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Theorem tmdcn 23807
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 23798 . 2 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
43simp3bi 1145 1 (𝐺 ∈ TopMnd β†’ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1539   ∈ wcel 2104  β€˜cfv 6542  (class class class)co 7411  TopOpenctopn 17371  +𝑓cplusf 18562  Mndcmnd 18659  TopSpctps 22654   Cn ccn 22948   Γ—t ctx 23284  TopMndctmd 23794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-rab 3431  df-v 3474  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 6494  df-fv 6550  df-ov 7414  df-tmd 23796
This theorem is referenced by:  tgpcn  23808  cnmpt1plusg  23811  cnmpt2plusg  23812  tmdcn2  23813  submtmd  23828  tsmsadd  23871  mulrcn  23903  mhmhmeotmd  33205  xrge0pluscn  33218
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