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Theorem tgpsubcn 22114
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 2771 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2771 . . 3 (+g𝐺) = (+g𝐺)
3 eqid 2771 . . 3 (invg𝐺) = (invg𝐺)
4 tgpsubcn.3 . . 3 = (-g𝐺)
51, 2, 3, 4grpsubfval 17672 . 2 = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)((invg𝐺)‘𝑦)))
6 tgpsubcn.2 . . 3 𝐽 = (TopOpen‘𝐺)
7 tgptmd 22103 . . 3 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
86, 1tgptopon 22106 . . 3 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
98, 8cnmpt1st 21692 . . 3 (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
108, 8cnmpt2nd 21693 . . . 4 (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
116, 3tgpinv 22109 . . . 4 (𝐺 ∈ TopGrp → (invg𝐺) ∈ (𝐽 Cn 𝐽))
128, 8, 10, 11cnmpt21f 21696 . . 3 (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
136, 2, 7, 8, 8, 9, 12cnmpt2plusg 22112 . 2 (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)((invg𝐺)‘𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
145, 13syl5eqel 2854 1 (𝐺 ∈ TopGrp → ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  cfv 6031  (class class class)co 6793  cmpt2 6795  Basecbs 16064  +gcplusg 16149  TopOpenctopn 16290  invgcminusg 17631  -gcsg 17632   Cn ccn 21249   ×t ctx 21584  TopGrpctgp 22095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-map 8011  df-topgen 16312  df-plusf 17449  df-sbg 17635  df-top 20919  df-topon 20936  df-topsp 20958  df-bases 20971  df-cn 21252  df-tx 21586  df-tmd 22096  df-tgp 22097
This theorem is referenced by:  istgp2  22115  clssubg  22132  clsnsg  22133  tgphaus  22140  tgpt0  22142  qustgplem  22144
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