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Theorem tmdtps 23379
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 2737 . . 3 (+𝑓𝐺) = (+𝑓𝐺)
2 eqid 2737 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
31, 2istmd 23377 . 2 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))))
43simp2bi 1146 1 (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cfv 6493  (class class class)co 7351  TopOpenctopn 17263  +𝑓cplusf 18454  Mndcmnd 18516  TopSpctps 22233   Cn ccn 22527   ×t ctx 22863  TopMndctmd 23373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-nul 5261
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-iota 6445  df-fv 6501  df-ov 7354  df-tmd 23375
This theorem is referenced by:  tgptps  23383  tmdtopon  23384  submtmd  23407  prdstmdd  23427  tsmsadd  23450  tsmssplit  23455  tlmtps  23491
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