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Theorem subgdmdprd 19217
 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 19180 . . . 4 Rel dom DProd
21brrelex2i 5579 . . 3 (𝐻dom DProd 𝑆𝑆 ∈ V)
32a1i 11 . 2 (𝐴 ∈ (SubGrp‘𝐺) → (𝐻dom DProd 𝑆𝑆 ∈ V))
41brrelex2i 5579 . . . 4 (𝐺dom DProd 𝑆𝑆 ∈ V)
54adantr 485 . . 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 6841 . . . . . . . . . . . . . . . 16 ((𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ 𝑥 ∈ dom 𝑆) → (𝑆𝑥) ∈ (SubGrp‘𝐻))
87ad2ant2lr 748 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆𝑥) ∈ (SubGrp‘𝐻))
9 eqid 2759 . . . . . . . . . . . . . . . 16 (Base‘𝐻) = (Base‘𝐻)
109subgss 18340 . . . . . . . . . . . . . . 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 18343 . . . . . . . . . . . . . . 15 (𝐴 ∈ (SubGrp‘𝐺) → 𝐴 = (Base‘𝐻))
1413ad2antrr 726 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝐴 = (Base‘𝐻))
1511, 14sseqtrrd 3934 . . . . . . . . . . . . 13 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆𝑥) ⊆ 𝐴)
1615biantrud 536 . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝐴 ∈ (SubGrp‘𝐺))
18 simplr 769 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝑆:dom 𝑆⟶(SubGrp‘𝐻))
19 eldifi 4033 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (dom 𝑆 ∖ {𝑥}) → 𝑦 ∈ dom 𝑆)
2019ad2antll 729 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝑦 ∈ dom 𝑆)
2118, 20ffvelrnd 6844 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆𝑦) ∈ (SubGrp‘𝐻))
229subgss 18340 . . . . . . . . . . . . . . . . 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, 14sseqtrrd 3934 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆𝑦) ⊆ 𝐴)
25 eqid 2759 . . . . . . . . . . . . . . . 16 (Cntz‘𝐺) = (Cntz‘𝐺)
26 eqid 2759 . . . . . . . . . . . . . . . 16 (Cntz‘𝐻) = (Cntz‘𝐻)
2712, 25, 26resscntz 18522 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (SubGrp‘𝐺) ∧ (𝑆𝑦) ⊆ 𝐴) → ((Cntz‘𝐻)‘(𝑆𝑦)) = (((Cntz‘𝐺)‘(𝑆𝑦)) ∩ 𝐴))
2817, 24, 27syl2anc 588 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → ((Cntz‘𝐻)‘(𝑆𝑦)) = (((Cntz‘𝐺)‘(𝑆𝑦)) ∩ 𝐴))
2928sseq2d 3925 . . . . . . . . . . . . 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 4136 . . . . . . . . . . . . 13 (((𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ (𝑆𝑥) ⊆ 𝐴) ↔ (𝑆𝑥) ⊆ (((Cntz‘𝐺)‘(𝑆𝑦)) ∩ 𝐴))
3129, 30bitr4di 293 . . . . . . . . . . . 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 285 . . . . . . . . . . 11 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → ((𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ↔ (𝑆𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆𝑦))))
3332anassrs 472 . . . . . . . . . 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 3126 . . . . . . . . 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 18344 . . . . . . . . . . . . . . 15 (𝐴 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
3635ad2antrr 726 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐺 ∈ Grp)
37 eqid 2759 . . . . . . . . . . . . . . 15 (Base‘𝐺) = (Base‘𝐺)
3837subgacs 18373 . . . . . . . . . . . . . 14 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
39 acsmre 16974 . . . . . . . . . . . . . 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 18342 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
4241ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐻 ∈ Grp)
439subgacs 18373 . . . . . . . . . . . . . . 15 (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
44 acsmre 16974 . . . . . . . . . . . . . . 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 2759 . . . . . . . . . . . . . 14 (mrCls‘(SubGrp‘𝐻)) = (mrCls‘(SubGrp‘𝐻))
47 imassrn 5913 . . . . . . . . . . . . . . . . 17 (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ ran 𝑆
48 frn 6505 . . . . . . . . . . . . . . . . . 18 (𝑆:dom 𝑆⟶(SubGrp‘𝐻) → ran 𝑆 ⊆ (SubGrp‘𝐻))
4948ad2antlr 727 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ran 𝑆 ⊆ (SubGrp‘𝐻))
5047, 49sstrid 3904 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (SubGrp‘𝐻))
51 mresspw 16914 . . . . . . . . . . . . . . . . 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 3903 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐻))
54 sspwuni 4988 . . . . . . . . . . . . . . 15 ((𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐻) ↔ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐻))
5553, 54sylib 221 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐻))
5645, 46, 55mrcssidd 16947 . . . . . . . . . . . . 13 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))))
5746mrccl 16933 . . . . . . . . . . . . . . . 16 (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐻)) → ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻))
5845, 55, 57syl2anc 588 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻))
5912subsubg 18362 . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . 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 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))
6261simpld 499 . . . . . . . . . . . . 13 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
63 eqid 2759 . . . . . . . . . . . . . 14 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
6463mrcsscl 16942 . . . . . . . . . . . . 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 1369 . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐴 = (Base‘𝐻))
6755, 66sseqtrrd 3934 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ 𝐴)
6837subgss 18340 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (SubGrp‘𝐺) → 𝐴 ⊆ (Base‘𝐺))
6968ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐴 ⊆ (Base‘𝐺))
7067, 69sstrd 3903 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐺))
7140, 63, 70mrcssidd 16947 . . . . . . . . . . . . 13 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))))
7263mrccl 16933 . . . . . . . . . . . . . . 15 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
7340, 70, 72syl2anc 588 . . . . . . . . . . . . . 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))
7563mrcsscl 16942 . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ⊆ 𝐴)
7712subsubg 18362 . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . 13 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻))
8046mrcsscl 16942 . . . . . . . . . . . . 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 1369 . . . . . . . . . . . 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 3910 . . . . . . . . . . 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 4118 . . . . . . . . . 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 2759 . . . . . . . . . . . . 13 (0g𝐺) = (0g𝐺)
8512, 84subg0 18345 . . . . . . . . . . . 12 (𝐴 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
8685ad2antrr 726 . . . . . . . . . . 11 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (0g𝐺) = (0g𝐻))
8786sneqd 4535 . . . . . . . . . 10 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → {(0g𝐺)} = {(0g𝐻)})
8883, 87eqeq12d 2775 . . . . . . . . 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 634 . . . . . . . 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 3126 . . . . . . 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 583 . . . . . 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 18362 . . . . . . . . . . . . 13 (𝐴 ∈ (SubGrp‘𝐺) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥𝐴)))
93 elin 3875 . . . . . . . . . . . . . 14 (𝑥 ∈ ((SubGrp‘𝐺) ∩ 𝒫 𝐴) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝒫 𝐴))
94 velpw 4500 . . . . . . . . . . . . . . 15 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
9594anbi2i 626 . . . . . . . . . . . . . 14 ((𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝒫 𝐴) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥𝐴))
9693, 95bitri 278 . . . . . . . . . . . . 13 (𝑥 ∈ ((SubGrp‘𝐺) ∩ 𝒫 𝐴) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥𝐴))
9792, 96bitr4di 293 . . . . . . . . . . . 12 (𝐴 ∈ (SubGrp‘𝐺) → (𝑥 ∈ (SubGrp‘𝐻) ↔ 𝑥 ∈ ((SubGrp‘𝐺) ∩ 𝒫 𝐴)))
9897eqrdv 2757 . . . . . . . . . . 11 (𝐴 ∈ (SubGrp‘𝐺) → (SubGrp‘𝐻) = ((SubGrp‘𝐺) ∩ 𝒫 𝐴))
9998sseq2d 3925 . . . . . . . . . 10 (𝐴 ∈ (SubGrp‘𝐺) → (ran 𝑆 ⊆ (SubGrp‘𝐻) ↔ ran 𝑆 ⊆ ((SubGrp‘𝐺) ∩ 𝒫 𝐴)))
100 ssin 4136 . . . . . . . . . 10 ((ran 𝑆 ⊆ (SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ↔ ran 𝑆 ⊆ ((SubGrp‘𝐺) ∩ 𝒫 𝐴))
10199, 100bitr4di 293 . . . . . . . . 9 (𝐴 ∈ (SubGrp‘𝐺) → (ran 𝑆 ⊆ (SubGrp‘𝐻) ↔ (ran 𝑆 ⊆ (SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
102101anbi2d 632 . . . . . . . 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 6340 . . . . . . . 8 (𝑆:dom 𝑆⟶(SubGrp‘𝐻) ↔ (𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ (SubGrp‘𝐻)))
104 df-f 6340 . . . . . . . . . 10 (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ↔ (𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ (SubGrp‘𝐺)))
105104anbi1i 627 . . . . . . . . 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)))
107105, 106bitri 278 . . . . . . . 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 318 . . . . . . 7 (𝐴 ∈ (SubGrp‘𝐺) → (𝑆:dom 𝑆⟶(SubGrp‘𝐻) ↔ (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
109108anbi1d 633 . . . . . 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 284 . . . . 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 485 . . . 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 7614 . . . . . 6 (𝑆 ∈ V → dom 𝑆 ∈ V)
113112adantl 486 . . . . 5 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → dom 𝑆 ∈ V)
114 eqidd 2760 . . . . 5 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → dom 𝑆 = dom 𝑆)
11541adantr 485 . . . . 5 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → 𝐻 ∈ Grp)
116 eqid 2759 . . . . . . . 8 (0g𝐻) = (0g𝐻)
11726, 116, 46dmdprd 19181 . . . . . . 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 1093 . . . . . . 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, 118bitrdi 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})(𝑆𝑥) ⊆ ((Cntz‘𝐻)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐻)})))))
120119baibd 544 . . . . 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 837 . . . 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 485 . . . . . . 7 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → 𝐺 ∈ Grp)
12325, 84, 63dmdprd 19181 . . . . . . . . 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 1093 . . . . . . . . 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, 124bitrdi 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})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)})))))
126125baibd 544 . . . . . . 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 837 . . . . . 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 633 . . . . 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 646 . . . . 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, 129bitrdi 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})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)}))))
131111, 121, 1303bitr4d 315 . . 3 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
132131ex 417 . 2 (𝐴 ∈ (SubGrp‘𝐺) → (𝑆 ∈ V → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴))))
1333, 6, 132pm5.21ndd 385 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 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3071  Vcvv 3410   ∖ cdif 3856   ∩ cin 3858   ⊆ wss 3859  𝒫 cpw 4495  {csn 4523  ∪ cuni 4799   class class class wbr 5033  dom cdm 5525  ran crn 5526   “ cima 5528   Fn wfn 6331  ⟶wf 6332  ‘cfv 6336  (class class class)co 7151  Basecbs 16534   ↾s cress 16535  0gc0g 16764  Moorecmre 16904  mrClscmrc 16905  ACScacs 16907  Grpcgrp 18162  SubGrpcsubg 18333  Cntzccntz 18505   DProd cdprd 19176 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-cnex 10624  ax-resscn 10625  ax-1cn 10626  ax-icn 10627  ax-addcl 10628  ax-addrcl 10629  ax-mulcl 10630  ax-mulrcl 10631  ax-mulcom 10632  ax-addass 10633  ax-mulass 10634  ax-distr 10635  ax-i2m1 10636  ax-1ne0 10637  ax-1rid 10638  ax-rnegex 10639  ax-rrecex 10640  ax-cnre 10641  ax-pre-lttri 10642  ax-pre-lttrn 10643  ax-pre-ltadd 10644  ax-pre-mulgt0 10645 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7581  df-1st 7694  df-2nd 7695  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-1o 8113  df-oadd 8117  df-er 8300  df-ixp 8481  df-en 8529  df-dom 8530  df-sdom 8531  df-fin 8532  df-pnf 10708  df-mnf 10709  df-xr 10710  df-ltxr 10711  df-le 10712  df-sub 10903  df-neg 10904  df-nn 11668  df-2 11730  df-ndx 16537  df-slot 16538  df-base 16540  df-sets 16541  df-ress 16542  df-plusg 16629  df-0g 16766  df-mre 16908  df-mrc 16909  df-acs 16911  df-mgm 17911  df-sgrp 17960  df-mnd 17971  df-submnd 18016  df-grp 18165  df-minusg 18166  df-subg 18336  df-cntz 18507  df-dprd 19178 This theorem is referenced by:  subgdprd  19218  ablfaclem3  19270
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