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Theorem tmdlactcn 23476
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 484 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) β†’ 𝐺 ∈ TopMnd)
5 tgplacthmeo.2 . . . . 5 𝑋 = (Baseβ€˜πΊ)
62, 5tmdtopon 23455 . . . 4 (𝐺 ∈ TopMnd β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
76adantr 482 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
8 simpr 486 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
97, 7, 8cnmptc 23036 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐽))
107cnmptid 23035 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↦ π‘₯) ∈ (𝐽 Cn 𝐽))
112, 3, 4, 7, 9, 10cnmpt1plusg 23461 . 2 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴 + π‘₯)) ∈ (𝐽 Cn 𝐽))
121, 11eqeltrid 2838 1 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) β†’ 𝐹 ∈ (𝐽 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ↦ cmpt 5192  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091  +gcplusg 17141  TopOpenctopn 17311  TopOnctopon 22282   Cn ccn 22598  TopMndctmd 23444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-map 8773  df-topgen 17333  df-plusf 18504  df-top 22266  df-topon 22283  df-topsp 22305  df-bases 22319  df-cn 22601  df-cnp 22602  df-tx 22936  df-tmd 23446
This theorem is referenced by:  tgplacthmeo  23477  ghmcnp  23489
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