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Theorem istgp2 22303
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 22290 . . 3 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2 tgptps 22292 . . 3 (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
3 tgpsubcn.2 . . . 4 𝐽 = (TopOpen‘𝐺)
4 tgpsubcn.3 . . . 4 = (-g𝐺)
53, 4tgpsubcn 22302 . . 3 (𝐺 ∈ TopGrp → ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
61, 2, 53jca 1119 . 2 (𝐺 ∈ TopGrp → (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
7 simp1 1127 . . 3 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ Grp)
8 grpmnd 17816 . . . . 5 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
983ad2ant1 1124 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ Mnd)
10 simp2 1128 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopSp)
11 eqid 2778 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
12 eqid 2778 . . . . . . . 8 (+g𝐺) = (+g𝐺)
13 eqid 2778 . . . . . . . 8 (invg𝐺) = (invg𝐺)
1473ad2ant1 1124 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → 𝐺 ∈ Grp)
15 simp2 1128 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
16 simp3 1129 . . . . . . . 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 17875 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥 ((invg𝐺)‘𝑦)) = (𝑥(+g𝐺)𝑦))
1817mpt2eq3dva 6996 . . . . . 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 2778 . . . . . . 7 (+𝑓𝐺) = (+𝑓𝐺)
2011, 12, 19plusffval 17633 . . . . . 6 (+𝑓𝐺) = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦))
2118, 20syl6eqr 2832 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥 ((invg𝐺)‘𝑦))) = (+𝑓𝐺))
2211, 3istps 21146 . . . . . . 7 (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝐺)))
2310, 22sylib 210 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
2423, 23cnmpt1st 21880 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
2523, 23cnmpt2nd 21881 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
2611, 13grpinvf 17853 . . . . . . . . . . 11 (𝐺 ∈ Grp → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
27263ad2ant1 1124 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
2827feqmptd 6509 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)))
29 eqid 2778 . . . . . . . . . . . 12 (0g𝐺) = (0g𝐺)
3011, 4, 13, 29grpinvval2 17885 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) = ((0g𝐺) 𝑥))
317, 30sylan 575 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) = ((0g𝐺) 𝑥))
3231mpteq2dva 4979 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)) = (𝑥 ∈ (Base‘𝐺) ↦ ((0g𝐺) 𝑥)))
3328, 32eqtrd 2814 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((0g𝐺) 𝑥)))
3411, 29grpidcl 17837 . . . . . . . . . . 11 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
35343ad2ant1 1124 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (0g𝐺) ∈ (Base‘𝐺))
3623, 23, 35cnmptc 21874 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ (0g𝐺)) ∈ (𝐽 Cn 𝐽))
3723cnmptid 21873 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
38 simp3 1129 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
3923, 36, 37, 38cnmpt12f 21878 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ ((0g𝐺) 𝑥)) ∈ (𝐽 Cn 𝐽))
4033, 39eqeltrd 2859 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg𝐺) ∈ (𝐽 Cn 𝐽))
4123, 23, 25, 40cnmpt21f 21884 . . . . . 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 21887 . . . . 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 2860 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
4419, 3istmd 22286 . . . 4 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
459, 10, 43, 44syl3anbrc 1400 . . 3 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopMnd)
463, 13istgp 22289 . . 3 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg𝐺) ∈ (𝐽 Cn 𝐽)))
477, 45, 40, 46syl3anbrc 1400 . 2 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopGrp)
486, 47impbii 201 1 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wb 198  w3a 1071   = wceq 1601  wcel 2107  cmpt 4965  wf 6131  cfv 6135  (class class class)co 6922  cmpt2 6924  Basecbs 16255  +gcplusg 16338  TopOpenctopn 16468  0gc0g 16486  +𝑓cplusf 17625  Mndcmnd 17680  Grpcgrp 17809  invgcminusg 17810  -gcsg 17811  TopOnctopon 21122  TopSpctps 21144   Cn ccn 21436   ×t ctx 21772  TopMndctmd 22282  TopGrpctgp 22283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-map 8142  df-0g 16488  df-topgen 16490  df-plusf 17627  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-grp 17812  df-minusg 17813  df-sbg 17814  df-top 21106  df-topon 21123  df-topsp 21145  df-bases 21158  df-cn 21439  df-cnp 21440  df-tx 21774  df-tmd 22284  df-tgp 22285
This theorem is referenced by:  distgp  22311  indistgp  22312  qustgplem  22332  ngptgp  22848  cnfldtgp  23080
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