Theorem tmdtps 24136

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 2762	. . 3 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
2		eqid 2762	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
3	1, 2	istmd 24134	. 2 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓‘𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))))
4	3	simp2bi 1159	1 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2142  ‘cfv 6521  (class class class)co 7396  TopOpenctopn 17450  +_𝑓cplusf 18671  Mndcmnd 18768  TopSpctps 22992   Cn ccn 23284   ×_t ctx 23620  TopMndctmd 24130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-nul 5256

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3456  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6477  df-fv 6529  df-ov 7399  df-tmd 24132

This theorem is referenced by:  tgptps  24140  tmdtopon  24141  submtmd  24164  prdstmdd  24184  tsmsadd  24207  tsmssplit  24212  tlmtps  24248