Theorem tgpsubcn 24464

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	⊢ − = (-g‘𝐺)

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.

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2763	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		eqid 2763	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
4		tgpsubcn.3	. . 3 ⊢ − = (-g‘𝐺)
5	1, 2, 3, 4	grpsubfval 19244	. 2 ⊢ − = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦)))
6		tgpsubcn.2	. . 3 ⊢ 𝐽 = (TopOpen‘𝐺)
7		tgptmd 24453	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
8	6, 1	tgptopon 24456	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
9	8, 8	cnmpt1st 24042	. . 3 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
10	8, 8	cnmpt2nd 24043	. . . 4 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
11	6, 3	tgpinv 24459	. . . 4 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺) ∈ (𝐽 Cn 𝐽))
12	8, 8, 10, 11	cnmpt21f 24046	. . 3 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ ((invg‘𝐺)‘𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
13	6, 2, 7, 8, 8, 9, 12	cnmpt2plusg 24462	. 2 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
14	5, 13	eqeltrid 2868	1 ⊢ (𝐺 ∈ TopGrp → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  Basecbs 17431  +_gcplusg 17472  TopOpenctopn 17642  inv_gcminusg 19195  -_gcsg 19196   Cn ccn 23597   ×_t ctx 23934  TopGrpctgp 24445

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fo 6572  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-map 8887  df-topgen 17664  df-plusf 18865  df-sbg 19199  df-top 23267  df-topon 23284  df-topsp 23306  df-bases 23319  df-cn 23600  df-tx 23936  df-tmd 24446  df-tgp 24447

This theorem is used by:  istgp2  24465  clssubg  24483  clsnsg  24484  tgphaus  24491  tgpt0  24493  qustgplem  24495