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Theorem cldsubg 22136
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 1228 . . . . . . . 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 22108 . . . . . . . 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 20942 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
75, 6syl 17 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑋 = 𝐽)
87difeq1d 3871 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 ((𝑋 / 𝑅) ∖ {𝑆})) = ( 𝐽 ((𝑋 / 𝑅) ∖ {𝑆})))
9 simpl2 1230 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ∈ (SubGrp‘𝐺))
10 unisng 4605 . . . . . . . . 9 (𝑆 ∈ (SubGrp‘𝐺) → {𝑆} = 𝑆)
119, 10syl 17 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → {𝑆} = 𝑆)
1211uneq2d 3911 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆))
13 uniun 4609 . . . . . . . 8 (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆})
14 undif1 4188 . . . . . . . . . . 11 (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ((𝑋 / 𝑅) ∪ {𝑆})
15 cldsubg.1 . . . . . . . . . . . . . . . 16 𝑅 = (𝐺 ~QG 𝑆)
16 eqid 2761 . . . . . . . . . . . . . . . 16 (0g𝐺) = (0g𝐺)
173, 15, 16eqgid 17868 . . . . . . . . . . . . . . 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 6843 . . . . . . . . . . . . . . . 16 (𝐺 ~QG 𝑆) ∈ V
2015, 19eqeltri 2836 . . . . . . . . . . . . . . 15 𝑅 ∈ V
21 tgpgrp 22104 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
221, 21syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝐺 ∈ Grp)
233, 16grpidcl 17672 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
2422, 23syl 17 . . . . . . . . . . . . . . 15 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (0g𝐺) ∈ 𝑋)
25 ecelqsg 7972 . . . . . . . . . . . . . . 15 ((𝑅 ∈ V ∧ (0g𝐺) ∈ 𝑋) → [(0g𝐺)]𝑅 ∈ (𝑋 / 𝑅))
2620, 24, 25sylancr 698 . . . . . . . . . . . . . 14 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → [(0g𝐺)]𝑅 ∈ (𝑋 / 𝑅))
2718, 26eqeltrrd 2841 . . . . . . . . . . . . 13 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ∈ (𝑋 / 𝑅))
2827snssd 4486 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → {𝑆} ⊆ (𝑋 / 𝑅))
29 ssequn2 3930 . . . . . . . . . . . 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 2807 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = (𝑋 / 𝑅))
3231unieqd 4599 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = (𝑋 / 𝑅))
333, 15eqger 17866 . . . . . . . . . . 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 7979 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 / 𝑅) = 𝑋)
3732, 36eqtrd 2795 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = 𝑋)
3813, 37syl5eqr 2809 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = 𝑋)
3912, 38eqtr3d 2797 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆) = 𝑋)
40 difss 3881 . . . . . . . . 9 ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (𝑋 / 𝑅)
4140unissi 4614 . . . . . . . 8 ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (𝑋 / 𝑅)
4241, 36syl5sseq 3795 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝑋)
43 df-ne 2934 . . . . . . . . . . . . 13 (𝑥𝑆 ↔ ¬ 𝑥 = 𝑆)
4434adantr 472 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → 𝑅 Er 𝑋)
45 simpr 479 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → 𝑥 ∈ (𝑋 / 𝑅))
4627adantr 472 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → 𝑆 ∈ (𝑋 / 𝑅))
4744, 45, 46qsdisj 7994 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (𝑥 = 𝑆 ∨ (𝑥𝑆) = ∅))
4847ord 391 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (¬ 𝑥 = 𝑆 → (𝑥𝑆) = ∅))
49 disj2 4169 . . . . . . . . . . . . . 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 630 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑥 ∈ (𝑋 / 𝑅) ∧ 𝑥𝑆) → 𝑥 ⊆ (V ∖ 𝑆)))
53 eldifsn 4463 . . . . . . . . . . 11 (𝑥 ∈ ((𝑋 / 𝑅) ∖ {𝑆}) ↔ (𝑥 ∈ (𝑋 / 𝑅) ∧ 𝑥𝑆))
54 selpw 4310 . . . . . . . . . . 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 3751 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝒫 (V ∖ 𝑆))
57 sspwuni 4764 . . . . . . . . 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 4169 . . . . . . . 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 4202 . . . . . . 7 (( ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝑋 ∧ ( ((𝑋 / 𝑅) ∖ {𝑆}) ∩ 𝑆) = ∅) → (( ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆) = 𝑋 ↔ (𝑋 ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆))
6242, 60, 61syl2anc 696 . . . . . 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 2797 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( 𝐽 ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆)
65 topontop 20941 . . . . . . 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 1232 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 / 𝑅) ∈ Fin)
68 diffi 8360 . . . . . . 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 3344 . . . . . . . . . 10 𝑥 ∈ V
7170elqs 7969 . . . . . . . . 9 (𝑥 ∈ (𝑋 / 𝑅) ↔ ∃𝑦𝑋 𝑥 = [𝑦]𝑅)
72 simpll2 1257 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑆 ∈ (SubGrp‘𝐺))
73 subgrcl 17821 . . . . . . . . . . . . . 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 17817 . . . . . . . . . . . . . . 15 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
769, 75syl 17 . . . . . . . . . . . . . 14 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆𝑋)
7776adantr 472 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑆𝑋)
78 simpr 479 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑦𝑋)
79 eqid 2761 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
803, 15, 79eqglact 17867 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝑆𝑋𝑦𝑋) → [𝑦]𝑅 = ((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) “ 𝑆))
8174, 77, 78, 80syl3anc 1477 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → [𝑦]𝑅 = ((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) “ 𝑆))
82 simplr 809 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑆 ∈ (Clsd‘𝐽))
83 eqid 2761 . . . . . . . . . . . . . . . 16 (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) = (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧))
8483, 3, 79, 2tgplacthmeo 22129 . . . . . . . . . . . . . . 15 ((𝐺 ∈ TopGrp ∧ 𝑦𝑋) → (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
851, 84sylan 489 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
8676, 7sseqtrd 3783 . . . . . . . . . . . . . . 15 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 𝐽)
8786adantr 472 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → 𝑆 𝐽)
88 eqid 2761 . . . . . . . . . . . . . . 15 𝐽 = 𝐽
8988hmeocld 21793 . . . . . . . . . . . . . 14 (((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽) ∧ 𝑆 𝐽) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)) “ 𝑆) ∈ (Clsd‘𝐽)))
9085, 87, 89syl2anc 696 . . . . . . . . . . . . 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 2840 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦𝑋) → [𝑦]𝑅 ∈ (Clsd‘𝐽))
93 eleq1 2828 . . . . . . . . . . 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 3170 . . . . . . . . 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 3751 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 / 𝑅) ⊆ (Clsd‘𝐽))
9897ssdifssd 3892 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (Clsd‘𝐽))
9988unicld 21073 . . . . . 6 ((𝐽 ∈ Top ∧ ((𝑋 / 𝑅) ∖ {𝑆}) ∈ Fin ∧ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ∈ (Clsd‘𝐽))
10066, 69, 98, 99syl3anc 1477 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ∈ (Clsd‘𝐽))
10188cldopn 21058 . . . . 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 2841 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆𝐽)
104103ex 449 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆 ∈ (Clsd‘𝐽) → 𝑆𝐽))
1052opnsubg 22133 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆 ∈ (Clsd‘𝐽))
1061053expia 1115 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆𝐽𝑆 ∈ (Clsd‘𝐽)))
1071063adant3 1127 . 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 383  w3a 1072   = wceq 1632  wcel 2140  wne 2933  wrex 3052  Vcvv 3341  cdif 3713  cun 3714  cin 3715  wss 3716  c0 4059  𝒫 cpw 4303  {csn 4322   cuni 4589  cmpt 4882  cima 5270  cfv 6050  (class class class)co 6815   Er wer 7911  [cec 7912   / cqs 7913  Fincfn 8124  Basecbs 16080  +gcplusg 16164  TopOpenctopn 16305  0gc0g 16323  Grpcgrp 17644  SubGrpcsubg 17810   ~QG cqg 17812  Topctop 20921  TopOnctopon 20938  Clsdccld 21043  Homeochmeo 21779  TopGrpctgp 22097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-iin 4676  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-1o 7731  df-oadd 7735  df-er 7914  df-ec 7916  df-qs 7920  df-map 8028  df-en 8125  df-dom 8126  df-sdom 8127  df-fin 8128  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-nn 11234  df-2 11292  df-ndx 16083  df-slot 16084  df-base 16086  df-sets 16087  df-ress 16088  df-plusg 16177  df-0g 16325  df-topgen 16327  df-plusf 17463  df-mgm 17464  df-sgrp 17506  df-mnd 17517  df-grp 17647  df-minusg 17648  df-sbg 17649  df-subg 17813  df-eqg 17815  df-top 20922  df-topon 20939  df-topsp 20960  df-bases 20973  df-cld 21046  df-cn 21254  df-cnp 21255  df-tx 21588  df-hmeo 21781  df-tmd 22098  df-tgp 22099
This theorem is referenced by: (None)
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