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Theorem tmdlactcn 21716
 Description: The left group action of element 𝐴 in a topological monoid 𝐺 is a continuous function. (Contributed by FL, 18-Mar-2008.) (Revised by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
tgplacthmeo.1 𝐹 = (𝑥𝑋 ↦ (𝐴 + 𝑥))
tgplacthmeo.2 𝑋 = (Base‘𝐺)
tgplacthmeo.3 + = (+g𝐺)
tgplacthmeo.4 𝐽 = (TopOpen‘𝐺)
Assertion
Ref Expression
tmdlactcn ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐺   𝑥,𝐽   𝑥, +   𝑥,𝑋
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem tmdlactcn
StepHypRef Expression
1 tgplacthmeo.1 . 2 𝐹 = (𝑥𝑋 ↦ (𝐴 + 𝑥))
2 tgplacthmeo.4 . . 3 𝐽 = (TopOpen‘𝐺)
3 tgplacthmeo.3 . . 3 + = (+g𝐺)
4 simpl 472 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐺 ∈ TopMnd)
5 tgplacthmeo.2 . . . . 5 𝑋 = (Base‘𝐺)
62, 5tmdtopon 21695 . . . 4 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))
76adantr 480 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
8 simpr 476 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐴𝑋)
97, 7, 8cnmptc 21275 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐽))
107cnmptid 21274 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
112, 3, 4, 7, 9, 10cnmpt1plusg 21701 . 2 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (𝐴 + 𝑥)) ∈ (𝐽 Cn 𝐽))
121, 11syl5eqel 2692 1 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  TopOpenctopn 15905  TopOnctopon 20518   Cn ccn 20838  TopMndctmd 21684 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-topgen 15927  df-plusf 17064  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cn 20841  df-cnp 20842  df-tx 21175  df-tmd 21686 This theorem is referenced by:  tgplacthmeo  21717  ghmcnp  21728
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