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Theorem istmd 17853
Description: The predicate "is a topological monoid". (Contributed by Mario Carneiro, 19-Sep-2015.)
Hypotheses
Ref Expression
istmd.1  |-  F  =  ( + f `  G )
istmd.2  |-  J  =  ( TopOpen `  G )
Assertion
Ref Expression
istmd  |-  ( G  e. TopMnd 
<->  ( G  e.  Mnd  /\  G  e.  TopSp  /\  F  e.  ( ( J  tX  J )  Cn  J
) ) )

Proof of Theorem istmd
Dummy variables  f 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3434 . . 3  |-  ( G  e.  ( Mnd  i^i  TopSp
)  <->  ( G  e. 
Mnd  /\  G  e.  TopSp
) )
21anbi1i 676 . 2  |-  ( ( G  e.  ( Mnd 
i^i  TopSp )  /\  F  e.  ( ( J  tX  J )  Cn  J
) )  <->  ( ( G  e.  Mnd  /\  G  e.  TopSp )  /\  F  e.  ( ( J  tX  J )  Cn  J
) ) )
3 fvex 5619 . . . . 5  |-  ( TopOpen `  f )  e.  _V
43a1i 10 . . . 4  |-  ( f  =  G  ->  ( TopOpen
`  f )  e. 
_V )
5 simpl 443 . . . . . . 7  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
f  =  G )
65fveq2d 5609 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( + f `  f )  =  ( + f `  G
) )
7 istmd.1 . . . . . 6  |-  F  =  ( + f `  G )
86, 7syl6eqr 2408 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( + f `  f )  =  F )
9 id 19 . . . . . . . 8  |-  ( j  =  ( TopOpen `  f
)  ->  j  =  ( TopOpen `  f )
)
10 fveq2 5605 . . . . . . . . 9  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  ( TopOpen `  G )
)
11 istmd.2 . . . . . . . . 9  |-  J  =  ( TopOpen `  G )
1210, 11syl6eqr 2408 . . . . . . . 8  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  J )
139, 12sylan9eqr 2412 . . . . . . 7  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
j  =  J )
1413, 13oveq12d 5960 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( j  tX  j
)  =  ( J 
tX  J ) )
1514, 13oveq12d 5960 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( ( j  tX  j )  Cn  j
)  =  ( ( J  tX  J )  Cn  J ) )
168, 15eleq12d 2426 . . . 4  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( ( + f `  f )  e.  ( ( j  tX  j
)  Cn  j )  <-> 
F  e.  ( ( J  tX  J )  Cn  J ) ) )
174, 16sbcied 3103 . . 3  |-  ( f  =  G  ->  ( [. ( TopOpen `  f )  /  j ]. ( + f `  f )  e.  ( ( j 
tX  j )  Cn  j )  <->  F  e.  ( ( J  tX  J )  Cn  J
) ) )
18 df-tmd 17851 . . 3  |- TopMnd  =  {
f  e.  ( Mnd 
i^i  TopSp )  |  [. ( TopOpen `  f )  /  j ]. ( + f `  f )  e.  ( ( j 
tX  j )  Cn  j ) }
1917, 18elrab2 3001 . 2  |-  ( G  e. TopMnd 
<->  ( G  e.  ( Mnd  i^i  TopSp )  /\  F  e.  ( ( J  tX  J )  Cn  J ) ) )
20 df-3an 936 . 2  |-  ( ( G  e.  Mnd  /\  G  e.  TopSp  /\  F  e.  ( ( J  tX  J )  Cn  J
) )  <->  ( ( G  e.  Mnd  /\  G  e.  TopSp )  /\  F  e.  ( ( J  tX  J )  Cn  J
) ) )
212, 19, 203bitr4i 268 1  |-  ( G  e. TopMnd 
<->  ( G  e.  Mnd  /\  G  e.  TopSp  /\  F  e.  ( ( J  tX  J )  Cn  J
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   _Vcvv 2864   [.wsbc 3067    i^i cin 3227   ` cfv 5334  (class class class)co 5942   TopOpenctopn 13419   Mndcmnd 14454   + fcplusf 14457   TopSpctps 16734    Cn ccn 17054    tX ctx 17355  TopMndctmd 17849
This theorem is referenced by:  tmdmnd  17854  tmdtps  17855  tmdcn  17862  istgp2  17870  oppgtmd  17876  symgtgp  17880  submtmd  17883  prdstmdd  17902  nrgtrg  18296  mhmhmeotmd  23469  xrge0tmdALT  23487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-nul 4228
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-iota 5298  df-fv 5342  df-ov 5945  df-tmd 17851
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