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Theorem subgdmdprd 18349
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 18312 . . . 4 Rel dom DProd
21brrelex2i 5124 . . 3 (𝐻dom DProd 𝑆𝑆 ∈ V)
32a1i 11 . 2 (𝐴 ∈ (SubGrp‘𝐺) → (𝐻dom DProd 𝑆𝑆 ∈ V))
41brrelex2i 5124 . . . 4 (𝐺dom DProd 𝑆𝑆 ∈ V)
54adantr 481 . . 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 6314 . . . . . . . . . . . . . . . 16 ((𝑆:dom 𝑆⟶(SubGrp‘𝐻) ∧ 𝑥 ∈ dom 𝑆) → (𝑆𝑥) ∈ (SubGrp‘𝐻))
87ad2ant2lr 783 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆𝑥) ∈ (SubGrp‘𝐻))
9 eqid 2626 . . . . . . . . . . . . . . . 16 (Base‘𝐻) = (Base‘𝐻)
109subgss 17511 . . . . . . . . . . . . . . 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 17514 . . . . . . . . . . . . . . 15 (𝐴 ∈ (SubGrp‘𝐺) → 𝐴 = (Base‘𝐻))
1413ad2antrr 761 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝐴 = (Base‘𝐻))
1511, 14sseqtr4d 3626 . . . . . . . . . . . . 13 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆𝑥) ⊆ 𝐴)
1615biantrud 528 . . . . . . . . . . . 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 789 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝐴 ∈ (SubGrp‘𝐺))
18 simplr 791 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝑆:dom 𝑆⟶(SubGrp‘𝐻))
19 eldifi 3715 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (dom 𝑆 ∖ {𝑥}) → 𝑦 ∈ dom 𝑆)
2019ad2antll 764 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → 𝑦 ∈ dom 𝑆)
2118, 20ffvelrnd 6317 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆𝑦) ∈ (SubGrp‘𝐻))
229subgss 17511 . . . . . . . . . . . . . . . . 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 3626 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → (𝑆𝑦) ⊆ 𝐴)
25 eqid 2626 . . . . . . . . . . . . . . . 16 (Cntz‘𝐺) = (Cntz‘𝐺)
26 eqid 2626 . . . . . . . . . . . . . . . 16 (Cntz‘𝐻) = (Cntz‘𝐻)
2712, 25, 26resscntz 17680 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (SubGrp‘𝐺) ∧ (𝑆𝑦) ⊆ 𝐴) → ((Cntz‘𝐻)‘(𝑆𝑦)) = (((Cntz‘𝐺)‘(𝑆𝑦)) ∩ 𝐴))
2817, 24, 27syl2anc 692 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ (𝑥 ∈ dom 𝑆𝑦 ∈ (dom 𝑆 ∖ {𝑥}))) → ((Cntz‘𝐻)‘(𝑆𝑦)) = (((Cntz‘𝐺)‘(𝑆𝑦)) ∩ 𝐴))
2928sseq2d 3617 . . . . . . . . . . . . 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 3818 . . . . . . . . . . . . 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 679 . . . . . . . . . 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 2984 . . . . . . . . 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 17515 . . . . . . . . . . . . . . 15 (𝐴 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
3635ad2antrr 761 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐺 ∈ Grp)
37 eqid 2626 . . . . . . . . . . . . . . 15 (Base‘𝐺) = (Base‘𝐺)
3837subgacs 17545 . . . . . . . . . . . . . 14 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
39 acsmre 16229 . . . . . . . . . . . . . 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 17513 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
4241ad2antrr 761 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐻 ∈ Grp)
439subgacs 17545 . . . . . . . . . . . . . . 15 (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
44 acsmre 16229 . . . . . . . . . . . . . . 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 2626 . . . . . . . . . . . . . 14 (mrCls‘(SubGrp‘𝐻)) = (mrCls‘(SubGrp‘𝐻))
47 imassrn 5440 . . . . . . . . . . . . . . . . 17 (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ ran 𝑆
48 frn 6012 . . . . . . . . . . . . . . . . . 18 (𝑆:dom 𝑆⟶(SubGrp‘𝐻) → ran 𝑆 ⊆ (SubGrp‘𝐻))
4948ad2antlr 762 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ran 𝑆 ⊆ (SubGrp‘𝐻))
5047, 49syl5ss 3599 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (SubGrp‘𝐻))
51 mresspw 16168 . . . . . . . . . . . . . . . . 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 3598 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐻))
54 sspwuni 4582 . . . . . . . . . . . . . . 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 16201 . . . . . . . . . . . . 13 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))))
5746mrccl 16187 . . . . . . . . . . . . . . . 16 (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐻)) → ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻))
5845, 55, 57syl2anc 692 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻))
5912subsubg 17533 . . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . 13 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐻))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
63 eqid 2626 . . . . . . . . . . . . . 14 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
6463mrcsscl 16196 . . . . . . . . . . . . 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 1323 . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐴 = (Base‘𝐻))
6755, 66sseqtr4d 3626 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ 𝐴)
6837subgss 17511 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (SubGrp‘𝐺) → 𝐴 ⊆ (Base‘𝐺))
6968ad2antrr 761 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐴 ⊆ (Base‘𝐺))
7067, 69sstrd 3598 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐺))
7140, 63, 70mrcssidd 16201 . . . . . . . . . . . . 13 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))))
7263mrccl 16187 . . . . . . . . . . . . . . 15 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ (𝑆 “ (dom 𝑆 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
7340, 70, 72syl2anc 692 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
74 simpll 789 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → 𝐴 ∈ (SubGrp‘𝐺))
7563mrcsscl 16196 . . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . . 14 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ⊆ 𝐴)
7712subsubg 17533 . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . 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 956 . . . . . . . . . . . . 13 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥}))) ∈ (SubGrp‘𝐻))
8046mrcsscl 16196 . . . . . . . . . . . . 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 1323 . . . . . . . . . . . 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 3605 . . . . . . . . . . 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 3797 . . . . . . . . . 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 2626 . . . . . . . . . . . . 13 (0g𝐺) = (0g𝐺)
8512, 84subg0 17516 . . . . . . . . . . . 12 (𝐴 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
8685ad2antrr 761 . . . . . . . . . . 11 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → (0g𝐺) = (0g𝐻))
8786sneqd 4165 . . . . . . . . . 10 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐻)) ∧ 𝑥 ∈ dom 𝑆) → {(0g𝐺)} = {(0g𝐻)})
8883, 87eqeq12d 2641 . . . . . . . . 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 746 . . . . . . . 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 2984 . . . . . . 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 672 . . . . . 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 17533 . . . . . . . . . . . . 13 (𝐴 ∈ (SubGrp‘𝐺) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥𝐴)))
93 elin 3779 . . . . . . . . . . . . . 14 (𝑥 ∈ ((SubGrp‘𝐺) ∩ 𝒫 𝐴) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝒫 𝐴))
94 selpw 4142 . . . . . . . . . . . . . . 15 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
9594anbi2i 729 . . . . . . . . . . . . . 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 2624 . . . . . . . . . . 11 (𝐴 ∈ (SubGrp‘𝐺) → (SubGrp‘𝐻) = ((SubGrp‘𝐺) ∩ 𝒫 𝐴))
9998sseq2d 3617 . . . . . . . . . 10 (𝐴 ∈ (SubGrp‘𝐺) → (ran 𝑆 ⊆ (SubGrp‘𝐻) ↔ ran 𝑆 ⊆ ((SubGrp‘𝐺) ∩ 𝒫 𝐴)))
100 ssin 3818 . . . . . . . . . 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 739 . . . . . . . 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 5854 . . . . . . . 8 (𝑆:dom 𝑆⟶(SubGrp‘𝐻) ↔ (𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ (SubGrp‘𝐻)))
104 df-f 5854 . . . . . . . . . 10 (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ↔ (𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ (SubGrp‘𝐺)))
105104anbi1i 730 . . . . . . . . 9 ((𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ran 𝑆 ⊆ 𝒫 𝐴) ↔ ((𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ (SubGrp‘𝐺)) ∧ ran 𝑆 ⊆ 𝒫 𝐴))
106 anass 680 . . . . . . . . 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 740 . . . . . 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 481 . . . 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 7045 . . . . . 6 (𝑆 ∈ V → dom 𝑆 ∈ V)
113112adantl 482 . . . . 5 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → dom 𝑆 ∈ V)
114 eqidd 2627 . . . . 5 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → dom 𝑆 = dom 𝑆)
11541adantr 481 . . . . 5 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → 𝐻 ∈ Grp)
116 eqid 2626 . . . . . . . 8 (0g𝐻) = (0g𝐻)
11726, 116, 46dmdprd 18313 . . . . . . 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 1040 . . . . . . 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 947 . . . . 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 1322 . . . 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 481 . . . . . . 7 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ V) → 𝐺 ∈ Grp)
12325, 84, 63dmdprd 18313 . . . . . . . . 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 1040 . . . . . . . . 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 947 . . . . . . 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 1322 . . . . . 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 740 . . . . 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 838 . . . . 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 450 . 2 (𝐴 ∈ (SubGrp‘𝐺) → (𝑆 ∈ V → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴))))
1333, 6, 132pm5.21ndd 369 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 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  Vcvv 3191  cdif 3557  cin 3559  wss 3560  𝒫 cpw 4135  {csn 4153   cuni 4407   class class class wbr 4618  dom cdm 5079  ran crn 5080  cima 5082   Fn wfn 5845  wf 5846  cfv 5850  (class class class)co 6605  Basecbs 15776  s cress 15777  0gc0g 16016  Moorecmre 16158  mrClscmrc 16159  ACScacs 16161  Grpcgrp 17338  SubGrpcsubg 17504  Cntzccntz 17664   DProd cdprd 18308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-ixp 7854  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-0g 16018  df-mre 16162  df-mrc 16163  df-acs 16165  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-submnd 17252  df-grp 17341  df-minusg 17342  df-subg 17507  df-cntz 17666  df-dprd 18310
This theorem is referenced by:  subgdprd  18350  ablfaclem3  18402
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