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Theorem tmdlactcn 18132
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 444 . . 3  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  G  e. TopMnd )
5 tgplacthmeo.2 . . . . 5  |-  X  =  ( Base `  G
)
62, 5tmdtopon 18111 . . . 4  |-  ( G  e. TopMnd  ->  J  e.  (TopOn `  X ) )
76adantr 452 . . 3  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
8 simpr 448 . . . 4  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  A  e.  X )
97, 7, 8cnmptc 17694 . . 3  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  (
x  e.  X  |->  A )  e.  ( J  Cn  J ) )
107cnmptid 17693 . . 3  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  (
x  e.  X  |->  x )  e.  ( J  Cn  J ) )
112, 3, 4, 7, 9, 10cnmpt1plusg 18117 . 2  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  (
x  e.  X  |->  ( A  .+  x ) )  e.  ( J  Cn  J ) )
121, 11syl5eqel 2520 1  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  F  e.  ( J  Cn  J
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   TopOpenctopn 13649  TopOnctopon 16959    Cn ccn 17288  TopMndctmd 18100
This theorem is referenced by:  tgplacthmeo  18133  ghmcnp  18144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-map 7020  df-topgen 13667  df-plusf 14691  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cn 17291  df-cnp 17292  df-tx 17594  df-tmd 18102
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