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Theorem istgp2 24115
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 24102 . . 3 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2 tgptps 24104 . . 3 (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
3 tgpsubcn.2 . . . 4 𝐽 = (TopOpen‘𝐺)
4 tgpsubcn.3 . . . 4 = (-g𝐺)
53, 4tgpsubcn 24114 . . 3 (𝐺 ∈ TopGrp → ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
61, 2, 53jca 1127 . 2 (𝐺 ∈ TopGrp → (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
7 simp1 1135 . . 3 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ Grp)
8 grpmnd 18971 . . . . 5 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
983ad2ant1 1132 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ Mnd)
10 simp2 1136 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopSp)
11 eqid 2735 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
12 eqid 2735 . . . . . . . 8 (+g𝐺) = (+g𝐺)
13 eqid 2735 . . . . . . . 8 (invg𝐺) = (invg𝐺)
1473ad2ant1 1132 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → 𝐺 ∈ Grp)
15 simp2 1136 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
16 simp3 1137 . . . . . . . 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 19043 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥 ((invg𝐺)‘𝑦)) = (𝑥(+g𝐺)𝑦))
1817mpoeq3dva 7510 . . . . . 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 2735 . . . . . . 7 (+𝑓𝐺) = (+𝑓𝐺)
2011, 12, 19plusffval 18672 . . . . . 6 (+𝑓𝐺) = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦))
2118, 20eqtr4di 2793 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥 ((invg𝐺)‘𝑦))) = (+𝑓𝐺))
2211, 3istps 22956 . . . . . . 7 (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝐺)))
2310, 22sylib 218 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
2423, 23cnmpt1st 23692 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
2523, 23cnmpt2nd 23693 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
2611, 13grpinvf 19017 . . . . . . . . . . 11 (𝐺 ∈ Grp → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
27263ad2ant1 1132 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
2827feqmptd 6977 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)))
29 eqid 2735 . . . . . . . . . . . 12 (0g𝐺) = (0g𝐺)
3011, 4, 13, 29grpinvval2 19054 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) = ((0g𝐺) 𝑥))
317, 30sylan 580 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) = ((0g𝐺) 𝑥))
3231mpteq2dva 5248 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)) = (𝑥 ∈ (Base‘𝐺) ↦ ((0g𝐺) 𝑥)))
3328, 32eqtrd 2775 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((0g𝐺) 𝑥)))
3411, 29grpidcl 18996 . . . . . . . . . . 11 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
35343ad2ant1 1132 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (0g𝐺) ∈ (Base‘𝐺))
3623, 23, 35cnmptc 23686 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ (0g𝐺)) ∈ (𝐽 Cn 𝐽))
3723cnmptid 23685 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
38 simp3 1137 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
3923, 36, 37, 38cnmpt12f 23690 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ ((0g𝐺) 𝑥)) ∈ (𝐽 Cn 𝐽))
4033, 39eqeltrd 2839 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg𝐺) ∈ (𝐽 Cn 𝐽))
4123, 23, 25, 40cnmpt21f 23696 . . . . . 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 23699 . . . . 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 2840 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
4419, 3istmd 24098 . . . 4 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
459, 10, 43, 44syl3anbrc 1342 . . 3 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopMnd)
463, 13istgp 24101 . . 3 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg𝐺) ∈ (𝐽 Cn 𝐽)))
477, 45, 40, 46syl3anbrc 1342 . 2 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopGrp)
486, 47impbii 209 1 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  w3a 1086   = wceq 1537  wcel 2106  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  Basecbs 17245  +gcplusg 17298  TopOpenctopn 17468  0gc0g 17486  +𝑓cplusf 18663  Mndcmnd 18760  Grpcgrp 18964  invgcminusg 18965  -gcsg 18966  TopOnctopon 22932  TopSpctps 22954   Cn ccn 23248   ×t ctx 23584  TopMndctmd 24094  TopGrpctgp 24095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-0g 17488  df-topgen 17490  df-plusf 18665  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cn 23251  df-cnp 23252  df-tx 23586  df-tmd 24096  df-tgp 24097
This theorem is referenced by:  distgp  24123  indistgp  24124  qustgplem  24145  ngptgp  24665  cnfldtgp  24907
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