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Theorem tgpsubcn 23464
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 2733 . . 3 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
2 eqid 2733 . . 3 (+gβ€˜πΊ) = (+gβ€˜πΊ)
3 eqid 2733 . . 3 (invgβ€˜πΊ) = (invgβ€˜πΊ)
4 tgpsubcn.3 . . 3 βˆ’ = (-gβ€˜πΊ)
51, 2, 3, 4grpsubfval 18802 . 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 23453 . . 3 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ TopMnd)
86, 1tgptopon 23456 . . 3 (𝐺 ∈ TopGrp β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
98, 8cnmpt1st 23042 . . 3 (𝐺 ∈ TopGrp β†’ (π‘₯ ∈ (Baseβ€˜πΊ), 𝑦 ∈ (Baseβ€˜πΊ) ↦ π‘₯) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
108, 8cnmpt2nd 23043 . . . 4 (𝐺 ∈ TopGrp β†’ (π‘₯ ∈ (Baseβ€˜πΊ), 𝑦 ∈ (Baseβ€˜πΊ) ↦ 𝑦) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
116, 3tgpinv 23459 . . . 4 (𝐺 ∈ TopGrp β†’ (invgβ€˜πΊ) ∈ (𝐽 Cn 𝐽))
128, 8, 10, 11cnmpt21f 23046 . . 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 23462 . 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 2838 1 (𝐺 ∈ TopGrp β†’ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  Basecbs 17091  +gcplusg 17141  TopOpenctopn 17311  invgcminusg 18757  -gcsg 18758   Cn ccn 22598   Γ—t ctx 22934  TopGrpctgp 23445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-map 8773  df-topgen 17333  df-plusf 18504  df-sbg 18761  df-top 22266  df-topon 22283  df-topsp 22305  df-bases 22319  df-cn 22601  df-tx 22936  df-tmd 23446  df-tgp 23447
This theorem is referenced by:  istgp2  23465  clssubg  23483  clsnsg  23484  tgphaus  23491  tgpt0  23493  qustgplem  23495
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