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Theorem cldsubg 21833
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 1062 . . . . . . . 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 21805 . . . . . . . 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 20647 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
75, 6syl 17 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑋 = 𝐽)
87difeq1d 3710 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 ((𝑋 / 𝑅) ∖ {𝑆})) = ( 𝐽 ((𝑋 / 𝑅) ∖ {𝑆})))
9 simpl2 1063 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ∈ (SubGrp‘𝐺))
10 unisng 4423 . . . . . . . . 9 (𝑆 ∈ (SubGrp‘𝐺) → {𝑆} = 𝑆)
119, 10syl 17 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → {𝑆} = 𝑆)
1211uneq2d 3750 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆))
13 uniun 4427 . . . . . . . 8 (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆})
14 undif1 4020 . . . . . . . . . . 11 (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ((𝑋 / 𝑅) ∪ {𝑆})
15 cldsubg.1 . . . . . . . . . . . . . . . 16 𝑅 = (𝐺 ~QG 𝑆)
16 eqid 2621 . . . . . . . . . . . . . . . 16 (0g𝐺) = (0g𝐺)
173, 15, 16eqgid 17574 . . . . . . . . . . . . . . 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)
19 ovex 6638 . . . . . . . . . . . . . . . 16 (𝐺 ~QG 𝑆) ∈ V
2015, 19eqeltri 2694 . . . . . . . . . . . . . . 15 𝑅 ∈ V
21 tgpgrp 21801 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
221, 21syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝐺 ∈ Grp)
233, 16grpidcl 17378 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
2422, 23syl 17 . . . . . . . . . . . . . . 15 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (0g𝐺) ∈ 𝑋)
25 ecelqsg 7754 . . . . . . . . . . . . . . 15 ((𝑅 ∈ V ∧ (0g𝐺) ∈ 𝑋) → [(0g𝐺)]𝑅 ∈ (𝑋 / 𝑅))
2620, 24, 25sylancr 694 . . . . . . . . . . . . . 14 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → [(0g𝐺)]𝑅 ∈ (𝑋 / 𝑅))
2718, 26eqeltrrd 2699 . . . . . . . . . . . . 13 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ∈ (𝑋 / 𝑅))
2827snssd 4314 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → {𝑆} ⊆ (𝑋 / 𝑅))
29 ssequn2 3769 . . . . . . . . . . . 12 ({𝑆} ⊆ (𝑋 / 𝑅) ↔ ((𝑋 / 𝑅) ∪ {𝑆}) = (𝑋 / 𝑅))
3028, 29sylib 208 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∪ {𝑆}) = (𝑋 / 𝑅))
3114, 30syl5eq 2667 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = (𝑋 / 𝑅))
3231unieqd 4417 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = (𝑋 / 𝑅))
333, 15eqger 17572 . . . . . . . . . . 11 (𝑆 ∈ (SubGrp‘𝐺) → 𝑅 Er 𝑋)
349, 33syl 17 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑅 Er 𝑋)
3520a1i 11 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑅 ∈ V)
3634, 35uniqs2 7761 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 / 𝑅) = 𝑋)
3732, 36eqtrd 2655 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = 𝑋)
3813, 37syl5eqr 2669 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = 𝑋)
3912, 38eqtr3d 2657 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆) = 𝑋)
40 difss 3720 . . . . . . . . 9 ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (𝑋 / 𝑅)
4140unissi 4432 . . . . . . . 8 ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (𝑋 / 𝑅)
4241, 36syl5sseq 3637 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝑋)
43 df-ne 2791 . . . . . . . . . . . . 13 (𝑥𝑆 ↔ ¬ 𝑥 = 𝑆)
4434adantr 481 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → 𝑅 Er 𝑋)
45 simpr 477 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → 𝑥 ∈ (𝑋 / 𝑅))
4627adantr 481 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → 𝑆 ∈ (𝑋 / 𝑅))
4744, 45, 46qsdisj 7776 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (𝑥 = 𝑆 ∨ (𝑥𝑆) = ∅))
4847ord 392 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (¬ 𝑥 = 𝑆 → (𝑥𝑆) = ∅))
49 disj2 4001 . . . . . . . . . . . . . 14 ((𝑥𝑆) = ∅ ↔ 𝑥 ⊆ (V ∖ 𝑆))
5048, 49syl6ib 241 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (¬ 𝑥 = 𝑆𝑥 ⊆ (V ∖ 𝑆)))
5143, 50syl5bi 232 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (𝑥𝑆𝑥 ⊆ (V ∖ 𝑆)))
5251expimpd 628 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑥 ∈ (𝑋 / 𝑅) ∧ 𝑥𝑆) → 𝑥 ⊆ (V ∖ 𝑆)))
53 eldifsn 4292 . . . . . . . . . . 11 (𝑥 ∈ ((𝑋 / 𝑅) ∖ {𝑆}) ↔ (𝑥 ∈ (𝑋 / 𝑅) ∧ 𝑥𝑆))
54 selpw 4142 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 (V ∖ 𝑆) ↔ 𝑥 ⊆ (V ∖ 𝑆))
5552, 53, 543imtr4g 285 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑥 ∈ ((𝑋 / 𝑅) ∖ {𝑆}) → 𝑥 ∈ 𝒫 (V ∖ 𝑆)))
5655ssrdv 3593 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝒫 (V ∖ 𝑆))
57 sspwuni 4582 . . . . . . . . 9 (((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝒫 (V ∖ 𝑆) ↔ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (V ∖ 𝑆))
5856, 57sylib 208 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (V ∖ 𝑆))
59 disj2 4001 . . . . . . . 8 (( ((𝑋 / 𝑅) ∖ {𝑆}) ∩ 𝑆) = ∅ ↔ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (V ∖ 𝑆))
6058, 59sylibr 224 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( ((𝑋 / 𝑅) ∖ {𝑆}) ∩ 𝑆) = ∅)
61 uneqdifeq 4034 . . . . . . 7 (( ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝑋 ∧ ( ((𝑋 / 𝑅) ∖ {𝑆}) ∩ 𝑆) = ∅) → (( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆) = 𝑋 ↔ (𝑋 ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆))
6242, 60, 61syl2anc 692 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆) = 𝑋 ↔ (𝑋 ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆))
6339, 62mpbid 222 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆)
648, 63eqtr3d 2657 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( 𝐽 ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆)
65 topontop 20646 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
665, 65syl 17 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
67 simpl3 1064 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 / 𝑅) ∈ Fin)
68 diffi 8143 . . . . . . 7 ((𝑋 / 𝑅) ∈ Fin → ((𝑋 / 𝑅) ∖ {𝑆}) ∈ Fin)
6967, 68syl 17 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ∈ Fin)
70 vex 3192 . . . . . . . . . 10 𝑥 ∈ V
7170elqs 7751 . . . . . . . . 9 (𝑥 ∈ (𝑋 / 𝑅) ↔ ∃𝑦𝑋 𝑥 = [𝑦]𝑅)
72 simpll2 1099 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑆 ∈ (SubGrp‘𝐺))
73 subgrcl 17527 . . . . . . . . . . . . . 14 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
7472, 73syl 17 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝐺 ∈ Grp)
753subgss 17523 . . . . . . . . . . . . . . 15 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
769, 75syl 17 . . . . . . . . . . . . . 14 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆𝑋)
7776adantr 481 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑆𝑋)
78 simpr 477 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑦𝑋)
79 eqid 2621 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
803, 15, 79eqglact 17573 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝑆𝑋𝑦𝑋) → [𝑦]𝑅 = ((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) “ 𝑆))
8174, 77, 78, 80syl3anc 1323 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → [𝑦]𝑅 = ((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) “ 𝑆))
82 simplr 791 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑆 ∈ (Clsd‘𝐽))
83 eqid 2621 . . . . . . . . . . . . . . . 16 (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) = (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧))
8483, 3, 79, 2tgplacthmeo 21826 . . . . . . . . . . . . . . 15 ((𝐺 ∈ TopGrp ∧ 𝑦𝑋) → (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
851, 84sylan 488 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
8676, 7sseqtrd 3625 . . . . . . . . . . . . . . 15 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 𝐽)
8786adantr 481 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑆 𝐽)
88 eqid 2621 . . . . . . . . . . . . . . 15 𝐽 = 𝐽
8988hmeocld 21489 . . . . . . . . . . . . . 14 (((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽) ∧ 𝑆 𝐽) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) “ 𝑆) ∈ (Clsd‘𝐽)))
9085, 87, 89syl2anc 692 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) “ 𝑆) ∈ (Clsd‘𝐽)))
9182, 90mpbid 222 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → ((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) “ 𝑆) ∈ (Clsd‘𝐽))
9281, 91eqeltrd 2698 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → [𝑦]𝑅 ∈ (Clsd‘𝐽))
93 eleq1 2686 . . . . . . . . . . 11 (𝑥 = [𝑦]𝑅 → (𝑥 ∈ (Clsd‘𝐽) ↔ [𝑦]𝑅 ∈ (Clsd‘𝐽)))
9492, 93syl5ibrcom 237 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → (𝑥 = [𝑦]𝑅𝑥 ∈ (Clsd‘𝐽)))
9594rexlimdva 3025 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∃𝑦𝑋 𝑥 = [𝑦]𝑅𝑥 ∈ (Clsd‘𝐽)))
9671, 95syl5bi 232 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑥 ∈ (𝑋 / 𝑅) → 𝑥 ∈ (Clsd‘𝐽)))
9796ssrdv 3593 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 / 𝑅) ⊆ (Clsd‘𝐽))
9897ssdifssd 3731 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (Clsd‘𝐽))
9988unicld 20769 . . . . . 6 ((𝐽 ∈ Top ∧ ((𝑋 / 𝑅) ∖ {𝑆}) ∈ Fin ∧ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ∈ (Clsd‘𝐽))
10066, 69, 98, 99syl3anc 1323 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ∈ (Clsd‘𝐽))
10188cldopn 20754 . . . . 5 ( ((𝑋 / 𝑅) ∖ {𝑆}) ∈ (Clsd‘𝐽) → ( 𝐽 ((𝑋 / 𝑅) ∖ {𝑆})) ∈ 𝐽)
102100, 101syl 17 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( 𝐽 ((𝑋 / 𝑅) ∖ {𝑆})) ∈ 𝐽)
10364, 102eqeltrrd 2699 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆𝐽)
104103ex 450 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆 ∈ (Clsd‘𝐽) → 𝑆𝐽))
1052opnsubg 21830 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆 ∈ (Clsd‘𝐽))
1061053expia 1264 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆𝐽𝑆 ∈ (Clsd‘𝐽)))
1071063adant3 1079 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆𝐽𝑆 ∈ (Clsd‘𝐽)))
108104, 107impbid 202 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2908  Vcvv 3189  cdif 3556  cun 3557  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135  {csn 4153   cuni 4407  cmpt 4678  cima 5082  cfv 5852  (class class class)co 6610   Er wer 7691  [cec 7692   / cqs 7693  Fincfn 7906  Basecbs 15788  +gcplusg 15869  TopOpenctopn 16010  0gc0g 16028  Grpcgrp 17350  SubGrpcsubg 17516   ~QG cqg 17518  Topctop 20626  TopOnctopon 20643  Clsdccld 20739  Homeochmeo 21475  TopGrpctgp 21794
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  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 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-ec 7696  df-qs 7700  df-map 7811  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-ndx 15791  df-slot 15792  df-base 15793  df-sets 15794  df-ress 15795  df-plusg 15882  df-0g 16030  df-topgen 16032  df-plusf 17169  df-mgm 17170  df-sgrp 17212  df-mnd 17223  df-grp 17353  df-minusg 17354  df-sbg 17355  df-subg 17519  df-eqg 17521  df-top 20627  df-topon 20644  df-topsp 20657  df-bases 20670  df-cld 20742  df-cn 20950  df-cnp 20951  df-tx 21284  df-hmeo 21477  df-tmd 21795  df-tgp 21796
This theorem is referenced by: (None)
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