Theorem tmdtps 24395

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 2761	. . 3 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
2		eqid 2761	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
3	1, 2	istmd 24393	. 2 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓‘𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))))
4	3	simp2bi 1164	1 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ‘cfv 6538  (class class class)co 7420  TopOpenctopn 17592  +_𝑓cplusf 18813  Mndcmnd 18923  TopSpctps 23250   Cn ccn 23542   ×_t ctx 23879  TopMndctmd 24389

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6494  df-fv 6546  df-ov 7423  df-tmd 24391

This theorem is used by:  tgptps  24399  tmdtopon  24400  submtmd  24423  prdstmdd  24443  tsmsadd  24466  tsmssplit  24471  tlmtps  24507