Theorem tmdtps 23902

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

Step	Hyp	Ref	Expression
1		eqid 2724	. . 3 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
2		eqid 2724	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
3	1, 2	istmd 23900	. 2 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓‘𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))))
4	3	simp2bi 1143	1 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2098  ‘cfv 6533  (class class class)co 7401  TopOpenctopn 17366  +_𝑓cplusf 18560  Mndcmnd 18657  TopSpctps 22756   Cn ccn 23050   ×_t ctx 23386  TopMndctmd 23896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-nul 5296

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541  df-ov 7404  df-tmd 23898

This theorem is referenced by:  tgptps  23906  tmdtopon  23907  submtmd  23930  prdstmdd  23950  tsmsadd  23973  tsmssplit  23978  tlmtps  24014