Theorem tmdcn 23279

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 23270	. 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 1539   ∈ wcel 2104  ‘cfv 6458  (class class class)co 7307  TopOpenctopn 17177  +_𝑓cplusf 18368  Mndcmnd 18430  TopSpctps 22126   Cn ccn 22420   ×_t ctx 22756  TopMndctmd 23266

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-nul 5239

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-rab 3287  df-v 3439  df-sbc 3722  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-iota 6410  df-fv 6466  df-ov 7310  df-tmd 23268

This theorem is referenced by:  tgpcn  23280  cnmpt1plusg  23283  cnmpt2plusg  23284  tmdcn2  23285  submtmd  23300  tsmsadd  23343  mulrcn  23375  mhmhmeotmd  31922  xrge0pluscn  31935