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Theorem tmdmnd 21860
Description: A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
Assertion
Ref Expression
tmdmnd (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)

Proof of Theorem tmdmnd
StepHypRef Expression
1 eqid 2620 . . 3 (+𝑓𝐺) = (+𝑓𝐺)
2 eqid 2620 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
31, 2istmd 21859 . 2 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))))
43simp1bi 1074 1 (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  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:  tmdmulg  21877  tmdgsum  21880  oppgtmd  21882  prdstmdd  21908  tsmsxp  21939  xrge0iifmhm  29959  esumcst  30099
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