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Theorem tmdlactcn 22325
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 476 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐺 ∈ TopMnd)
5 tgplacthmeo.2 . . . . 5 𝑋 = (Base‘𝐺)
62, 5tmdtopon 22304 . . . 4 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))
76adantr 474 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
8 simpr 479 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐴𝑋)
97, 7, 8cnmptc 21885 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐽))
107cnmptid 21884 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
112, 3, 4, 7, 9, 10cnmpt1plusg 22310 . 2 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (𝐴 + 𝑥)) ∈ (𝐽 Cn 𝐽))
121, 11syl5eqel 2863 1 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  cmpt 4967  cfv 6137  (class class class)co 6924  Basecbs 16266  +gcplusg 16349  TopOpenctopn 16479  TopOnctopon 21133   Cn ccn 21447  TopMndctmd 22293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-map 8144  df-topgen 16501  df-plusf 17638  df-top 21117  df-topon 21134  df-topsp 21156  df-bases 21169  df-cn 21450  df-cnp 21451  df-tx 21785  df-tmd 22295
This theorem is referenced by:  tgplacthmeo  22326  ghmcnp  22337
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