MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cldsubg Structured version   Visualization version   GIF version

Theorem cldsubg 23485
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 1192 . . . . . . . 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 23456 . . . . . . . 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 22286 . . . . . . 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 4085 . . . . 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 1193 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))
10 unisng 4890 . . . . . . . . 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 4127 . . . . . . 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 4895 . . . . . . . 8 βˆͺ (((𝑋 / 𝑅) βˆ– {𝑆}) βˆͺ {𝑆}) = (βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) βˆͺ βˆͺ {𝑆})
14 undif1 4439 . . . . . . . . . . 11 (((𝑋 / 𝑅) βˆ– {𝑆}) βˆͺ {𝑆}) = ((𝑋 / 𝑅) βˆͺ {𝑆})
15 cldsubg.1 . . . . . . . . . . . . . . . 16 𝑅 = (𝐺 ~QG 𝑆)
16 eqid 2733 . . . . . . . . . . . . . . . 16 (0gβ€˜πΊ) = (0gβ€˜πΊ)
173, 15, 16eqgid 18990 . . . . . . . . . . . . . . 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 7395 . . . . . . . . . . . . . . 15 𝑅 ∈ V
20 tgpgrp 23452 . . . . . . . . . . . . . . . . 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 18786 . . . . . . . . . . . . . . . 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 8717 . . . . . . . . . . . . . . 15 ((𝑅 ∈ V ∧ (0gβ€˜πΊ) ∈ 𝑋) β†’ [(0gβ€˜πΊ)]𝑅 ∈ (𝑋 / 𝑅))
2519, 23, 24sylancr 588 . . . . . . . . . . . . . 14 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ [(0gβ€˜πΊ)]𝑅 ∈ (𝑋 / 𝑅))
2618, 25eqeltrrd 2835 . . . . . . . . . . . . 13 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ 𝑆 ∈ (𝑋 / 𝑅))
2726snssd 4773 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ {𝑆} βŠ† (𝑋 / 𝑅))
28 ssequn2 4147 . . . . . . . . . . . 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 2785 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (((𝑋 / 𝑅) βˆ– {𝑆}) βˆͺ {𝑆}) = (𝑋 / 𝑅))
3130unieqd 4883 . . . . . . . . 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 18988 . . . . . . . . . . 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 8724 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ βˆͺ (𝑋 / 𝑅) = 𝑋)
3631, 35eqtrd 2773 . . . . . . . 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 2787 . . . . . . 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 2775 . . . . . 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 4095 . . . . . . . . 9 ((𝑋 / 𝑅) βˆ– {𝑆}) βŠ† (𝑋 / 𝑅)
4039unissi 4878 . . . . . . . 8 βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) βŠ† βˆͺ (𝑋 / 𝑅)
4140, 35sseqtrid 4000 . . . . . . 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 2941 . . . . . . . . . . . . 13 (π‘₯ β‰  𝑆 ↔ Β¬ π‘₯ = 𝑆)
4333adantr 482 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ π‘₯ ∈ (𝑋 / 𝑅)) β†’ 𝑅 Er 𝑋)
44 simpr 486 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ π‘₯ ∈ (𝑋 / 𝑅)) β†’ π‘₯ ∈ (𝑋 / 𝑅))
4526adantr 482 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ π‘₯ ∈ (𝑋 / 𝑅)) β†’ 𝑆 ∈ (𝑋 / 𝑅))
4643, 44, 45qsdisj 8739 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ π‘₯ ∈ (𝑋 / 𝑅)) β†’ (π‘₯ = 𝑆 ∨ (π‘₯ ∩ 𝑆) = βˆ…))
4746ord 863 . . . . . . . . . . . . . 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 4421 . . . . . . . . . . . . . 14 ((π‘₯ ∩ 𝑆) = βˆ… ↔ π‘₯ βŠ† (V βˆ– 𝑆))
4947, 48syl6ib 251 . . . . . . . . . . . . 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 455 . . . . . . . . . . 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 4751 . . . . . . . . . . 11 (π‘₯ ∈ ((𝑋 / 𝑅) βˆ– {𝑆}) ↔ (π‘₯ ∈ (𝑋 / 𝑅) ∧ π‘₯ β‰  𝑆))
53 velpw 4569 . . . . . . . . . . 11 (π‘₯ ∈ 𝒫 (V βˆ– 𝑆) ↔ π‘₯ βŠ† (V βˆ– 𝑆))
5451, 52, 533imtr4g 296 . . . . . . . . . 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 3954 . . . . . . . . 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 5064 . . . . . . . . 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 4421 . . . . . . . 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 4454 . . . . . . 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 585 . . . . . 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 2775 . . . 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 22285 . . . . . . 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 1194 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (𝑋 / 𝑅) ∈ Fin)
67 diffi 9129 . . . . . . 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 3451 . . . . . . . . . 10 π‘₯ ∈ V
7069elqs 8714 . . . . . . . . 9 (π‘₯ ∈ (𝑋 / 𝑅) ↔ βˆƒπ‘¦ ∈ 𝑋 π‘₯ = [𝑦]𝑅)
71 simpll2 1214 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ 𝑦 ∈ 𝑋) β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))
72 subgrcl 18941 . . . . . . . . . . . . . 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 18937 . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ 𝑦 ∈ 𝑋) β†’ 𝑆 βŠ† 𝑋)
77 simpr 486 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ 𝑦 ∈ 𝑋) β†’ 𝑦 ∈ 𝑋)
78 eqid 2733 . . . . . . . . . . . . . 14 (+gβ€˜πΊ) = (+gβ€˜πΊ)
793, 15, 78eqglact 18989 . . . . . . . . . . . . 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 1372 . . . . . . . . . . . 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 768 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ 𝑦 ∈ 𝑋) β†’ 𝑆 ∈ (Clsdβ€˜π½))
82 eqid 2733 . . . . . . . . . . . . . . . 16 (𝑧 ∈ 𝑋 ↦ (𝑦(+gβ€˜πΊ)𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝑦(+gβ€˜πΊ)𝑧))
8382, 3, 78, 2tgplacthmeo 23477 . . . . . . . . . . . . . . 15 ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑋) β†’ (𝑧 ∈ 𝑋 ↦ (𝑦(+gβ€˜πΊ)𝑧)) ∈ (𝐽Homeo𝐽))
841, 83sylan 581 . . . . . . . . . . . . . 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 3988 . . . . . . . . . . . . . . 15 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ 𝑆 βŠ† βˆͺ 𝐽)
8685adantr 482 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ 𝑦 ∈ 𝑋) β†’ 𝑆 βŠ† βˆͺ 𝐽)
87 eqid 2733 . . . . . . . . . . . . . . 15 βˆͺ 𝐽 = βˆͺ 𝐽
8887hmeocld 23141 . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . 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 2834 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) ∧ 𝑦 ∈ 𝑋) β†’ [𝑦]𝑅 ∈ (Clsdβ€˜π½))
92 eleq1 2822 . . . . . . . . . . 11 (π‘₯ = [𝑦]𝑅 β†’ (π‘₯ ∈ (Clsdβ€˜π½) ↔ [𝑦]𝑅 ∈ (Clsdβ€˜π½)))
9391, 92syl5ibrcom 247 . . . . . . . . . 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 3149 . . . . . . . . 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 3954 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (𝑋 / 𝑅) βŠ† (Clsdβ€˜π½))
9796ssdifssd 4106 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ ((𝑋 / 𝑅) βˆ– {𝑆}) βŠ† (Clsdβ€˜π½))
9887unicld 22420 . . . . . 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 1372 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ βˆͺ ((𝑋 / 𝑅) βˆ– {𝑆}) ∈ (Clsdβ€˜π½))
10087cldopn 22405 . . . . 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 2835 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ 𝑆 ∈ 𝐽)
103102ex 414 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑋 / 𝑅) ∈ Fin) β†’ (𝑆 ∈ (Clsdβ€˜π½) β†’ 𝑆 ∈ 𝐽))
1042opnsubg 23482 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) β†’ 𝑆 ∈ (Clsdβ€˜π½))
1051043expia 1122 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ (𝑆 ∈ 𝐽 β†’ 𝑆 ∈ (Clsdβ€˜π½)))
1061053adant3 1133 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆƒwrex 3070  Vcvv 3447   βˆ– cdif 3911   βˆͺ cun 3912   ∩ cin 3913   βŠ† wss 3914  βˆ…c0 4286  π’« cpw 4564  {csn 4590  βˆͺ cuni 4869   ↦ cmpt 5192   β€œ cima 5640  β€˜cfv 6500  (class class class)co 7361   Er wer 8651  [cec 8652   / cqs 8653  Fincfn 8889  Basecbs 17091  +gcplusg 17141  TopOpenctopn 17311  0gc0g 17329  Grpcgrp 18756  SubGrpcsubg 18930   ~QG cqg 18932  Topctop 22265  TopOnctopon 22282  Clsdccld 22390  Homeochmeo 23127  TopGrpctgp 23445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-ec 8656  df-qs 8660  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-0g 17331  df-topgen 17333  df-plusf 18504  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-grp 18759  df-minusg 18760  df-sbg 18761  df-subg 18933  df-eqg 18935  df-top 22266  df-topon 22283  df-topsp 22305  df-bases 22319  df-cld 22393  df-cn 22601  df-cnp 22602  df-tx 22936  df-hmeo 23129  df-tmd 23446  df-tgp 23447
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator