Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tmdtps Structured version   Visualization version   GIF version

Theorem tmdtps 22679
 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
StepHypRef Expression
1 eqid 2822 . . 3 (+𝑓𝐺) = (+𝑓𝐺)
2 eqid 2822 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
31, 2istmd 22677 . 2 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))))
43simp2bi 1143 1 (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2114  ‘cfv 6334  (class class class)co 7140  TopOpenctopn 16686  +𝑓cplusf 17840  Mndcmnd 17902  TopSpctps 21535   Cn ccn 21827   ×t ctx 22163  TopMndctmd 22673 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-nul 5186 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342  df-ov 7143  df-tmd 22675 This theorem is referenced by:  tgptps  22683  tmdtopon  22684  submtmd  22707  prdstmdd  22727  tsmsadd  22750  tsmssplit  22755  tlmtps  22791
 Copyright terms: Public domain W3C validator