Theorem tgpsubcn 22692

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 2821	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2821	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		eqid 2821	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
4		tgpsubcn.3	. . 3 ⊢ − = (-g‘𝐺)
5	1, 2, 3, 4	grpsubfval 18141	. 2 ⊢ − = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦)))
6		tgpsubcn.2	. . 3 ⊢ 𝐽 = (TopOpen‘𝐺)
7		tgptmd 22681	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
8	6, 1	tgptopon 22684	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
9	8, 8	cnmpt1st 22270	. . 3 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
10	8, 8	cnmpt2nd 22271	. . . 4 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
11	6, 3	tgpinv 22687	. . . 4 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺) ∈ (𝐽 Cn 𝐽))
12	8, 8, 10, 11	cnmpt21f 22274	. . 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 22690	. 2 ⊢ (𝐺 ∈ TopGrp → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
14	5, 13	eqeltrid 2917	1 ⊢ (𝐺 ∈ TopGrp → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  ‘cfv 6350  (class class class)co 7150   ∈ cmpo 7152  Basecbs 16477  +_gcplusg 16559  TopOpenctopn 16689  inv_gcminusg 18098  -_gcsg 18099   Cn ccn 21826   ×_t ctx 22162  TopGrpctgp 22673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fo 6356  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-map 8402  df-topgen 16711  df-plusf 17845  df-sbg 18102  df-top 21496  df-topon 21513  df-topsp 21535  df-bases 21548  df-cn 21829  df-tx 22164  df-tmd 22674  df-tgp 22675

This theorem is referenced by:  istgp2  22693  clssubg  22711  clsnsg  22712  tgphaus  22719  tgpt0  22721  qustgplem  22723