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Theorem istgp2 21805
Description: A group with a topology is a topological group iff the subtraction operation is continuous. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypotheses
Ref Expression
tgpsubcn.2 𝐽 = (TopOpen‘𝐺)
tgpsubcn.3 = (-g𝐺)
Assertion
Ref Expression
istgp2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))

Proof of Theorem istgp2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tgpgrp 21792 . . 3 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2 tgptps 21794 . . 3 (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
3 tgpsubcn.2 . . . 4 𝐽 = (TopOpen‘𝐺)
4 tgpsubcn.3 . . . 4 = (-g𝐺)
53, 4tgpsubcn 21804 . . 3 (𝐺 ∈ TopGrp → ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
61, 2, 53jca 1240 . 2 (𝐺 ∈ TopGrp → (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
7 simp1 1059 . . 3 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ Grp)
8 grpmnd 17350 . . . . 5 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
983ad2ant1 1080 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ Mnd)
10 simp2 1060 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopSp)
11 eqid 2621 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
12 eqid 2621 . . . . . . . 8 (+g𝐺) = (+g𝐺)
13 eqid 2621 . . . . . . . 8 (invg𝐺) = (invg𝐺)
1473ad2ant1 1080 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → 𝐺 ∈ Grp)
15 simp2 1060 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
16 simp3 1061 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → 𝑦 ∈ (Base‘𝐺))
1711, 12, 4, 13, 14, 15, 16grpsubinv 17409 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥 ((invg𝐺)‘𝑦)) = (𝑥(+g𝐺)𝑦))
1817mpt2eq3dva 6672 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥 ((invg𝐺)‘𝑦))) = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)))
19 eqid 2621 . . . . . . 7 (+𝑓𝐺) = (+𝑓𝐺)
2011, 12, 19plusffval 17168 . . . . . 6 (+𝑓𝐺) = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦))
2118, 20syl6eqr 2673 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥 ((invg𝐺)‘𝑦))) = (+𝑓𝐺))
2211, 3istps 20651 . . . . . . 7 (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝐺)))
2310, 22sylib 208 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
2423, 23cnmpt1st 21381 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
2523, 23cnmpt2nd 21382 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
2611, 13grpinvf 17387 . . . . . . . . . . 11 (𝐺 ∈ Grp → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
27263ad2ant1 1080 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
2827feqmptd 6206 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)))
29 eqid 2621 . . . . . . . . . . . 12 (0g𝐺) = (0g𝐺)
3011, 4, 13, 29grpinvval2 17419 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) = ((0g𝐺) 𝑥))
317, 30sylan 488 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) = ((0g𝐺) 𝑥))
3231mpteq2dva 4704 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)) = (𝑥 ∈ (Base‘𝐺) ↦ ((0g𝐺) 𝑥)))
3328, 32eqtrd 2655 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((0g𝐺) 𝑥)))
3411, 29grpidcl 17371 . . . . . . . . . . 11 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
35343ad2ant1 1080 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (0g𝐺) ∈ (Base‘𝐺))
3623, 23, 35cnmptc 21375 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ (0g𝐺)) ∈ (𝐽 Cn 𝐽))
3723cnmptid 21374 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
38 simp3 1061 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
3923, 36, 37, 38cnmpt12f 21379 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ ((0g𝐺) 𝑥)) ∈ (𝐽 Cn 𝐽))
4033, 39eqeltrd 2698 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg𝐺) ∈ (𝐽 Cn 𝐽))
4123, 23, 25, 40cnmpt21f 21385 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
4223, 23, 24, 41, 38cnmpt22f 21388 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥 ((invg𝐺)‘𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
4321, 42eqeltrrd 2699 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
4419, 3istmd 21788 . . . 4 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
459, 10, 43, 44syl3anbrc 1244 . . 3 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopMnd)
463, 13istgp 21791 . . 3 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg𝐺) ∈ (𝐽 Cn 𝐽)))
477, 45, 40, 46syl3anbrc 1244 . 2 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopGrp)
486, 47impbii 199 1 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  w3a 1036   = wceq 1480  wcel 1987  cmpt 4673  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  Basecbs 15781  +gcplusg 15862  TopOpenctopn 16003  0gc0g 16021  +𝑓cplusf 17160  Mndcmnd 17215  Grpcgrp 17343  invgcminusg 17344  -gcsg 17345  TopOnctopon 20618  TopSpctps 20619   Cn ccn 20938   ×t ctx 21273  TopMndctmd 21784  TopGrpctgp 21785
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804  df-0g 16023  df-topgen 16025  df-plusf 17162  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-sbg 17348  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cn 20941  df-cnp 20942  df-tx 21275  df-tmd 21786  df-tgp 21787
This theorem is referenced by:  distgp  21813  indistgp  21814  qustgplem  21834  ngptgp  22350  cnfldtgp  22580
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