subgdmdprd - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > subgdmdprd		Structured version Visualization version GIF version

Theorem subgdmdprd 20101

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 20064	. . . 4 ⊢ Rel dom DProd
2	1	brrelex2i 5718	. . 3 ⊢ (𝐻dom DProd 𝑆 → 𝑆 ∈ V)
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝐻dom DProd 𝑆 → 𝑆 ∈ V))
4	1	brrelex2i 5718	. . . 4 ⊢ (𝐺dom DProd 𝑆 → 𝑆 ∈ V)
5	4	adantr 485	. . 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 7076	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ 𝑥 ∈ dom 𝑆) → (𝑆‘𝑥) ∈ (SubGrp‘𝐻))
8	7	ad2ant2lr 760	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆‘𝑥) ∈ (SubGrp‘𝐻))
9		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐻) = (Base‘𝐻)
10	9	subgss 19188	. . . . . . . . . . . . . . 15 ⊢ ((𝑆‘𝑥) ∈ (SubGrp‘𝐻) → (𝑆‘𝑥) ⊆ (Base‘𝐻))
11	8, 10	syl 18	. . . . . . . . . . . . . 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 19191	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐴 = (Base‘𝐻))
14	13	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝐴 = (Base‘𝐻))
15	11, 14	sseqtrrd 3974	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆‘𝑥) ⊆ 𝐴)
16	15	biantrud 540	. . . . . . . . . . . 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 778	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝐴 ∈ (SubGrp‘𝐺))
18		simplr 780	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝑆:dom 𝑆⟶(SubGrp‘𝐻))
19		eldifi 4085	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (dom 𝑆 ∖ {𝑥}) → 𝑦 ∈ dom 𝑆)
20	19	ad2antll 741	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝑦 ∈ dom 𝑆)
21	18, 20	ffvelcdmd 7080	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆‘𝑦) ∈ (SubGrp‘𝐻))
22	9	subgss 19188	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆‘𝑦) ∈ (SubGrp‘𝐻) → (𝑆‘𝑦) ⊆ (Base‘𝐻))
23	21, 22	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆‘𝑦) ⊆ (Base‘𝐻))
24	23, 14	sseqtrrd 3974	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆‘𝑦) ⊆ 𝐴)
25		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
26		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (Cntz‘𝐻) = (Cntz‘𝐻)
27	12, 25, 26	resscntz 19398	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ (𝑆‘𝑦) ⊆ 𝐴) → ((Cntz‘𝐻)‘(𝑆‘𝑦)) = (((Cntz‘𝐺)‘(𝑆‘𝑦)) ∩ 𝐴))
28	17, 24, 27	syl2anc 595	. . . . . . . . . . . . . 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 3969	. . . . . . . . . . . . 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 4191	. . . . . . . . . . . . 13 ⊢ (((𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ (𝑆‘𝑥) ⊆ 𝐴) ↔ (𝑆‘𝑥) ⊆ (((Cntz‘𝐺)‘(𝑆‘𝑦)) ∩ 𝐴))
31	29, 30	bitr4di 292	. . . . . . . . . . . 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 285	. . . . . . . . . . 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 472	. . . . . . . . . 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 3186	. . . . . . . . 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 19192	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
36	35	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐺 ∈ Grp)
37		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐺) = (Base‘𝐺)
38	37	subgacs 19222	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
39		acsmre 17703	. . . . . . . . . . . . . 14 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
40	36, 38, 39	3syl 19	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
41	12	subggrp 19190	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
42	41	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐻 ∈ Grp)
43	9	subgacs 19222	. . . . . . . . . . . . . . 15 ⊢ (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
44		acsmre 17703	. . . . . . . . . . . . . . 15 ⊢ ((SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)) → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
45	42, 43, 44	3syl 19	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
46		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (mrCls‘(SubGrp‘𝐻)) = (mrCls‘(SubGrp‘𝐻))
47		imassrn 6073	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ ran 𝑆
48		frn 6713	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆:dom 𝑆⟶(SubGrp‘𝐻) → ran 𝑆 ⊆ (SubGrp‘𝐻))
49	48	ad2antlr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ran 𝑆 ⊆ (SubGrp‘𝐻))
50	47, 49	sstrid 3948	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (SubGrp‘𝐻))
51		mresspw 17639	. . . . . . . . . . . . . . . . 17 ⊢ ((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
52	45, 51	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
53	50, 52	sstrd 3947	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐻))
54		sspwuni 5066	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐻) ↔ ∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐻))
55	53, 54	sylib 221	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐻))
56	45, 46, 55	mrcssidd 17676	. . . . . . . . . . . . 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 17662	. . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻))
59	12	subsubg 19211	. . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . 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 499	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐻))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
63		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
64	63	mrcsscl 17671	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐴 = (Base‘𝐻))
67	55, 66	sseqtrrd 3974	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ 𝐴)
68	37	subgss 19188	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐴 ⊆ (Base‘𝐺))
69	68	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐴 ⊆ (Base‘𝐺))
70	67, 69	sstrd 3947	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐺))
71	40, 63, 70	mrcssidd 17676	. . . . . . . . . . . . 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 17662	. . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
74		simpll 778	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐴 ∈ (SubGrp‘𝐺))
75	63	mrcsscl 17671	. . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ⊆ 𝐴)
77	12	subsubg 19211	. . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . 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 725	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻))
80	46	mrcsscl 17671	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 3954	. . . . . . . . . . 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 4173	. . . . . . . . . 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 2763	. . . . . . . . . . . . 13 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
85	12, 84	subg0 19193	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (0_g‘𝐺) = (0_g‘𝐻))
86	85	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (0_g‘𝐺) = (0_g‘𝐻))
87	86	sneqd 4601	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → {(0_g‘𝐺)} = {(0_g‘𝐻)})
88	83, 87	eqeq12d 2779	. . . . . . . . 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 643	. . . . . . . 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 3186	. . . . . . 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 589	. . . . . 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 19211	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ⊆ 𝐴)))
93		elin 3921	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((SubGrp‘𝐺) ∩ 𝒫 𝐴) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝒫 𝐴))
94		velpw 4567	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
95	94	anbi2i 634	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝒫 𝐴) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ⊆ 𝐴))
96	93, 95	bitri 278	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ((SubGrp‘𝐺) ∩ 𝒫 𝐴) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ⊆ 𝐴))
97	92, 96	bitr4di 292	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝑥 ∈ (SubGrp‘𝐻) ↔ 𝑥 ∈ ((SubGrp‘𝐺) ∩ 𝒫 𝐴)))
98	97	eqrdv 2761	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (SubGrp‘𝐻) = ((SubGrp‘𝐺) ∩ 𝒫 𝐴))
99	98	sseq2d 3969	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (ran 𝑆 ⊆ (SubGrp‘𝐻) ↔ ran 𝑆 ⊆ ((SubGrp‘𝐺) ∩ 𝒫 𝐴)))
100		ssin 4191	. . . . . . . . . 10 ⊢ ((ran 𝑆 ⊆ (SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ↔ ran 𝑆 ⊆ ((SubGrp‘𝐺) ∩ 𝒫 𝐴))
101	99, 100	bitr4di 292	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (ran 𝑆 ⊆ (SubGrp‘𝐻) ↔ (ran 𝑆 ⊆ (SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
102	101	anbi2d 641	. . . . . . . 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 6540	. . . . . . . 8 ⊢ (𝑆:dom 𝑆⟶(SubGrp‘𝐻) ↔ (𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ (SubGrp‘𝐻)))
104		df-f 6540	. . . . . . . . . 10 ⊢ (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ↔ (𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ (SubGrp‘𝐺)))
105	104	anbi1i 635	. . . . . . . . 9 ⊢ ((𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ↔ ((𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ (SubGrp‘𝐺)) ∧ ran 𝑆 ⊆ 𝒫 𝐴))
106		anass 473	. . . . . . . . 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 278	. . . . . . . 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 317	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝑆:dom 𝑆⟶(SubGrp‘𝐻) ↔ (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
109	108	anbi1d 642	. . . . . 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 284	. . . . 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 485	. . . 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 7894	. . . . . 6 ⊢ (𝑆 ∈ V → dom 𝑆 ∈ V)
113	112	adantl 486	. . . . 5 ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → dom 𝑆 ∈ V)
114		eqidd 2764	. . . . 5 ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → dom 𝑆 = dom 𝑆)
115	41	adantr 485	. . . . 5 ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → 𝐻 ∈ Grp)
116		eqid 2763	. . . . . . . 8 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
117	26, 116, 46	dmdprd 20065	. . . . . . 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 1111	. . . . . . 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 290	. . . . . 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 548	. . . . 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 850	. . . 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 485	. . . . . . 7 ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → 𝐺 ∈ Grp)
123	25, 84, 63	dmdprd 20065	. . . . . . . . 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 1111	. . . . . . . . 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 290	. . . . . . . 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 548	. . . . . . 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 850	. . . . . 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 642	. . . . 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 658	. . . . 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 290	. . . 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 314	. . 3 ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
132	131	ex 417	. 2 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝑆 ∈ V → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴))))
133	3, 6, 132	pm5.21ndd 382	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 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ∀wral 3079 Vcvv 3455 ∖ cdif 3902 ∩ cin 3904 ⊆ wss 3905 𝒫 cpw 4562 {csn 4589 ∪ cuni 4872 class class class wbr 5109 dom cdm 5661 ran crn 5662 “ cima 5664 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 Basecbs 17264 ↾_s cress 17285 0_gc0g 17487 Moorecmre 17629 mrClscmrc 17630 ACScacs 17632 Grpcgrp 18995 SubGrpcsubg 19181 Cntzccntz 19380 DProd cdprd 20060

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 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 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-nn 12229 df-2 12298 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-0g 17489 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-grp 18998 df-minusg 18999 df-subg 19184 df-cntz 19382 df-dprd 20062

This theorem is referenced by: subgdprd 20102 ablfaclem3 20154

W3C validator