MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tmdcn Structured version   Visualization version   GIF version

Theorem tmdcn 22685
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 = (TopOpen‘𝐺)
tgpcn.1 𝐹 = (+𝑓𝐺)
Assertion
Ref Expression
tmdcn (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Proof of Theorem tmdcn
StepHypRef Expression
1 tgpcn.1 . . 3 𝐹 = (+𝑓𝐺)
2 tgpcn.j . . 3 𝐽 = (TopOpen‘𝐺)
31, 2istmd 22676 . 2 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
43simp3bi 1143 1 (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2110  cfv 6349  (class class class)co 7150  TopOpenctopn 16689  +𝑓cplusf 17843  Mndcmnd 17905  TopSpctps 21534   Cn ccn 21826   ×t ctx 22162  TopMndctmd 22672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5202
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-iota 6308  df-fv 6357  df-ov 7153  df-tmd 22674
This theorem is referenced by:  tgpcn  22686  cnmpt1plusg  22689  cnmpt2plusg  22690  tmdcn2  22691  submtmd  22706  tsmsadd  22749  mulrcn  22781  mhmhmeotmd  31165  xrge0pluscn  31178
  Copyright terms: Public domain W3C validator