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Theorem tgpsubcn 24014
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 2728 . . 3 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
2 eqid 2728 . . 3 (+gβ€˜πΊ) = (+gβ€˜πΊ)
3 eqid 2728 . . 3 (invgβ€˜πΊ) = (invgβ€˜πΊ)
4 tgpsubcn.3 . . 3 βˆ’ = (-gβ€˜πΊ)
51, 2, 3, 4grpsubfval 18947 . 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 24003 . . 3 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ TopMnd)
86, 1tgptopon 24006 . . 3 (𝐺 ∈ TopGrp β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
98, 8cnmpt1st 23592 . . 3 (𝐺 ∈ TopGrp β†’ (π‘₯ ∈ (Baseβ€˜πΊ), 𝑦 ∈ (Baseβ€˜πΊ) ↦ π‘₯) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
108, 8cnmpt2nd 23593 . . . 4 (𝐺 ∈ TopGrp β†’ (π‘₯ ∈ (Baseβ€˜πΊ), 𝑦 ∈ (Baseβ€˜πΊ) ↦ 𝑦) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
116, 3tgpinv 24009 . . . 4 (𝐺 ∈ TopGrp β†’ (invgβ€˜πΊ) ∈ (𝐽 Cn 𝐽))
128, 8, 10, 11cnmpt21f 23596 . . 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 24012 . 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 2833 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 6553  (class class class)co 7426   ∈ cmpo 7428  Basecbs 17187  +gcplusg 17240  TopOpenctopn 17410  invgcminusg 18898  -gcsg 18899   Cn ccn 23148   Γ—t ctx 23484  TopGrpctgp 23995
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fo 6559  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-map 8853  df-topgen 17432  df-plusf 18606  df-sbg 18902  df-top 22816  df-topon 22833  df-topsp 22855  df-bases 22869  df-cn 23151  df-tx 23486  df-tmd 23996  df-tgp 23997
This theorem is referenced by:  istgp2  24015  clssubg  24033  clsnsg  24034  tgphaus  24041  tgpt0  24043  qustgplem  24045
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