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Theorem cldsubg 23462
Description: A subgroup of finite index is closed iff it is open. (Contributed by Mario Carneiro, 20-Sep-2015.)
Hypotheses
Ref Expression
subgntr.h 𝐽 = (TopOpen‘𝐺)
cldsubg.1 𝑅 = (𝐺 ~QG 𝑆)
cldsubg.2 𝑋 = (Base‘𝐺)
Assertion
Ref Expression
cldsubg ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆 ∈ (Clsd‘𝐽) ↔ 𝑆𝐽))

Proof of Theorem cldsubg
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 1191 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝐺 ∈ TopGrp)
2 subgntr.h . . . . . . . . 9 𝐽 = (TopOpen‘𝐺)
3 cldsubg.2 . . . . . . . . 9 𝑋 = (Base‘𝐺)
42, 3tgptopon 23433 . . . . . . . 8 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
51, 4syl 17 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝐽 ∈ (TopOn‘𝑋))
6 toponuni 22263 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
75, 6syl 17 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑋 = 𝐽)
87difeq1d 4081 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 ((𝑋 / 𝑅) ∖ {𝑆})) = ( 𝐽 ((𝑋 / 𝑅) ∖ {𝑆})))
9 simpl2 1192 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ∈ (SubGrp‘𝐺))
10 unisng 4886 . . . . . . . . 9 (𝑆 ∈ (SubGrp‘𝐺) → {𝑆} = 𝑆)
119, 10syl 17 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → {𝑆} = 𝑆)
1211uneq2d 4123 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆))
13 uniun 4891 . . . . . . . 8 (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆})
14 undif1 4435 . . . . . . . . . . 11 (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ((𝑋 / 𝑅) ∪ {𝑆})
15 cldsubg.1 . . . . . . . . . . . . . . . 16 𝑅 = (𝐺 ~QG 𝑆)
16 eqid 2736 . . . . . . . . . . . . . . . 16 (0g𝐺) = (0g𝐺)
173, 15, 16eqgid 18982 . . . . . . . . . . . . . . 15 (𝑆 ∈ (SubGrp‘𝐺) → [(0g𝐺)]𝑅 = 𝑆)
189, 17syl 17 . . . . . . . . . . . . . 14 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → [(0g𝐺)]𝑅 = 𝑆)
1915ovexi 7391 . . . . . . . . . . . . . . 15 𝑅 ∈ V
20 tgpgrp 23429 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
211, 20syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝐺 ∈ Grp)
223, 16grpidcl 18778 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
2321, 22syl 17 . . . . . . . . . . . . . . 15 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (0g𝐺) ∈ 𝑋)
24 ecelqsg 8711 . . . . . . . . . . . . . . 15 ((𝑅 ∈ V ∧ (0g𝐺) ∈ 𝑋) → [(0g𝐺)]𝑅 ∈ (𝑋 / 𝑅))
2519, 23, 24sylancr 587 . . . . . . . . . . . . . 14 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → [(0g𝐺)]𝑅 ∈ (𝑋 / 𝑅))
2618, 25eqeltrrd 2839 . . . . . . . . . . . . 13 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ∈ (𝑋 / 𝑅))
2726snssd 4769 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → {𝑆} ⊆ (𝑋 / 𝑅))
28 ssequn2 4143 . . . . . . . . . . . 12 ({𝑆} ⊆ (𝑋 / 𝑅) ↔ ((𝑋 / 𝑅) ∪ {𝑆}) = (𝑋 / 𝑅))
2927, 28sylib 217 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∪ {𝑆}) = (𝑋 / 𝑅))
3014, 29eqtrid 2788 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = (𝑋 / 𝑅))
3130unieqd 4879 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = (𝑋 / 𝑅))
323, 15eqger 18980 . . . . . . . . . . 11 (𝑆 ∈ (SubGrp‘𝐺) → 𝑅 Er 𝑋)
339, 32syl 17 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑅 Er 𝑋)
3419a1i 11 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑅 ∈ V)
3533, 34uniqs2 8718 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 / 𝑅) = 𝑋)
3631, 35eqtrd 2776 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = 𝑋)
3713, 36eqtr3id 2790 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = 𝑋)
3812, 37eqtr3d 2778 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆) = 𝑋)
39 difss 4091 . . . . . . . . 9 ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (𝑋 / 𝑅)
4039unissi 4874 . . . . . . . 8 ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (𝑋 / 𝑅)
4140, 35sseqtrid 3996 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝑋)
42 df-ne 2944 . . . . . . . . . . . . 13 (𝑥𝑆 ↔ ¬ 𝑥 = 𝑆)
4333adantr 481 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → 𝑅 Er 𝑋)
44 simpr 485 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → 𝑥 ∈ (𝑋 / 𝑅))
4526adantr 481 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → 𝑆 ∈ (𝑋 / 𝑅))
4643, 44, 45qsdisj 8733 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (𝑥 = 𝑆 ∨ (𝑥𝑆) = ∅))
4746ord 862 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (¬ 𝑥 = 𝑆 → (𝑥𝑆) = ∅))
48 disj2 4417 . . . . . . . . . . . . . 14 ((𝑥𝑆) = ∅ ↔ 𝑥 ⊆ (V ∖ 𝑆))
4947, 48syl6ib 250 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (¬ 𝑥 = 𝑆𝑥 ⊆ (V ∖ 𝑆)))
5042, 49biimtrid 241 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (𝑥𝑆𝑥 ⊆ (V ∖ 𝑆)))
5150expimpd 454 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑥 ∈ (𝑋 / 𝑅) ∧ 𝑥𝑆) → 𝑥 ⊆ (V ∖ 𝑆)))
52 eldifsn 4747 . . . . . . . . . . 11 (𝑥 ∈ ((𝑋 / 𝑅) ∖ {𝑆}) ↔ (𝑥 ∈ (𝑋 / 𝑅) ∧ 𝑥𝑆))
53 velpw 4565 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 (V ∖ 𝑆) ↔ 𝑥 ⊆ (V ∖ 𝑆))
5451, 52, 533imtr4g 295 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑥 ∈ ((𝑋 / 𝑅) ∖ {𝑆}) → 𝑥 ∈ 𝒫 (V ∖ 𝑆)))
5554ssrdv 3950 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝒫 (V ∖ 𝑆))
56 sspwuni 5060 . . . . . . . . 9 (((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝒫 (V ∖ 𝑆) ↔ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (V ∖ 𝑆))
5755, 56sylib 217 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (V ∖ 𝑆))
58 disj2 4417 . . . . . . . 8 (( ((𝑋 / 𝑅) ∖ {𝑆}) ∩ 𝑆) = ∅ ↔ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (V ∖ 𝑆))
5957, 58sylibr 233 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( ((𝑋 / 𝑅) ∖ {𝑆}) ∩ 𝑆) = ∅)
60 uneqdifeq 4450 . . . . . . 7 (( ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝑋 ∧ ( ((𝑋 / 𝑅) ∖ {𝑆}) ∩ 𝑆) = ∅) → (( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆) = 𝑋 ↔ (𝑋 ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆))
6141, 59, 60syl2anc 584 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆) = 𝑋 ↔ (𝑋 ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆))
6238, 61mpbid 231 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆)
638, 62eqtr3d 2778 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( 𝐽 ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆)
64 topontop 22262 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
655, 64syl 17 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
66 simpl3 1193 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 / 𝑅) ∈ Fin)
67 diffi 9123 . . . . . . 7 ((𝑋 / 𝑅) ∈ Fin → ((𝑋 / 𝑅) ∖ {𝑆}) ∈ Fin)
6866, 67syl 17 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ∈ Fin)
69 vex 3449 . . . . . . . . . 10 𝑥 ∈ V
7069elqs 8708 . . . . . . . . 9 (𝑥 ∈ (𝑋 / 𝑅) ↔ ∃𝑦𝑋 𝑥 = [𝑦]𝑅)
71 simpll2 1213 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑆 ∈ (SubGrp‘𝐺))
72 subgrcl 18933 . . . . . . . . . . . . . 14 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
7371, 72syl 17 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝐺 ∈ Grp)
743subgss 18929 . . . . . . . . . . . . . . 15 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
759, 74syl 17 . . . . . . . . . . . . . 14 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆𝑋)
7675adantr 481 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑆𝑋)
77 simpr 485 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑦𝑋)
78 eqid 2736 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
793, 15, 78eqglact 18981 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝑆𝑋𝑦𝑋) → [𝑦]𝑅 = ((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) “ 𝑆))
8073, 76, 77, 79syl3anc 1371 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → [𝑦]𝑅 = ((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) “ 𝑆))
81 simplr 767 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑆 ∈ (Clsd‘𝐽))
82 eqid 2736 . . . . . . . . . . . . . . . 16 (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) = (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧))
8382, 3, 78, 2tgplacthmeo 23454 . . . . . . . . . . . . . . 15 ((𝐺 ∈ TopGrp ∧ 𝑦𝑋) → (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
841, 83sylan 580 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
8575, 7sseqtrd 3984 . . . . . . . . . . . . . . 15 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 𝐽)
8685adantr 481 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑆 𝐽)
87 eqid 2736 . . . . . . . . . . . . . . 15 𝐽 = 𝐽
8887hmeocld 23118 . . . . . . . . . . . . . 14 (((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽) ∧ 𝑆 𝐽) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) “ 𝑆) ∈ (Clsd‘𝐽)))
8984, 86, 88syl2anc 584 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) “ 𝑆) ∈ (Clsd‘𝐽)))
9081, 89mpbid 231 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → ((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) “ 𝑆) ∈ (Clsd‘𝐽))
9180, 90eqeltrd 2838 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → [𝑦]𝑅 ∈ (Clsd‘𝐽))
92 eleq1 2825 . . . . . . . . . . 11 (𝑥 = [𝑦]𝑅 → (𝑥 ∈ (Clsd‘𝐽) ↔ [𝑦]𝑅 ∈ (Clsd‘𝐽)))
9391, 92syl5ibrcom 246 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → (𝑥 = [𝑦]𝑅𝑥 ∈ (Clsd‘𝐽)))
9493rexlimdva 3152 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∃𝑦𝑋 𝑥 = [𝑦]𝑅𝑥 ∈ (Clsd‘𝐽)))
9570, 94biimtrid 241 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑥 ∈ (𝑋 / 𝑅) → 𝑥 ∈ (Clsd‘𝐽)))
9695ssrdv 3950 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 / 𝑅) ⊆ (Clsd‘𝐽))
9796ssdifssd 4102 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (Clsd‘𝐽))
9887unicld 22397 . . . . . 6 ((𝐽 ∈ Top ∧ ((𝑋 / 𝑅) ∖ {𝑆}) ∈ Fin ∧ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ∈ (Clsd‘𝐽))
9965, 68, 97, 98syl3anc 1371 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ∈ (Clsd‘𝐽))
10087cldopn 22382 . . . . 5 ( ((𝑋 / 𝑅) ∖ {𝑆}) ∈ (Clsd‘𝐽) → ( 𝐽 ((𝑋 / 𝑅) ∖ {𝑆})) ∈ 𝐽)
10199, 100syl 17 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( 𝐽 ((𝑋 / 𝑅) ∖ {𝑆})) ∈ 𝐽)
10263, 101eqeltrrd 2839 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆𝐽)
103102ex 413 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆 ∈ (Clsd‘𝐽) → 𝑆𝐽))
1042opnsubg 23459 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆 ∈ (Clsd‘𝐽))
1051043expia 1121 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆𝐽𝑆 ∈ (Clsd‘𝐽)))
1061053adant3 1132 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆𝐽𝑆 ∈ (Clsd‘𝐽)))
107103, 106impbid 211 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆 ∈ (Clsd‘𝐽) ↔ 𝑆𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wrex 3073  Vcvv 3445  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586   cuni 4865  cmpt 5188  cima 5636  cfv 6496  (class class class)co 7357   Er wer 8645  [cec 8646   / cqs 8647  Fincfn 8883  Basecbs 17083  +gcplusg 17133  TopOpenctopn 17303  0gc0g 17321  Grpcgrp 18748  SubGrpcsubg 18922   ~QG cqg 18924  Topctop 22242  TopOnctopon 22259  Clsdccld 22367  Homeochmeo 23104  TopGrpctgp 23422
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-ec 8650  df-qs 8654  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-topgen 17325  df-plusf 18496  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-sbg 18753  df-subg 18925  df-eqg 18927  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-cn 22578  df-cnp 22579  df-tx 22913  df-hmeo 23106  df-tmd 23423  df-tgp 23424
This theorem is referenced by: (None)
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