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Theorem subgdmdprd 18554
Description: A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
Hypothesis
Ref Expression
subgdprd.1 𝐻 = (𝐺s 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.
StepHypRef Expression
1 reldmdprd 18517 . . . 4 Rel dom DProd
21brrelex2i 5268 . . 3 (𝐻dom DProd 𝑆𝑆 ∈ V)
32a1i 11 . 2 (𝐴 ∈ (SubGrp‘𝐺) → (𝐻dom DProd 𝑆𝑆 ∈ V))
41brrelex2i 5268 . . . 4 (𝐺dom DProd 𝑆𝑆 ∈ V)
54adantr 472 . . 3 ((𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴) → 𝑆 ∈ V)
65a1i 11 . 2 (𝐴 ∈ (SubGrp‘𝐺) → ((𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴) → 𝑆 ∈ V))
7 ffvelrn 6472 . . . . . . . . . . . . . . . 16 ((𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ 𝑥 ∈ dom 𝑆) → (𝑆𝑥) ∈ (SubGrp‘𝐻))
87ad2ant2lr 801 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆𝑥) ∈ (SubGrp‘𝐻))
9 eqid 2724 . . . . . . . . . . . . . . . 16 (Base‘𝐻) = (Base‘𝐻)
109subgss 17717 . . . . . . . . . . . . . . 15 ((𝑆𝑥) ∈ (SubGrp‘𝐻) → (𝑆𝑥) ⊆ (Base‘𝐻))
118, 10syl 17 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆𝑥) ⊆ (Base‘𝐻))
12 subgdprd.1 . . . . . . . . . . . . . . . 16 𝐻 = (𝐺s 𝐴)
1312subgbas 17720 . . . . . . . . . . . . . . 15 (𝐴 ∈ (SubGrp‘𝐺) → 𝐴 = (Base‘𝐻))
1413ad2antrr 764 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝐴 = (Base‘𝐻))
1511, 14sseqtr4d 3748 . . . . . . . . . . . . 13 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆𝑥) ⊆ 𝐴)
1615biantrud 529 . . . . . . . . . . . 12 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → ((𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ↔ ((𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ (𝑆𝑥) ⊆ 𝐴)))
17 simpll 807 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝐴 ∈ (SubGrp‘𝐺))
18 simplr 809 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝑆:dom 𝑆⟶(SubGrp‘𝐻))
19 eldifi 3840 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (dom 𝑆 ∖ {𝑥}) → 𝑦 ∈ dom 𝑆)
2019ad2antll 767 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝑦 ∈ dom 𝑆)
2118, 20ffvelrnd 6475 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆𝑦) ∈ (SubGrp‘𝐻))
229subgss 17717 . . . . . . . . . . . . . . . . 17 ((𝑆𝑦) ∈ (SubGrp‘𝐻) → (𝑆𝑦) ⊆ (Base‘𝐻))
2321, 22syl 17 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆𝑦) ⊆ (Base‘𝐻))
2423, 14sseqtr4d 3748 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆𝑦) ⊆ 𝐴)
25 eqid 2724 . . . . . . . . . . . . . . . 16 (Cntz‘𝐺) = (Cntz‘𝐺)
26 eqid 2724 . . . . . . . . . . . . . . . 16 (Cntz‘𝐻) = (Cntz‘𝐻)
2712, 25, 26resscntz 17885 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (SubGrp‘𝐺) ∧ (𝑆𝑦) ⊆ 𝐴) → ((Cntz‘𝐻)‘(𝑆𝑦)) = (((Cntz‘𝐺)‘(𝑆𝑦)) ∩ 𝐴))
2817, 24, 27syl2anc 696 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → ((Cntz‘𝐻)‘(𝑆𝑦)) = (((Cntz‘𝐺)‘(𝑆𝑦)) ∩ 𝐴))
2928sseq2d 3739 . . . . . . . . . . . . 13 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → ((𝑆𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆𝑦)) ↔ (𝑆𝑥) ⊆ (((Cntz‘𝐺)‘(𝑆𝑦)) ∩ 𝐴)))
30 ssin 3943 . . . . . . . . . . . . 13 (((𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ (𝑆𝑥) ⊆ 𝐴) ↔ (𝑆𝑥) ⊆ (((Cntz‘𝐺)‘(𝑆𝑦)) ∩ 𝐴))
3129, 30syl6bbr 278 . . . . . . . . . . . 12 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → ((𝑆𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆𝑦)) ↔ ((𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ (𝑆𝑥) ⊆ 𝐴)))
3216, 31bitr4d 271 . . . . . . . . . . 11 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → ((𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ↔ (𝑆𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆𝑦))))
3332anassrs 683 . . . . . . . . . 10 ((((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) ∧ 𝑦 ∈ (dom 𝑆 ∖ {𝑥})) → ((𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ↔ (𝑆𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆𝑦))))
3433ralbidva 3087 . . . . . . . . 9 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ↔ ∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆𝑦))))
35 subgrcl 17721 . . . . . . . . . . . . . . 15 (𝐴 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
3635ad2antrr 764 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐺 ∈ Grp)
37 eqid 2724 . . . . . . . . . . . . . . 15 (Base‘𝐺) = (Base‘𝐺)
3837subgacs 17751 . . . . . . . . . . . . . 14 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
39 acsmre 16435 . . . . . . . . . . . . . 14 ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
4036, 38, 393syl 18 . . . . . . . . . . . . 13 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
4112subggrp 17719 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
4241ad2antrr 764 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐻 ∈ Grp)
439subgacs 17751 . . . . . . . . . . . . . . 15 (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
44 acsmre 16435 . . . . . . . . . . . . . . 15 ((SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)) → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
4542, 43, 443syl 18 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
46 eqid 2724 . . . . . . . . . . . . . 14 (mrCls‘(SubGrp‘𝐻)) = (mrCls‘(SubGrp‘𝐻))
47 imassrn 5587 . . . . . . . . . . . . . . . . 17 (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ ran 𝑆
48 frn 6166 . . . . . . . . . . . . . . . . . 18 (𝑆:dom 𝑆⟶(SubGrp‘𝐻) → ran 𝑆 ⊆ (SubGrp‘𝐻))
4948ad2antlr 765 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ran 𝑆 ⊆ (SubGrp‘𝐻))
5047, 49syl5ss 3720 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (SubGrp‘𝐻))
51 mresspw 16375 . . . . . . . . . . . . . . . . 17 ((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
5245, 51syl 17 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (SubGrp‘𝐻) ⊆ 𝒫 (Base‘𝐻))
5350, 52sstrd 3719 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐻))
54 sspwuni 4719 . . . . . . . . . . . . . . 15 ((𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐻) ↔ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐻))
5553, 54sylib 208 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐻))
5645, 46, 55mrcssidd 16408 . . . . . . . . . . . . 13 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))))
5746mrccl 16394 . . . . . . . . . . . . . . . 16 (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐻)) → ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻))
5845, 55, 57syl2anc 696 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻))
5912subsubg 17739 . . . . . . . . . . . . . . . 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)))
6059ad2antrr 764 . . . . . . . . . . . . . . 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)))
6158, 60mpbid 222 . . . . . . . . . . . . . 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))
6261simpld 477 . . . . . . . . . . . . 13 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
63 eqid 2724 . . . . . . . . . . . . . 14 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
6463mrcsscl 16403 . . . . . . . . . . . . 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}))))
6540, 56, 62, 64syl3anc 1439 . . . . . . . . . . . 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}))))
6613ad2antrr 764 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐴 = (Base‘𝐻))
6755, 66sseqtr4d 3748 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ 𝐴)
6837subgss 17717 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (SubGrp‘𝐺) → 𝐴 ⊆ (Base‘𝐺))
6968ad2antrr 764 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐴 ⊆ (Base‘𝐺))
7067, 69sstrd 3719 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐺))
7140, 63, 70mrcssidd 16408 . . . . . . . . . . . . 13 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))))
7263mrccl 16394 . . . . . . . . . . . . . . 15 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
7340, 70, 72syl2anc 696 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
74 simpll 807 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐴 ∈ (SubGrp‘𝐺))
7563mrcsscl 16403 . . . . . . . . . . . . . . 15 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ 𝐴𝐴 ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ⊆ 𝐴)
7640, 67, 74, 75syl3anc 1439 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ⊆ 𝐴)
7712subsubg 17739 . . . . . . . . . . . . . . 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)))
7877ad2antrr 764 . . . . . . . . . . . . . 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)))
7973, 76, 78mpbir2and 995 . . . . . . . . . . . . 13 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻))
8046mrcsscl 16403 . . . . . . . . . . . . 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}))))
8145, 71, 79, 80syl3anc 1439 . . . . . . . . . . . 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}))))
8265, 81eqssd 3726 . . . . . . . . . . 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}))))
8382ineq2d 3922 . . . . . . . . . 10 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))))
84 eqid 2724 . . . . . . . . . . . . 13 (0g𝐺) = (0g𝐺)
8512, 84subg0 17722 . . . . . . . . . . . 12 (𝐴 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
8685ad2antrr 764 . . . . . . . . . . 11 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (0g𝐺) = (0g𝐻))
8786sneqd 4297 . . . . . . . . . 10 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → {(0g𝐺)} = {(0g𝐻)})
8883, 87eqeq12d 2739 . . . . . . . . 9 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)} ↔ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐻)}))
8934, 88anbi12d 749 . . . . . . . 8 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)}) ↔ (∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐻)})))
9089ralbidva 3087 . . . . . . 7 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) → (∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)}) ↔ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐻)})))
9190pm5.32da 676 . . . . . 6 (𝐴 ∈ (SubGrp‘𝐺) → ((𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)})) ↔ (𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐻)}))))
9212subsubg 17739 . . . . . . . . . . . . 13 (𝐴 ∈ (SubGrp‘𝐺) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥𝐴)))
93 elin 3904 . . . . . . . . . . . . . 14 (𝑥 ∈ ((SubGrp‘𝐺) ∩ 𝒫 𝐴) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝒫 𝐴))
94 selpw 4273 . . . . . . . . . . . . . . 15 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
9594anbi2i 732 . . . . . . . . . . . . . 14 ((𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝒫 𝐴) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥𝐴))
9693, 95bitri 264 . . . . . . . . . . . . 13 (𝑥 ∈ ((SubGrp‘𝐺) ∩ 𝒫 𝐴) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥𝐴))
9792, 96syl6bbr 278 . . . . . . . . . . . 12 (𝐴 ∈ (SubGrp‘𝐺) → (𝑥 ∈ (SubGrp‘𝐻) ↔ 𝑥 ∈ ((SubGrp‘𝐺) ∩ 𝒫 𝐴)))
9897eqrdv 2722 . . . . . . . . . . 11 (𝐴 ∈ (SubGrp‘𝐺) → (SubGrp‘𝐻) = ((SubGrp‘𝐺) ∩ 𝒫 𝐴))
9998sseq2d 3739 . . . . . . . . . 10 (𝐴 ∈ (SubGrp‘𝐺) → (ran 𝑆 ⊆ (SubGrp‘𝐻) ↔ ran 𝑆 ⊆ ((SubGrp‘𝐺) ∩ 𝒫 𝐴)))
100 ssin 3943 . . . . . . . . . 10 ((ran 𝑆 ⊆ (SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ↔ ran 𝑆 ⊆ ((SubGrp‘𝐺) ∩ 𝒫 𝐴))
10199, 100syl6bbr 278 . . . . . . . . 9 (𝐴 ∈ (SubGrp‘𝐺) → (ran 𝑆 ⊆ (SubGrp‘𝐻) ↔ (ran 𝑆 ⊆ (SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
102101anbi2d 742 . . . . . . . 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 6005 . . . . . . . 8 (𝑆:dom 𝑆⟶(SubGrp‘𝐻) ↔ (𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ (SubGrp‘𝐻)))
104 df-f 6005 . . . . . . . . . 10 (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ↔ (𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ (SubGrp‘𝐺)))
105104anbi1i 733 . . . . . . . . 9 ((𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ↔ ((𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ (SubGrp‘𝐺)) ∧ ran 𝑆 ⊆ 𝒫 𝐴))
106 anass 684 . . . . . . . . 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)))
107105, 106bitri 264 . . . . . . . 8 ((𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ↔ (𝑆 Fn dom 𝑆 ∧ (ran 𝑆 ⊆ (SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
108102, 103, 1073bitr4g 303 . . . . . . 7 (𝐴 ∈ (SubGrp‘𝐺) → (𝑆:dom 𝑆⟶(SubGrp‘𝐻) ↔ (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
109108anbi1d 743 . . . . . 6 (𝐴 ∈ (SubGrp‘𝐺) → ((𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)})) ↔ ((𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)}))))
11091, 109bitr3d 270 . . . . 5 (𝐴 ∈ (SubGrp‘𝐺) → ((𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐻)})) ↔ ((𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)}))))
111110adantr 472 . . . 4 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → ((𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐻)})) ↔ ((𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)}))))
112 dmexg 7214 . . . . . 6 (𝑆 ∈ V → dom 𝑆 ∈ V)
113112adantl 473 . . . . 5 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → dom 𝑆 ∈ V)
114 eqidd 2725 . . . . 5 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → dom 𝑆 = dom 𝑆)
11541adantr 472 . . . . 5 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → 𝐻 ∈ Grp)
116 eqid 2724 . . . . . . . 8 (0g𝐻) = (0g𝐻)
11726, 116, 46dmdprd 18518 . . . . . . 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})(𝑆𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐻)}))))
118 3anass 1081 . . . . . . 7 ((𝐻 ∈ Grp ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐻)})) ↔ (𝐻 ∈ Grp ∧ (𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐻)}))))
119117, 118syl6bb 276 . . . . . 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})(𝑆𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐻)})))))
120119baibd 986 . . . . 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})(𝑆𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐻)}))))
121113, 114, 115, 120syl21anc 1438 . . . 4 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → (𝐻dom DProd 𝑆 ↔ (𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐻)}))))
12235adantr 472 . . . . . . 7 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → 𝐺 ∈ Grp)
12325, 84, 63dmdprd 18518 . . . . . . . . 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})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)}))))
124 3anass 1081 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)})) ↔ (𝐺 ∈ Grp ∧ (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)}))))
125123, 124syl6bb 276 . . . . . . . 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})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)})))))
126125baibd 986 . . . . . . 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})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)}))))
127113, 114, 122, 126syl21anc 1438 . . . . . 6 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → (𝐺dom DProd 𝑆 ↔ (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)}))))
128127anbi1d 743 . . . . 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})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)})) ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
129 an32 874 . . . . 5 (((𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)})) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ↔ ((𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)})))
130128, 129syl6bb 276 . . . 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})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)}))))
131111, 121, 1303bitr4d 300 . . 3 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
132131ex 449 . 2 (𝐴 ∈ (SubGrp‘𝐺) → (𝑆 ∈ V → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴))))
1333, 6, 132pm5.21ndd 368 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 196  wa 383  w3a 1072   = wceq 1596  wcel 2103  wral 3014  Vcvv 3304  cdif 3677  cin 3679  wss 3680  𝒫 cpw 4266  {csn 4285   cuni 4544   class class class wbr 4760  dom cdm 5218  ran crn 5219  cima 5221   Fn wfn 5996  wf 5997  cfv 6001  (class class class)co 6765  Basecbs 15980  s cress 15981  0gc0g 16223  Moorecmre 16365  mrClscmrc 16366  ACScacs 16368  Grpcgrp 17544  SubGrpcsubg 17710  Cntzccntz 17869   DProd cdprd 18513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-iin 4631  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-ixp 8026  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-2 11192  df-ndx 15983  df-slot 15984  df-base 15986  df-sets 15987  df-ress 15988  df-plusg 16077  df-0g 16225  df-mre 16369  df-mrc 16370  df-acs 16372  df-mgm 17364  df-sgrp 17406  df-mnd 17417  df-submnd 17458  df-grp 17547  df-minusg 17548  df-subg 17713  df-cntz 17871  df-dprd 18515
This theorem is referenced by:  subgdprd  18555  ablfaclem3  18607
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