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Theorem tgpsubcn 23944
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 = (TopOpenβ€˜πΊ)
tgpsubcn.3 βˆ’ = (-gβ€˜πΊ)
Assertion
Ref Expression
tgpsubcn (𝐺 ∈ TopGrp β†’ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))

Proof of Theorem tgpsubcn
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2726 . . 3 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
2 eqid 2726 . . 3 (+gβ€˜πΊ) = (+gβ€˜πΊ)
3 eqid 2726 . . 3 (invgβ€˜πΊ) = (invgβ€˜πΊ)
4 tgpsubcn.3 . . 3 βˆ’ = (-gβ€˜πΊ)
51, 2, 3, 4grpsubfval 18910 . 2 βˆ’ = (π‘₯ ∈ (Baseβ€˜πΊ), 𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯(+gβ€˜πΊ)((invgβ€˜πΊ)β€˜π‘¦)))
6 tgpsubcn.2 . . 3 𝐽 = (TopOpenβ€˜πΊ)
7 tgptmd 23933 . . 3 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ TopMnd)
86, 1tgptopon 23936 . . 3 (𝐺 ∈ TopGrp β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
98, 8cnmpt1st 23522 . . 3 (𝐺 ∈ TopGrp β†’ (π‘₯ ∈ (Baseβ€˜πΊ), 𝑦 ∈ (Baseβ€˜πΊ) ↦ π‘₯) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
108, 8cnmpt2nd 23523 . . . 4 (𝐺 ∈ TopGrp β†’ (π‘₯ ∈ (Baseβ€˜πΊ), 𝑦 ∈ (Baseβ€˜πΊ) ↦ 𝑦) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
116, 3tgpinv 23939 . . . 4 (𝐺 ∈ TopGrp β†’ (invgβ€˜πΊ) ∈ (𝐽 Cn 𝐽))
128, 8, 10, 11cnmpt21f 23526 . . 3 (𝐺 ∈ TopGrp β†’ (π‘₯ ∈ (Baseβ€˜πΊ), 𝑦 ∈ (Baseβ€˜πΊ) ↦ ((invgβ€˜πΊ)β€˜π‘¦)) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
136, 2, 7, 8, 8, 9, 12cnmpt2plusg 23942 . 2 (𝐺 ∈ TopGrp β†’ (π‘₯ ∈ (Baseβ€˜πΊ), 𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯(+gβ€˜πΊ)((invgβ€˜πΊ)β€˜π‘¦))) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
145, 13eqeltrid 2831 1 (𝐺 ∈ TopGrp β†’ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  β€˜cfv 6536  (class class class)co 7404   ∈ cmpo 7406  Basecbs 17150  +gcplusg 17203  TopOpenctopn 17373  invgcminusg 18861  -gcsg 18862   Cn ccn 23078   Γ—t ctx 23414  TopGrpctgp 23925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-map 8821  df-topgen 17395  df-plusf 18569  df-sbg 18865  df-top 22746  df-topon 22763  df-topsp 22785  df-bases 22799  df-cn 23081  df-tx 23416  df-tmd 23926  df-tgp 23927
This theorem is referenced by:  istgp2  23945  clssubg  23963  clsnsg  23964  tgphaus  23971  tgpt0  23973  qustgplem  23975
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