ↔ (𝑆 Fn dom 𝑆 ∧ (ran 𝑆 ⊆ (SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
107	105, 106	bitri 275	. . . . . . . 8 ⊢ ((𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ↔ (𝑆 Fn dom 𝑆 ∧ (ran 𝑆 ⊆ (SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
108	102, 103, 107	3bitr4g 314	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝑆:dom 𝑆⟶(SubGrp‘𝐻) ↔ (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
109	108	anbi1d 632	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → ((𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐺)})) ↔ ((𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐺)}))))
110	91, 109	bitr3d 281	. . . . 5 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → ((𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐻)})) ↔ ((𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐺)}))))
111	110	adantr 480	. . . 4 ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → ((𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐻)})) ↔ ((𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐺)}))))
112		dmexg 7845	. . . . . 6 ⊢ (𝑆 ∈ V → dom 𝑆 ∈ V)
113	112	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → dom 𝑆 ∈ V)
114		eqidd 2738	. . . . 5 ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → dom 𝑆 = dom 𝑆)
115	41	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → 𝐻 ∈ Grp)
116		eqid 2737	. . . . . . . 8 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
117	26, 116, 46	dmdprd 19966	. . . . . . 7 ⊢ ((dom 𝑆 ∈ V ∧ dom 𝑆 = dom 𝑆) → (𝐻dom DProd 𝑆 ↔ (𝐻 ∈ Grp ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐻)}))))
118		3anass 1095	. . . . . . 7 ⊢ ((𝐻 ∈ Grp ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐻)})) ↔ (𝐻 ∈ Grp ∧ (𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐻)}))))
119	117, 118	bitrdi 287	. . . . . 6 ⊢ ((dom 𝑆 ∈ V ∧ dom 𝑆 = dom 𝑆) → (𝐻dom DProd 𝑆 ↔ (𝐻 ∈ Grp ∧ (𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐻)})))))
120	119	baibd 539	. . . . 5 ⊢ (((dom 𝑆 ∈ V ∧ dom 𝑆 = dom 𝑆) ∧ 𝐻 ∈ Grp) → (𝐻dom DProd 𝑆 ↔ (𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐻)}))))
121	113, 114, 115, 120	syl21anc 838	. . . 4 ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → (𝐻dom DProd 𝑆 ↔ (𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐻)}))))
122	35	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → 𝐺 ∈ Grp)
123	25, 84, 63	dmdprd 19966	. . . . . . . . 9 ⊢ ((dom 𝑆 ∈ V ∧ dom 𝑆 = dom 𝑆) → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐺)}))))
124		3anass 1095	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐺)})) ↔ (𝐺 ∈ Grp ∧ (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐺)}))))
125	123, 124	bitrdi 287	. . . . . . . 8 ⊢ ((dom 𝑆 ∈ V ∧ dom 𝑆 = dom 𝑆) → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐺)})))))
126	125	baibd 539	. . . . . . 7 ⊢ (((dom 𝑆 ∈ V ∧ dom 𝑆 = dom 𝑆) ∧ 𝐺 ∈ Grp) → (𝐺dom DProd 𝑆 ↔ (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐺)}))))
127	113, 114, 122, 126	syl21anc 838	. . . . . 6 ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → (𝐺dom DProd 𝑆 ↔ (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐺)}))))
128	127	anbi1d 632	. . . . 5 ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → ((𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴) ↔ ((𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐺)})) ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
129		an32 647	. . . . 5 ⊢ (((𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐺)})) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ↔ ((𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐺)})))
130	128, 129	bitrdi 287	. . . 4 ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → ((𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴) ↔ ((𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0_g‘𝐺)}))))
131	111, 121, 130	3bitr4d 311	. . 3 ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
132	131	ex 412	. 2 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝑆 ∈ V → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴))))
133	3, 6, 132	pm5.21ndd 379	1 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3430 ∖ cdif 3887 ∩ cin 3889 ⊆ wss 3890 𝒫 cpw 4542 {csn 4568 ∪ cuni 4851 class class class wbr 5086 dom cdm 5624 ran crn 5625 “ cima 5627 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 Basecbs 17170 ↾_s cress 17191 0_gc0g 17393 Moorecmre 17535 mrClscmrc 17536 ACScacs 17538 Grpcgrp 18900 SubGrpcsubg 19087 Cntzccntz 19281 DProd cdprd 19961

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 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

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 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-0g 17395 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-grp 18903 df-minusg 18904 df-subg 19090 df-cntz 19283 df-dprd 19963

This theorem is referenced by: subgdprd 20003 ablfaclem3 20055

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