Theorem tmdtps 24020

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 2736	. . 3 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
2		eqid 2736	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
3	1, 2	istmd 24018	. 2 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓‘𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))))
4	3	simp2bi 1146	1 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2113  ‘cfv 6492  (class class class)co 7358  TopOpenctopn 17341  +_𝑓cplusf 18562  Mndcmnd 18659  TopSpctps 22876   Cn ccn 23168   ×_t ctx 23504  TopMndctmd 24014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-nul 5251

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361  df-tmd 24016

This theorem is referenced by:  tgptps  24024  tmdtopon  24025  submtmd  24048  prdstmdd  24068  tsmsadd  24091  tsmssplit  24096  tlmtps  24132