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Theorem tmdlactcn 24111
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 24090 . . . 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 23671 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐽))
107cnmptid 23670 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
112, 3, 4, 7, 9, 10cnmpt1plusg 24096 . 2 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (𝐴 + 𝑥)) ∈ (𝐽 Cn 𝐽))
121, 11eqeltrid 2844 1 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  cmpt 5224  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  TopOpenctopn 17467  TopOnctopon 22917   Cn ccn 23233  TopMndctmd 24079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-map 8869  df-topgen 17489  df-plusf 18653  df-top 22901  df-topon 22918  df-topsp 22940  df-bases 22954  df-cn 23236  df-cnp 23237  df-tx 23571  df-tmd 24081
This theorem is referenced by:  tgplacthmeo  24112  ghmcnp  24124
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