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Theorem tmdlactcn 23961
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 = (π‘₯ ∈ 𝑋 ↦ (𝐴 + π‘₯))
tgplacthmeo.2 𝑋 = (Baseβ€˜πΊ)
tgplacthmeo.3 + = (+gβ€˜πΊ)
tgplacthmeo.4 𝐽 = (TopOpenβ€˜πΊ)
Assertion
Ref Expression
tmdlactcn ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) β†’ 𝐹 ∈ (𝐽 Cn 𝐽))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐺   π‘₯,𝐽   π‘₯, +   π‘₯,𝑋
Allowed substitution hint:   𝐹(π‘₯)

Proof of Theorem tmdlactcn
StepHypRef Expression
1 tgplacthmeo.1 . 2 𝐹 = (π‘₯ ∈ 𝑋 ↦ (𝐴 + π‘₯))
2 tgplacthmeo.4 . . 3 𝐽 = (TopOpenβ€˜πΊ)
3 tgplacthmeo.3 . . 3 + = (+gβ€˜πΊ)
4 simpl 482 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) β†’ 𝐺 ∈ TopMnd)
5 tgplacthmeo.2 . . . . 5 𝑋 = (Baseβ€˜πΊ)
62, 5tmdtopon 23940 . . . 4 (𝐺 ∈ TopMnd β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
76adantr 480 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
8 simpr 484 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
97, 7, 8cnmptc 23521 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐽))
107cnmptid 23520 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↦ π‘₯) ∈ (𝐽 Cn 𝐽))
112, 3, 4, 7, 9, 10cnmpt1plusg 23946 . 2 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴 + π‘₯)) ∈ (𝐽 Cn 𝐽))
121, 11eqeltrid 2831 1 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) β†’ 𝐹 ∈ (𝐽 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ↦ cmpt 5224  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  +gcplusg 17206  TopOpenctopn 17376  TopOnctopon 22767   Cn ccn 23083  TopMndctmd 23929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-map 8824  df-topgen 17398  df-plusf 18572  df-top 22751  df-topon 22768  df-topsp 22790  df-bases 22804  df-cn 23086  df-cnp 23087  df-tx 23421  df-tmd 23931
This theorem is referenced by:  tgplacthmeo  23962  ghmcnp  23974
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