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Theorem tmdlactcn 20767
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 455 . . 3  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  G  e. TopMnd )
5 tgplacthmeo.2 . . . . 5  |-  X  =  ( Base `  G
)
62, 5tmdtopon 20746 . . . 4  |-  ( G  e. TopMnd  ->  J  e.  (TopOn `  X ) )
76adantr 463 . . 3  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
8 simpr 459 . . . 4  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  A  e.  X )
97, 7, 8cnmptc 20329 . . 3  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  (
x  e.  X  |->  A )  e.  ( J  Cn  J ) )
107cnmptid 20328 . . 3  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  (
x  e.  X  |->  x )  e.  ( J  Cn  J ) )
112, 3, 4, 7, 9, 10cnmpt1plusg 20752 . 2  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  (
x  e.  X  |->  ( A  .+  x ) )  e.  ( J  Cn  J ) )
121, 11syl5eqel 2546 1  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  F  e.  ( J  Cn  J
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   Basecbs 14716   +g cplusg 14784   TopOpenctopn 14911  TopOnctopon 19562    Cn ccn 19892  TopMndctmd 20735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-map 7414  df-topgen 14933  df-plusf 16070  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cn 19895  df-cnp 19896  df-tx 20229  df-tmd 20737
This theorem is referenced by:  tgplacthmeo  20768  ghmcnp  20779
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