Theorem tmdcn 24091

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 24082	. 2 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
4	3	simp3bi 1148	1 ⊢ (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ‘cfv 6561  (class class class)co 7431  TopOpenctopn 17466  +_𝑓cplusf 18650  Mndcmnd 18747  TopSpctps 22938   Cn ccn 23232   ×_t ctx 23568  TopMndctmd 24078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-tmd 24080

This theorem is referenced by:  tgpcn  24092  cnmpt1plusg  24095  cnmpt2plusg  24096  tmdcn2  24097  submtmd  24112  tsmsadd  24155  mulrcn  24187  mhmhmeotmd  33926  xrge0pluscn  33939