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Theorem tmdtps 21790
Description: A topological monoid is a topological space. (Contributed by Mario Carneiro, 19-Sep-2015.)
Assertion
Ref Expression
tmdtps (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)

Proof of Theorem tmdtps
StepHypRef Expression
1 eqid 2621 . . 3 (+𝑓𝐺) = (+𝑓𝐺)
2 eqid 2621 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
31, 2istmd 21788 . 2 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))))
43simp2bi 1075 1 (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  cfv 5847  (class class class)co 6604  TopOpenctopn 16003  +𝑓cplusf 17160  Mndcmnd 17215  TopSpctps 20619   Cn ccn 20938   ×t ctx 21273  TopMndctmd 21784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607  df-tmd 21786
This theorem is referenced by:  tgptps  21794  tmdtopon  21795  submtmd  21818  prdstmdd  21837  tsmsadd  21860  tsmssplit  21865  tlmtps  21901
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