Theorem tmdcn 23332

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 23323	. 2 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
4	3	simp3bi 1146	1 ⊢ (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  ‘cfv 6473  (class class class)co 7329  TopOpenctopn 17221  +_𝑓cplusf 18412  Mndcmnd 18474  TopSpctps 22179   Cn ccn 22473   ×_t ctx 22809  TopMndctmd 23319

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-nul 5247

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-rab 3404  df-v 3443  df-sbc 3727  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-iota 6425  df-fv 6481  df-ov 7332  df-tmd 23321

This theorem is referenced by:  tgpcn  23333  cnmpt1plusg  23336  cnmpt2plusg  23337  tmdcn2  23338  submtmd  23353  tsmsadd  23396  mulrcn  23428  mhmhmeotmd  32116  xrge0pluscn  32129