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Theorem tgpsubcn 21652
Description: In a topological group, the "subtraction" (or "division") is continuous. Axiom GT' of [BourbakiTop1] p. III.1. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 19-Mar-2015.)
Hypotheses
Ref Expression
tgpsubcn.2 𝐽 = (TopOpen‘𝐺)
tgpsubcn.3 = (-g𝐺)
Assertion
Ref Expression
tgpsubcn (𝐺 ∈ TopGrp → ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Proof of Theorem tgpsubcn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2610 . . 3 (+g𝐺) = (+g𝐺)
3 eqid 2610 . . 3 (invg𝐺) = (invg𝐺)
4 tgpsubcn.3 . . 3 = (-g𝐺)
51, 2, 3, 4grpsubfval 17236 . 2 = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)((invg𝐺)‘𝑦)))
6 tgpsubcn.2 . . 3 𝐽 = (TopOpen‘𝐺)
7 tgptmd 21641 . . 3 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
86, 1tgptopon 21644 . . 3 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
98, 8cnmpt1st 21229 . . 3 (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
108, 8cnmpt2nd 21230 . . . 4 (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
116, 3tgpinv 21647 . . . 4 (𝐺 ∈ TopGrp → (invg𝐺) ∈ (𝐽 Cn 𝐽))
128, 8, 10, 11cnmpt21f 21233 . . 3 (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
136, 2, 7, 8, 8, 9, 12cnmpt2plusg 21650 . 2 (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)((invg𝐺)‘𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
145, 13syl5eqel 2692 1 (𝐺 ∈ TopGrp → ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  cfv 5790  (class class class)co 6527  cmpt2 6529  Basecbs 15644  +gcplusg 15717  TopOpenctopn 15854  invgcminusg 17195  -gcsg 17196   Cn ccn 20786   ×t ctx 21121  TopGrpctgp 21633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7037  df-2nd 7038  df-map 7724  df-topgen 15876  df-plusf 17013  df-sbg 17199  df-top 20469  df-bases 20470  df-topon 20471  df-topsp 20472  df-cn 20789  df-tx 21123  df-tmd 21634  df-tgp 21635
This theorem is referenced by:  istgp2  21653  clssubg  21670  clsnsg  21671  tgphaus  21678  tgpt0  21680  qustgplem  21682
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