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Theorem cldsubg 23835
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 = (TopOpenβ€˜πΊ)
cldsubg.1 𝑅 = (𝐺 ~QG 𝑆)
cldsubg.2 𝑋 = (Baseβ€˜πΊ)
Assertion
Ref Expression
cldsubg ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) β†’ (𝑆 ∈ (Clsdβ€˜π½) ↔ 𝑆 ∈ 𝐽))
Proof of Theorem cldsubg
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 1189 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ 𝐺 ∈ TopGrp)
2 subgntr.h . . . . . . . . 9 𝐽 = (TopOpenβ€˜πΊ)
3 cldsubg.2 . . . . . . . . 9 𝑋 = (Baseβ€˜πΊ)
42, 3tgptopon 23806 . . . . . . . 8 (𝐺 ∈ TopGrp β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
51, 4syl 17 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
6 toponuni 22636 . . . . . . 7 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
75, 6syl 17 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ 𝑋 = βˆͺ 𝐽)
87difeq1d 4120 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (𝑋 βˆ– βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆})) = (βˆͺ 𝐽 βˆ– βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆})))
9 simpl2 1190 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))
10 unisng 4928 . . . . . . . . 9 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ βˆͺ {𝑆} = 𝑆)
119, 10syl 17 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ βˆͺ {𝑆} = 𝑆)
1211uneq2d 4162 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) βˆͺ βˆͺ {𝑆}) = (βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) βˆͺ 𝑆))
13 uniun 4933 . . . . . . . 8 βˆͺ (((𝑋 / 𝑅) βˆ– {𝑆}) βˆͺ {𝑆}) = (βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) βˆͺ βˆͺ {𝑆})
14 undif1 4474 . . . . . . . . . . 11 (((𝑋 / 𝑅) βˆ– {𝑆}) βˆͺ {𝑆}) = ((𝑋 / 𝑅) βˆͺ {𝑆})
15 cldsubg.1 . . . . . . . . . . . . . . . 16 𝑅 = (𝐺 ~QG 𝑆)
16 eqid 2730 . . . . . . . . . . . . . . . 16 (0gβ€˜πΊ) = (0gβ€˜πΊ)
173, 15, 16eqgid 19096 . . . . . . . . . . . . . . 15 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ [(0gβ€˜πΊ)]𝑅 = 𝑆)
189, 17syl 17 . . . . . . . . . . . . . 14 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ [(0gβ€˜πΊ)]𝑅 = 𝑆)
1915ovexi 7445 . . . . . . . . . . . . . . 15 𝑅 ∈ V
20 tgpgrp 23802 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ Grp)
211, 20syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ 𝐺 ∈ Grp)
223, 16grpidcl 18886 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Grp β†’ (0gβ€˜πΊ) ∈ 𝑋)
2321, 22syl 17 . . . . . . . . . . . . . . 15 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (0gβ€˜πΊ) ∈ 𝑋)
24 ecelqsg 8768 . . . . . . . . . . . . . . 15 ((𝑅 ∈ V ∧ (0gβ€˜πΊ) ∈ 𝑋) β†’ [(0gβ€˜πΊ)]𝑅 ∈ (𝑋 / 𝑅))
2519, 23, 24sylancr 585 . . . . . . . . . . . . . 14 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ [(0gβ€˜πΊ)]𝑅 ∈ (𝑋 / 𝑅))
2618, 25eqeltrrd 2832 . . . . . . . . . . . . 13 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ 𝑆 ∈ (𝑋 / 𝑅))
2726snssd 4811 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ {𝑆} βŠ† (𝑋 / 𝑅))
28 ssequn2 4182 . . . . . . . . . . . 12 ({𝑆} βŠ† (𝑋 / 𝑅) ↔ ((𝑋 / 𝑅) βˆͺ {𝑆}) = (𝑋 / 𝑅))
2927, 28sylib 217 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ ((𝑋 / 𝑅) βˆͺ {𝑆}) = (𝑋 / 𝑅))
3014, 29eqtrid 2782 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (((𝑋 / 𝑅) βˆ– {𝑆}) βˆͺ {𝑆}) = (𝑋 / 𝑅))
3130unieqd 4921 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ βˆͺ (((𝑋 / 𝑅) βˆ– {𝑆}) βˆͺ {𝑆}) = βˆͺ (𝑋 / 𝑅))
323, 15eqger 19094 . . . . . . . . . . 11 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ 𝑅 Er 𝑋)
339, 32syl 17 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ 𝑅 Er 𝑋)
3419a1i 11 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ 𝑅 ∈ V)
3533, 34uniqs2 8775 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ βˆͺ (𝑋 / 𝑅) = 𝑋)
3631, 35eqtrd 2770 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ βˆͺ (((𝑋 / 𝑅) βˆ– {𝑆}) βˆͺ {𝑆}) = 𝑋)
3713, 36eqtr3id 2784 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) βˆͺ βˆͺ {𝑆}) = 𝑋)
3812, 37eqtr3d 2772 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) βˆͺ 𝑆) = 𝑋)
39 difss 4130 . . . . . . . . 9 ((𝑋 / 𝑅) βˆ– {𝑆}) βŠ† (𝑋 / 𝑅)
4039unissi 4916 . . . . . . . 8 βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) βŠ† βˆͺ (𝑋 / 𝑅)
4140, 35sseqtrid 4033 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) βŠ† 𝑋)
42 df-ne 2939 . . . . . . . . . . . . 13 (π‘₯ β‰  𝑆 ↔ Β¬ π‘₯ = 𝑆)
4333adantr 479 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ π‘₯ ∈ (𝑋 / 𝑅)) β†’ 𝑅 Er 𝑋)
44 simpr 483 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ π‘₯ ∈ (𝑋 / 𝑅)) β†’ π‘₯ ∈ (𝑋 / 𝑅))
4526adantr 479 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ π‘₯ ∈ (𝑋 / 𝑅)) β†’ 𝑆 ∈ (𝑋 / 𝑅))
4643, 44, 45qsdisj 8790 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ π‘₯ ∈ (𝑋 / 𝑅)) β†’ (π‘₯ = 𝑆 ∨ (π‘₯ ∩ 𝑆) = βˆ…))
4746ord 860 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ π‘₯ ∈ (𝑋 / 𝑅)) β†’ (Β¬ π‘₯ = 𝑆 β†’ (π‘₯ ∩ 𝑆) = βˆ…))
48 disj2 4456 . . . . . . . . . . . . . 14 ((π‘₯ ∩ 𝑆) = βˆ… ↔ π‘₯ βŠ† (V βˆ– 𝑆))
4947, 48imbitrdi 250 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ π‘₯ ∈ (𝑋 / 𝑅)) β†’ (Β¬ π‘₯ = 𝑆 β†’ π‘₯ βŠ† (V βˆ– 𝑆)))
5042, 49biimtrid 241 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ π‘₯ ∈ (𝑋 / 𝑅)) β†’ (π‘₯ β‰  𝑆 β†’ π‘₯ βŠ† (V βˆ– 𝑆)))
5150expimpd 452 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ ((π‘₯ ∈ (𝑋 / 𝑅) ∧ π‘₯ β‰  𝑆) β†’ π‘₯ βŠ† (V βˆ– 𝑆)))
52 eldifsn 4789 . . . . . . . . . . 11 (π‘₯ ∈ ((𝑋 / 𝑅) βˆ– {𝑆}) ↔ (π‘₯ ∈ (𝑋 / 𝑅) ∧ π‘₯ β‰  𝑆))
53 velpw 4606 . . . . . . . . . . 11 (π‘₯ ∈ 𝒫 (V βˆ– 𝑆) ↔ π‘₯ βŠ† (V βˆ– 𝑆))
5451, 52, 533imtr4g 295 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (π‘₯ ∈ ((𝑋 / 𝑅) βˆ– {𝑆}) β†’ π‘₯ ∈ 𝒫 (V βˆ– 𝑆)))
5554ssrdv 3987 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ ((𝑋 / 𝑅) βˆ– {𝑆}) βŠ† 𝒫 (V βˆ– 𝑆))
56 sspwuni 5102 . . . . . . . . 9 (((𝑋 / 𝑅) βˆ– {𝑆}) βŠ† 𝒫 (V βˆ– 𝑆) ↔ βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) βŠ† (V βˆ– 𝑆))
5755, 56sylib 217 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) βŠ† (V βˆ– 𝑆))
58 disj2 4456 . . . . . . . 8 ((βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) ∩ 𝑆) = βˆ… ↔ βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) βŠ† (V βˆ– 𝑆))
5957, 58sylibr 233 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) ∩ 𝑆) = βˆ…)
60 uneqdifeq 4491 . . . . . . 7 ((βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) βŠ† 𝑋 ∧ (βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) ∩ 𝑆) = βˆ…) β†’ ((βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) βˆͺ 𝑆) = 𝑋 ↔ (𝑋 βˆ– βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆})) = 𝑆))
6141, 59, 60syl2anc 582 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ ((βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) βˆͺ 𝑆) = 𝑋 ↔ (𝑋 βˆ– βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆})) = 𝑆))
6238, 61mpbid 231 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (𝑋 βˆ– βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆})) = 𝑆)
638, 62eqtr3d 2772 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (βˆͺ 𝐽 βˆ– βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆})) = 𝑆)
64 topontop 22635 . . . . . . 7 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
655, 64syl 17 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ 𝐽 ∈ Top)
66 simpl3 1191 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (𝑋 / 𝑅) ∈ Fin)
67 diffi 9181 . . . . . . 7 ((𝑋 / 𝑅) ∈ Fin β†’ ((𝑋 / 𝑅) βˆ– {𝑆}) ∈ Fin)
6866, 67syl 17 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ ((𝑋 / 𝑅) βˆ– {𝑆}) ∈ Fin)
69 vex 3476 . . . . . . . . . 10 π‘₯ ∈ V
7069elqs 8765 . . . . . . . . 9 (π‘₯ ∈ (𝑋 / 𝑅) ↔ βˆƒπ‘¦ ∈ 𝑋 π‘₯ = [𝑦]𝑅)
71 simpll2 1211 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ 𝑦 ∈ 𝑋) β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))
72 subgrcl 19047 . . . . . . . . . . . . . 14 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ 𝐺 ∈ Grp)
7371, 72syl 17 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ 𝑦 ∈ 𝑋) β†’ 𝐺 ∈ Grp)
743subgss 19043 . . . . . . . . . . . . . . 15 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ 𝑆 βŠ† 𝑋)
759, 74syl 17 . . . . . . . . . . . . . 14 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ 𝑆 βŠ† 𝑋)
7675adantr 479 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ 𝑦 ∈ 𝑋) β†’ 𝑆 βŠ† 𝑋)
77 simpr 483 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ 𝑦 ∈ 𝑋) β†’ 𝑦 ∈ 𝑋)
78 eqid 2730 . . . . . . . . . . . . . 14 (+gβ€˜πΊ) = (+gβ€˜πΊ)
793, 15, 78eqglact 19095 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝑆 βŠ† 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ [𝑦]𝑅 = ((𝑧 ∈ 𝑋 ↦ (𝑦(+gβ€˜πΊ)𝑧)) β€œ 𝑆))
8073, 76, 77, 79syl3anc 1369 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ 𝑦 ∈ 𝑋) β†’ [𝑦]𝑅 = ((𝑧 ∈ 𝑋 ↦ (𝑦(+gβ€˜πΊ)𝑧)) β€œ 𝑆))
81 simplr 765 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ 𝑦 ∈ 𝑋) β†’ 𝑆 ∈ (Clsdβ€˜π½))
82 eqid 2730 . . . . . . . . . . . . . . . 16 (𝑧 ∈ 𝑋 ↦ (𝑦(+gβ€˜πΊ)𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝑦(+gβ€˜πΊ)𝑧))
8382, 3, 78, 2tgplacthmeo 23827 . . . . . . . . . . . . . . 15 ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑋) β†’ (𝑧 ∈ 𝑋 ↦ (𝑦(+gβ€˜πΊ)𝑧)) ∈ (𝐽Homeo𝐽))
841, 83sylan 578 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ 𝑦 ∈ 𝑋) β†’ (𝑧 ∈ 𝑋 ↦ (𝑦(+gβ€˜πΊ)𝑧)) ∈ (𝐽Homeo𝐽))
8575, 7sseqtrd 4021 . . . . . . . . . . . . . . 15 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ 𝑆 βŠ† βˆͺ 𝐽)
8685adantr 479 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ 𝑦 ∈ 𝑋) β†’ 𝑆 βŠ† βˆͺ 𝐽)
87 eqid 2730 . . . . . . . . . . . . . . 15 βˆͺ 𝐽 = βˆͺ 𝐽
8887hmeocld 23491 . . . . . . . . . . . . . 14 (((𝑧 ∈ 𝑋 ↦ (𝑦(+gβ€˜πΊ)𝑧)) ∈ (𝐽Homeo𝐽) ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ (𝑆 ∈ (Clsdβ€˜π½) ↔ ((𝑧 ∈ 𝑋 ↦ (𝑦(+gβ€˜πΊ)𝑧)) β€œ 𝑆) ∈ (Clsdβ€˜π½)))
8984, 86, 88syl2anc 582 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ 𝑦 ∈ 𝑋) β†’ (𝑆 ∈ (Clsdβ€˜π½) ↔ ((𝑧 ∈ 𝑋 ↦ (𝑦(+gβ€˜πΊ)𝑧)) β€œ 𝑆) ∈ (Clsdβ€˜π½)))
9081, 89mpbid 231 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ 𝑦 ∈ 𝑋) β†’ ((𝑧 ∈ 𝑋 ↦ (𝑦(+gβ€˜πΊ)𝑧)) β€œ 𝑆) ∈ (Clsdβ€˜π½))
9180, 90eqeltrd 2831 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ 𝑦 ∈ 𝑋) β†’ [𝑦]𝑅 ∈ (Clsdβ€˜π½))
92 eleq1 2819 . . . . . . . . . . 11 (π‘₯ = [𝑦]𝑅 β†’ (π‘₯ ∈ (Clsdβ€˜π½) ↔ [𝑦]𝑅 ∈ (Clsdβ€˜π½)))
9391, 92syl5ibrcom 246 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯ = [𝑦]𝑅 β†’ π‘₯ ∈ (Clsdβ€˜π½)))
9493rexlimdva 3153 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (βˆƒπ‘¦ ∈ 𝑋 π‘₯ = [𝑦]𝑅 β†’ π‘₯ ∈ (Clsdβ€˜π½)))
9570, 94biimtrid 241 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (π‘₯ ∈ (𝑋 / 𝑅) β†’ π‘₯ ∈ (Clsdβ€˜π½)))
9695ssrdv 3987 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (𝑋 / 𝑅) βŠ† (Clsdβ€˜π½))
9796ssdifssd 4141 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ ((𝑋 / 𝑅) βˆ– {𝑆}) βŠ† (Clsdβ€˜π½))
9887unicld 22770 . . . . . 6 ((𝐽 ∈ Top ∧ ((𝑋 / 𝑅) βˆ– {𝑆}) ∈ Fin ∧ ((𝑋 / 𝑅) βˆ– {𝑆}) βŠ† (Clsdβ€˜π½)) β†’ βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) ∈ (Clsdβ€˜π½))
9965, 68, 97, 98syl3anc 1369 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) ∈ (Clsdβ€˜π½))
10087cldopn 22755 . . . . 5 (βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) ∈ (Clsdβ€˜π½) β†’ (βˆͺ 𝐽 βˆ– βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆})) ∈ 𝐽)
10199, 100syl 17 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (βˆͺ 𝐽 βˆ– βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆})) ∈ 𝐽)
10263, 101eqeltrrd 2832 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ 𝑆 ∈ 𝐽)
103102ex 411 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) β†’ (𝑆 ∈ (Clsdβ€˜π½) β†’ 𝑆 ∈ 𝐽))
1042opnsubg 23832 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) β†’ 𝑆 ∈ (Clsdβ€˜π½))
1051043expia 1119 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ (𝑆 ∈ 𝐽 β†’ 𝑆 ∈ (Clsdβ€˜π½)))
1061053adant3 1130 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) β†’ (𝑆 ∈ 𝐽 β†’ 𝑆 ∈ (Clsdβ€˜π½)))
107103, 106impbid 211 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) β†’ (𝑆 ∈ (Clsdβ€˜π½) ↔ 𝑆 ∈ 𝐽))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆƒwrex 3068  Vcvv 3472   βˆ– cdif 3944   βˆͺ cun 3945   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  {csn 4627  βˆͺ cuni 4907   ↦ cmpt 5230   β€œ cima 5678  β€˜cfv 6542  (class class class)co 7411   Er wer 8702  [cec 8703   / cqs 8704  Fincfn 8941  Basecbs 17148  +gcplusg 17201  TopOpenctopn 17371  0gc0g 17389  Grpcgrp 18855  SubGrpcsubg 19036   ~QG cqg 19038  Topctop 22615  TopOnctopon 22632  Clsdccld 22740  Homeochmeo 23477  TopGrpctgp 23795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-ec 8707  df-qs 8711  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-0g 17391  df-topgen 17393  df-plusf 18564  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-sbg 18860  df-subg 19039  df-eqg 19041  df-top 22616  df-topon 22633  df-topsp 22655  df-bases 22669  df-cld 22743  df-cn 22951  df-cnp 22952  df-tx 23286  df-hmeo 23479  df-tmd 23796  df-tgp 23797
This theorem is referenced by: (None)
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