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Theorem tgpsubcn 21941
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 2651 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2651 . . 3 (+g𝐺) = (+g𝐺)
3 eqid 2651 . . 3 (invg𝐺) = (invg𝐺)
4 tgpsubcn.3 . . 3 = (-g𝐺)
51, 2, 3, 4grpsubfval 17511 . 2 = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)((invg𝐺)‘𝑦)))
6 tgpsubcn.2 . . 3 𝐽 = (TopOpen‘𝐺)
7 tgptmd 21930 . . 3 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
86, 1tgptopon 21933 . . 3 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
98, 8cnmpt1st 21519 . . 3 (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
108, 8cnmpt2nd 21520 . . . 4 (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
116, 3tgpinv 21936 . . . 4 (𝐺 ∈ TopGrp → (invg𝐺) ∈ (𝐽 Cn 𝐽))
128, 8, 10, 11cnmpt21f 21523 . . 3 (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
136, 2, 7, 8, 8, 9, 12cnmpt2plusg 21939 . 2 (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)((invg𝐺)‘𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
145, 13syl5eqel 2734 1 (𝐺 ∈ TopGrp → ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  cfv 5926  (class class class)co 6690  cmpt2 6692  Basecbs 15904  +gcplusg 15988  TopOpenctopn 16129  invgcminusg 17470  -gcsg 17471   Cn ccn 21076   ×t ctx 21411  TopGrpctgp 21922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-topgen 16151  df-plusf 17288  df-sbg 17474  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cn 21079  df-tx 21413  df-tmd 21923  df-tgp 21924
This theorem is referenced by:  istgp2  21942  clssubg  21959  clsnsg  21960  tgphaus  21967  tgpt0  21969  qustgplem  21971
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