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Theorem tmdtps 22612
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 2818 . . 3 (+𝑓𝐺) = (+𝑓𝐺)
2 eqid 2818 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
31, 2istmd 22610 . 2 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))))
43simp2bi 1138 1 (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  cfv 6348  (class class class)co 7145  TopOpenctopn 16683  +𝑓cplusf 17837  Mndcmnd 17899  TopSpctps 21468   Cn ccn 21760   ×t ctx 22096  TopMndctmd 22606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-tmd 22608
This theorem is referenced by:  tgptps  22616  tmdtopon  22617  submtmd  22640  prdstmdd  22659  tsmsadd  22682  tsmssplit  22687  tlmtps  22723
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