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Theorem tmdlactcn 17785
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  =  ( x  e.  X  |->  ( A  .+  x
) )
tgplacthmeo.2  |-  X  =  ( Base `  G
)
tgplacthmeo.3  |-  .+  =  ( +g  `  G )
tgplacthmeo.4  |-  J  =  ( TopOpen `  G )
Assertion
Ref Expression
tmdlactcn  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  F  e.  ( 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  =  ( x  e.  X  |->  ( A  .+  x
) )
2 tgplacthmeo.4 . . 3  |-  J  =  ( TopOpen `  G )
3 tgplacthmeo.3 . . 3  |-  .+  =  ( +g  `  G )
4 simpl 443 . . 3  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  G  e. TopMnd )
5 tgplacthmeo.2 . . . . 5  |-  X  =  ( Base `  G
)
62, 5tmdtopon 17764 . . . 4  |-  ( G  e. TopMnd  ->  J  e.  (TopOn `  X ) )
76adantr 451 . . 3  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
8 simpr 447 . . . 4  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  A  e.  X )
97, 7, 8cnmptc 17356 . . 3  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  (
x  e.  X  |->  A )  e.  ( J  Cn  J ) )
107cnmptid 17355 . . 3  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  (
x  e.  X  |->  x )  e.  ( J  Cn  J ) )
112, 3, 4, 7, 9, 10cnmpt1plusg 17770 . 2  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  (
x  e.  X  |->  ( A  .+  x ) )  e.  ( J  Cn  J ) )
121, 11syl5eqel 2367 1  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  F  e.  ( J  Cn  J
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   TopOpenctopn 13326  TopOnctopon 16632    Cn ccn 16954  TopMndctmd 17753
This theorem is referenced by:  tgplacthmeo  17786  ghmcnp  17797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-topgen 13344  df-plusf 14368  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-cnp 16958  df-tx 17257  df-tmd 17755
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