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Theorem istmd 18135
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 3516 . . 3  |-  ( G  e.  ( Mnd  i^i  TopSp
)  <->  ( G  e. 
Mnd  /\  G  e.  TopSp
) )
21anbi1i 678 . 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 5771 . . . . 5  |-  ( TopOpen `  f )  e.  _V
43a1i 11 . . . 4  |-  ( f  =  G  ->  ( TopOpen
`  f )  e. 
_V )
5 simpl 445 . . . . . . 7  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
f  =  G )
65fveq2d 5761 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( + f `  f )  =  ( + f `  G
) )
7 istmd.1 . . . . . 6  |-  F  =  ( + f `  G )
86, 7syl6eqr 2492 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( + f `  f )  =  F )
9 id 21 . . . . . . . 8  |-  ( j  =  ( TopOpen `  f
)  ->  j  =  ( TopOpen `  f )
)
10 fveq2 5757 . . . . . . . . 9  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  ( TopOpen `  G )
)
11 istmd.2 . . . . . . . . 9  |-  J  =  ( TopOpen `  G )
1210, 11syl6eqr 2492 . . . . . . . 8  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  J )
139, 12sylan9eqr 2496 . . . . . . 7  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
j  =  J )
1413, 13oveq12d 6128 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( j  tX  j
)  =  ( J 
tX  J ) )
1514, 13oveq12d 6128 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( ( j  tX  j )  Cn  j
)  =  ( ( J  tX  J )  Cn  J ) )
168, 15eleq12d 2510 . . . 4  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( ( + f `  f )  e.  ( ( j  tX  j
)  Cn  j )  <-> 
F  e.  ( ( J  tX  J )  Cn  J ) ) )
174, 16sbcied 3203 . . 3  |-  ( f  =  G  ->  ( [. ( TopOpen `  f )  /  j ]. ( + f `  f )  e.  ( ( j 
tX  j )  Cn  j )  <->  F  e.  ( ( J  tX  J )  Cn  J
) ) )
18 df-tmd 18133 . . 3  |- TopMnd  =  {
f  e.  ( Mnd 
i^i  TopSp )  |  [. ( TopOpen `  f )  /  j ]. ( + f `  f )  e.  ( ( j 
tX  j )  Cn  j ) }
1917, 18elrab2 3100 . 2  |-  ( G  e. TopMnd 
<->  ( G  e.  ( Mnd  i^i  TopSp )  /\  F  e.  ( ( J  tX  J )  Cn  J ) ) )
20 df-3an 939 . 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 270 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   _Vcvv 2962   [.wsbc 3167    i^i cin 3305   ` cfv 5483  (class class class)co 6110   TopOpenctopn 13680   Mndcmnd 14715   + fcplusf 14718   TopSpctps 16992    Cn ccn 17319    tX ctx 17623  TopMndctmd 18131
This theorem is referenced by:  tmdmnd  18136  tmdtps  18137  tmdcn  18144  istgp2  18152  oppgtmd  18158  symgtgp  18162  submtmd  18165  prdstmdd  18184  nrgtrg  18756  mhmhmeotmd  24344  xrge0tmdOLD  24362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-nul 4363
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-iota 5447  df-fv 5491  df-ov 6113  df-tmd 18133
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