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Theorem tmdlactcn 22705
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 486 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐺 ∈ TopMnd)
5 tgplacthmeo.2 . . . . 5 𝑋 = (Base‘𝐺)
62, 5tmdtopon 22684 . . . 4 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))
76adantr 484 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
8 simpr 488 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐴𝑋)
97, 7, 8cnmptc 22265 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐽))
107cnmptid 22264 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
112, 3, 4, 7, 9, 10cnmpt1plusg 22690 . 2 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (𝐴 + 𝑥)) ∈ (𝐽 Cn 𝐽))
121, 11eqeltrid 2918 1 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  cmpt 5122  cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  TopOpenctopn 16686  TopOnctopon 21513   Cn ccn 21827  TopMndctmd 22673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-map 8395  df-topgen 16708  df-plusf 17842  df-top 21497  df-topon 21514  df-topsp 21536  df-bases 21549  df-cn 21830  df-cnp 21831  df-tx 22165  df-tmd 22675
This theorem is referenced by:  tgplacthmeo  22706  ghmcnp  22718
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