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Theorem tmdcn 23578
Description: In a topological monoid, the operation 𝐹 representing the functionalization of the operator slot +g is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
Hypotheses
Ref Expression
tgpcn.j 𝐽 = (TopOpenβ€˜πΊ)
tgpcn.1 𝐹 = (+π‘“β€˜πΊ)
Assertion
Ref Expression
tmdcn (𝐺 ∈ TopMnd β†’ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))

Proof of Theorem tmdcn
StepHypRef Expression
1 tgpcn.1 . . 3 𝐹 = (+π‘“β€˜πΊ)
2 tgpcn.j . . 3 𝐽 = (TopOpenβ€˜πΊ)
31, 2istmd 23569 . 2 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
43simp3bi 1147 1 (𝐺 ∈ TopMnd β†’ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  β€˜cfv 6540  (class class class)co 7405  TopOpenctopn 17363  +𝑓cplusf 18554  Mndcmnd 18621  TopSpctps 22425   Cn ccn 22719   Γ—t ctx 23055  TopMndctmd 23565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-ov 7408  df-tmd 23567
This theorem is referenced by:  tgpcn  23579  cnmpt1plusg  23582  cnmpt2plusg  23583  tmdcn2  23584  submtmd  23599  tsmsadd  23642  mulrcn  23674  mhmhmeotmd  32895  xrge0pluscn  32908
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