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Theorem tmdmnd 21689
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 2610 . . 3 (+𝑓𝐺) = (+𝑓𝐺)
2 eqid 2610 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
31, 2istmd 21688 . 2 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))))
43simp1bi 1069 1 (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1977  cfv 5804  (class class class)co 6549  TopOpenctopn 15905  +𝑓cplusf 17062  Mndcmnd 17117  TopSpctps 20519   Cn ccn 20838   ×t ctx 21173  TopMndctmd 21684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-tmd 21686
This theorem is referenced by:  tmdmulg  21706  tmdgsum  21709  oppgtmd  21711  prdstmdd  21737  tsmsxp  21768  xrge0iifmhm  29313  esumcst  29452
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