Theorem tmdmnd 24040

Description: A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)

Assertion

Ref	Expression
tmdmnd	⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)

Proof of Theorem tmdmnd

Step	Hyp	Ref	Expression
1		eqid 2736	. . 3 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
2		eqid 2736	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
3	1, 2	istmd 24039	. 2 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓‘𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))))
4	3	simp1bi 1146	1 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)

Syntax hints:   → wi 4   ∈ wcel 2114  ‘cfv 6498  (class class class)co 7367  TopOpenctopn 17384  +_𝑓cplusf 18605  Mndcmnd 18702  TopSpctps 22897   Cn ccn 23189   ×_t ctx 23525  TopMndctmd 24035

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-tmd 24037

This theorem is referenced by:  tmdmulg  24057  tmdgsum  24060  oppgtmd  24062  prdstmdd  24089  tsmsxp  24120  xrge0iifmhm  34083  esumcst  34207