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Theorem tmdlactcn 24126
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 482 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐺 ∈ TopMnd)
5 tgplacthmeo.2 . . . . 5 𝑋 = (Base‘𝐺)
62, 5tmdtopon 24105 . . . 4 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))
76adantr 480 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
8 simpr 484 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐴𝑋)
97, 7, 8cnmptc 23686 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐽))
107cnmptid 23685 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
112, 3, 4, 7, 9, 10cnmpt1plusg 24111 . 2 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (𝐴 + 𝑥)) ∈ (𝐽 Cn 𝐽))
121, 11eqeltrid 2843 1 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cmpt 5231  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  TopOpenctopn 17468  TopOnctopon 22932   Cn ccn 23248  TopMndctmd 24094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-topgen 17490  df-plusf 18665  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cn 23251  df-cnp 23252  df-tx 23586  df-tmd 24096
This theorem is referenced by:  tgplacthmeo  24127  ghmcnp  24139
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