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Theorem cldsubg 22719
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 1188 . . . . . . . 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 22690 . . . . . . . 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 21522 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
75, 6syl 17 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑋 = 𝐽)
87difeq1d 4084 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 ((𝑋 / 𝑅) ∖ {𝑆})) = ( 𝐽 ((𝑋 / 𝑅) ∖ {𝑆})))
9 simpl2 1189 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ∈ (SubGrp‘𝐺))
10 unisng 4843 . . . . . . . . 9 (𝑆 ∈ (SubGrp‘𝐺) → {𝑆} = 𝑆)
119, 10syl 17 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → {𝑆} = 𝑆)
1211uneq2d 4125 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆))
13 uniun 4847 . . . . . . . 8 (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆})
14 undif1 4407 . . . . . . . . . . 11 (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ((𝑋 / 𝑅) ∪ {𝑆})
15 cldsubg.1 . . . . . . . . . . . . . . . 16 𝑅 = (𝐺 ~QG 𝑆)
16 eqid 2824 . . . . . . . . . . . . . . . 16 (0g𝐺) = (0g𝐺)
173, 15, 16eqgid 18332 . . . . . . . . . . . . . . 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 7183 . . . . . . . . . . . . . . 15 𝑅 ∈ V
20 tgpgrp 22686 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
211, 20syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝐺 ∈ Grp)
223, 16grpidcl 18131 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
2321, 22syl 17 . . . . . . . . . . . . . . 15 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (0g𝐺) ∈ 𝑋)
24 ecelqsg 8348 . . . . . . . . . . . . . . 15 ((𝑅 ∈ V ∧ (0g𝐺) ∈ 𝑋) → [(0g𝐺)]𝑅 ∈ (𝑋 / 𝑅))
2519, 23, 24sylancr 590 . . . . . . . . . . . . . 14 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → [(0g𝐺)]𝑅 ∈ (𝑋 / 𝑅))
2618, 25eqeltrrd 2917 . . . . . . . . . . . . 13 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ∈ (𝑋 / 𝑅))
2726snssd 4726 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → {𝑆} ⊆ (𝑋 / 𝑅))
28 ssequn2 4145 . . . . . . . . . . . 12 ({𝑆} ⊆ (𝑋 / 𝑅) ↔ ((𝑋 / 𝑅) ∪ {𝑆}) = (𝑋 / 𝑅))
2927, 28sylib 221 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∪ {𝑆}) = (𝑋 / 𝑅))
3014, 29syl5eq 2871 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = (𝑋 / 𝑅))
3130unieqd 4838 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = (𝑋 / 𝑅))
323, 15eqger 18330 . . . . . . . . . . 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 8355 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 / 𝑅) = 𝑋)
3631, 35eqtrd 2859 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = 𝑋)
3713, 36syl5eqr 2873 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = 𝑋)
3812, 37eqtr3d 2861 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆) = 𝑋)
39 difss 4094 . . . . . . . . 9 ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (𝑋 / 𝑅)
4039unissi 4833 . . . . . . . 8 ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (𝑋 / 𝑅)
4140, 35sseqtrid 4005 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝑋)
42 df-ne 3015 . . . . . . . . . . . . 13 (𝑥𝑆 ↔ ¬ 𝑥 = 𝑆)
4333adantr 484 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → 𝑅 Er 𝑋)
44 simpr 488 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → 𝑥 ∈ (𝑋 / 𝑅))
4526adantr 484 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → 𝑆 ∈ (𝑋 / 𝑅))
4643, 44, 45qsdisj 8370 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (𝑥 = 𝑆 ∨ (𝑥𝑆) = ∅))
4746ord 861 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (¬ 𝑥 = 𝑆 → (𝑥𝑆) = ∅))
48 disj2 4390 . . . . . . . . . . . . . 14 ((𝑥𝑆) = ∅ ↔ 𝑥 ⊆ (V ∖ 𝑆))
4947, 48syl6ib 254 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (¬ 𝑥 = 𝑆𝑥 ⊆ (V ∖ 𝑆)))
5042, 49syl5bi 245 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (𝑥𝑆𝑥 ⊆ (V ∖ 𝑆)))
5150expimpd 457 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑥 ∈ (𝑋 / 𝑅) ∧ 𝑥𝑆) → 𝑥 ⊆ (V ∖ 𝑆)))
52 eldifsn 4704 . . . . . . . . . . 11 (𝑥 ∈ ((𝑋 / 𝑅) ∖ {𝑆}) ↔ (𝑥 ∈ (𝑋 / 𝑅) ∧ 𝑥𝑆))
53 velpw 4527 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 (V ∖ 𝑆) ↔ 𝑥 ⊆ (V ∖ 𝑆))
5451, 52, 533imtr4g 299 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑥 ∈ ((𝑋 / 𝑅) ∖ {𝑆}) → 𝑥 ∈ 𝒫 (V ∖ 𝑆)))
5554ssrdv 3959 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝒫 (V ∖ 𝑆))
56 sspwuni 5008 . . . . . . . . 9 (((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝒫 (V ∖ 𝑆) ↔ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (V ∖ 𝑆))
5755, 56sylib 221 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (V ∖ 𝑆))
58 disj2 4390 . . . . . . . 8 (( ((𝑋 / 𝑅) ∖ {𝑆}) ∩ 𝑆) = ∅ ↔ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (V ∖ 𝑆))
5957, 58sylibr 237 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( ((𝑋 / 𝑅) ∖ {𝑆}) ∩ 𝑆) = ∅)
60 uneqdifeq 4421 . . . . . . 7 (( ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝑋 ∧ ( ((𝑋 / 𝑅) ∖ {𝑆}) ∩ 𝑆) = ∅) → (( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆) = 𝑋 ↔ (𝑋 ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆))
6141, 59, 60syl2anc 587 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆) = 𝑋 ↔ (𝑋 ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆))
6238, 61mpbid 235 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆)
638, 62eqtr3d 2861 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( 𝐽 ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆)
64 topontop 21521 . . . . . . 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 1190 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 / 𝑅) ∈ Fin)
67 diffi 8747 . . . . . . 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 3483 . . . . . . . . . 10 𝑥 ∈ V
7069elqs 8345 . . . . . . . . 9 (𝑥 ∈ (𝑋 / 𝑅) ↔ ∃𝑦𝑋 𝑥 = [𝑦]𝑅)
71 simpll2 1210 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑆 ∈ (SubGrp‘𝐺))
72 subgrcl 18284 . . . . . . . . . . . . . 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 18280 . . . . . . . . . . . . . . 15 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
759, 74syl 17 . . . . . . . . . . . . . 14 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆𝑋)
7675adantr 484 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑆𝑋)
77 simpr 488 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑦𝑋)
78 eqid 2824 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
793, 15, 78eqglact 18331 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝑆𝑋𝑦𝑋) → [𝑦]𝑅 = ((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) “ 𝑆))
8073, 76, 77, 79syl3anc 1368 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → [𝑦]𝑅 = ((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) “ 𝑆))
81 simplr 768 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑆 ∈ (Clsd‘𝐽))
82 eqid 2824 . . . . . . . . . . . . . . . 16 (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) = (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧))
8382, 3, 78, 2tgplacthmeo 22711 . . . . . . . . . . . . . . 15 ((𝐺 ∈ TopGrp ∧ 𝑦𝑋) → (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
841, 83sylan 583 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
8575, 7sseqtrd 3993 . . . . . . . . . . . . . . 15 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 𝐽)
8685adantr 484 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑆 𝐽)
87 eqid 2824 . . . . . . . . . . . . . . 15 𝐽 = 𝐽
8887hmeocld 22375 . . . . . . . . . . . . . 14 (((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽) ∧ 𝑆 𝐽) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) “ 𝑆) ∈ (Clsd‘𝐽)))
8984, 86, 88syl2anc 587 . . . . . . . . . . . . 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 235 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → ((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) “ 𝑆) ∈ (Clsd‘𝐽))
9180, 90eqeltrd 2916 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → [𝑦]𝑅 ∈ (Clsd‘𝐽))
92 eleq1 2903 . . . . . . . . . . 11 (𝑥 = [𝑦]𝑅 → (𝑥 ∈ (Clsd‘𝐽) ↔ [𝑦]𝑅 ∈ (Clsd‘𝐽)))
9391, 92syl5ibrcom 250 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → (𝑥 = [𝑦]𝑅𝑥 ∈ (Clsd‘𝐽)))
9493rexlimdva 3276 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∃𝑦𝑋 𝑥 = [𝑦]𝑅𝑥 ∈ (Clsd‘𝐽)))
9570, 94syl5bi 245 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑥 ∈ (𝑋 / 𝑅) → 𝑥 ∈ (Clsd‘𝐽)))
9695ssrdv 3959 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 / 𝑅) ⊆ (Clsd‘𝐽))
9796ssdifssd 4105 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (Clsd‘𝐽))
9887unicld 21654 . . . . . 6 ((𝐽 ∈ Top ∧ ((𝑋 / 𝑅) ∖ {𝑆}) ∈ Fin ∧ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ∈ (Clsd‘𝐽))
9965, 68, 97, 98syl3anc 1368 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ∈ (Clsd‘𝐽))
10087cldopn 21639 . . . . 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 2917 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆𝐽)
103102ex 416 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆 ∈ (Clsd‘𝐽) → 𝑆𝐽))
1042opnsubg 22716 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆 ∈ (Clsd‘𝐽))
1051043expia 1118 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆𝐽𝑆 ∈ (Clsd‘𝐽)))
1061053adant3 1129 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆𝐽𝑆 ∈ (Clsd‘𝐽)))
107103, 106impbid 215 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014  wrex 3134  Vcvv 3480  cdif 3916  cun 3917  cin 3918  wss 3919  c0 4276  𝒫 cpw 4522  {csn 4550   cuni 4824  cmpt 5132  cima 5545  cfv 6343  (class class class)co 7149   Er wer 8282  [cec 8283   / cqs 8284  Fincfn 8505  Basecbs 16483  +gcplusg 16565  TopOpenctopn 16695  0gc0g 16713  Grpcgrp 18103  SubGrpcsubg 18273   ~QG cqg 18275  Topctop 21501  TopOnctopon 21518  Clsdccld 21624  Homeochmeo 22361  TopGrpctgp 22679
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-ec 8287  df-qs 8291  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-0g 16715  df-topgen 16717  df-plusf 17851  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-minusg 18107  df-sbg 18108  df-subg 18276  df-eqg 18278  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-cld 21627  df-cn 21835  df-cnp 21836  df-tx 22170  df-hmeo 22363  df-tmd 22680  df-tgp 22681
This theorem is referenced by: (None)
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