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Theorem tmdlactcn 17837
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 17816 . . . 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 17412 . . 3  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  (
x  e.  X  |->  A )  e.  ( J  Cn  J ) )
107cnmptid 17411 . . 3  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  (
x  e.  X  |->  x )  e.  ( J  Cn  J ) )
112, 3, 4, 7, 9, 10cnmpt1plusg 17822 . 2  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  (
x  e.  X  |->  ( A  .+  x ) )  e.  ( J  Cn  J ) )
121, 11syl5eqel 2400 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 1633    e. wcel 1701    e. cmpt 4114   ` cfv 5292  (class class class)co 5900   Basecbs 13195   +g cplusg 13255   TopOpenctopn 13375  TopOnctopon 16688    Cn ccn 17010  TopMndctmd 17805
This theorem is referenced by:  tgplacthmeo  17838  ghmcnp  17849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-map 6817  df-topgen 13393  df-plusf 14417  df-top 16692  df-bases 16694  df-topon 16695  df-topsp 16696  df-cn 17013  df-cnp 17014  df-tx 17313  df-tmd 17807
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