Theorem tmdcn 24025

Description: In a topological monoid, the operation 𝐹 representing the functionalization of the operator slot +_g is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)

Hypotheses

Ref	Expression
tgpcn.j	⊢ 𝐽 = (TopOpen‘𝐺)
tgpcn.1	⊢ 𝐹 = (+𝑓‘𝐺)

Assertion

Ref	Expression
tmdcn	⊢ (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Proof of Theorem tmdcn

Step	Hyp	Ref	Expression
1		tgpcn.1	. . 3 ⊢ 𝐹 = (+𝑓‘𝐺)
2		tgpcn.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝐺)
3	1, 2	istmd 24016	. 2 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
4	3	simp3bi 1147	1 ⊢ (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ‘cfv 6490  (class class class)co 7356  TopOpenctopn 17339  +_𝑓cplusf 18560  Mndcmnd 18657  TopSpctps 22874   Cn ccn 23166   ×_t ctx 23502  TopMndctmd 24012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-nul 5249

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-iota 6446  df-fv 6498  df-ov 7359  df-tmd 24014

This theorem is referenced by:  tgpcn  24026  cnmpt1plusg  24029  cnmpt2plusg  24030  tmdcn2  24031  submtmd  24046  tsmsadd  24089  mulrcn  24121  mhmhmeotmd  34033  xrge0pluscn  34046