Theorem tmdcn 23215

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 23206	. 2 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
4	3	simp3bi 1145	1 ⊢ (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2109  ‘cfv 6430  (class class class)co 7268  TopOpenctopn 17113  +_𝑓cplusf 18304  Mndcmnd 18366  TopSpctps 22062   Cn ccn 22356   ×_t ctx 22692  TopMndctmd 23202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-nul 5233

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-iota 6388  df-fv 6438  df-ov 7271  df-tmd 23204

This theorem is referenced by:  tgpcn  23216  cnmpt1plusg  23219  cnmpt2plusg  23220  tmdcn2  23221  submtmd  23236  tsmsadd  23279  mulrcn  23311  mhmhmeotmd  31856  xrge0pluscn  31869