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Theorem tgpsubcn 24082
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 2726 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2726 . . 3 (+g𝐺) = (+g𝐺)
3 eqid 2726 . . 3 (invg𝐺) = (invg𝐺)
4 tgpsubcn.3 . . 3 = (-g𝐺)
51, 2, 3, 4grpsubfval 18973 . 2 = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)((invg𝐺)‘𝑦)))
6 tgpsubcn.2 . . 3 𝐽 = (TopOpen‘𝐺)
7 tgptmd 24071 . . 3 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
86, 1tgptopon 24074 . . 3 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
98, 8cnmpt1st 23660 . . 3 (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
108, 8cnmpt2nd 23661 . . . 4 (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
116, 3tgpinv 24077 . . . 4 (𝐺 ∈ TopGrp → (invg𝐺) ∈ (𝐽 Cn 𝐽))
128, 8, 10, 11cnmpt21f 23664 . . 3 (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
136, 2, 7, 8, 8, 9, 12cnmpt2plusg 24080 . 2 (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)((invg𝐺)‘𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
145, 13eqeltrid 2830 1 (𝐺 ∈ TopGrp → ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cfv 6546  (class class class)co 7416  cmpo 7418  Basecbs 17208  +gcplusg 17261  TopOpenctopn 17431  invgcminusg 18924  -gcsg 18925   Cn ccn 23216   ×t ctx 23552  TopGrpctgp 24063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-fo 6552  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7995  df-2nd 7996  df-map 8849  df-topgen 17453  df-plusf 18627  df-sbg 18928  df-top 22884  df-topon 22901  df-topsp 22923  df-bases 22937  df-cn 23219  df-tx 23554  df-tmd 24064  df-tgp 24065
This theorem is referenced by:  istgp2  24083  clssubg  24101  clsnsg  24102  tgphaus  24109  tgpt0  24111  qustgplem  24113
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