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

Theorem tmdcn 21868
 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 21859 . 2 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
43simp3bi 1076 1 (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1481   ∈ wcel 1988  ‘cfv 5876  (class class class)co 6635  TopOpenctopn 16063  +𝑓cplusf 17220  Mndcmnd 17275  TopSpctps 20717   Cn ccn 21009   ×t ctx 21344  TopMndctmd 21855 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-nul 4780 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ov 6638  df-tmd 21857 This theorem is referenced by:  tgpcn  21869  cnmpt1plusg  21872  cnmpt2plusg  21873  tmdcn2  21874  submtmd  21889  tsmsadd  21931  mulrcn  21963  mhmhmeotmd  29947  xrge0pluscn  29960
 Copyright terms: Public domain W3C validator