Theorem tmdtps 24404

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 2760	. . 3 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
2		eqid 2760	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
3	1, 2	istmd 24402	. 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 2144  ‘cfv 6543  (class class class)co 7424  TopOpenctopn 17601  +_𝑓cplusf 18822  Mndcmnd 18932  TopSpctps 23259   Cn ccn 23551   ×_t ctx 23888  TopMndctmd 24398

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6497  df-fv 6551  df-ov 7427  df-tmd 24400

This theorem is used by:  tgptps  24408  tmdtopon  24409  submtmd  24432  prdstmdd  24452  tsmsadd  24475  tsmssplit  24480  tlmtps  24516