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Theorem istmd 17757
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 3358 . . 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 5539 . . . . 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 5529 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( + f `  f )  =  ( + f `  G
) )
7 istmd.1 . . . . . 6  |-  F  =  ( + f `  G )
86, 7syl6eqr 2333 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( + f `  f )  =  F )
9 id 19 . . . . . . . 8  |-  ( j  =  ( TopOpen `  f
)  ->  j  =  ( TopOpen `  f )
)
10 fveq2 5525 . . . . . . . . 9  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  ( TopOpen `  G )
)
11 istmd.2 . . . . . . . . 9  |-  J  =  ( TopOpen `  G )
1210, 11syl6eqr 2333 . . . . . . . 8  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  J )
139, 12sylan9eqr 2337 . . . . . . 7  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
j  =  J )
1413, 13oveq12d 5876 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( j  tX  j
)  =  ( J 
tX  J ) )
1514, 13oveq12d 5876 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( ( j  tX  j )  Cn  j
)  =  ( ( J  tX  J )  Cn  J ) )
168, 15eleq12d 2351 . . . 4  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( ( + f `  f )  e.  ( ( j  tX  j
)  Cn  j )  <-> 
F  e.  ( ( J  tX  J )  Cn  J ) ) )
174, 16sbcied 3027 . . 3  |-  ( f  =  G  ->  ( [. ( TopOpen `  f )  /  j ]. ( + f `  f )  e.  ( ( j 
tX  j )  Cn  j )  <->  F  e.  ( ( J  tX  J )  Cn  J
) ) )
18 df-tmd 17755 . . 3  |- TopMnd  =  {
f  e.  ( Mnd 
i^i  TopSp )  |  [. ( TopOpen `  f )  /  j ]. ( + f `  f )  e.  ( ( j 
tX  j )  Cn  j ) }
1917, 18elrab2 2925 . 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 1623    e. wcel 1684   _Vcvv 2788   [.wsbc 2991    i^i cin 3151   ` cfv 5255  (class class class)co 5858   TopOpenctopn 13326   Mndcmnd 14361   + fcplusf 14364   TopSpctps 16634    Cn ccn 16954    tX ctx 17255  TopMndctmd 17753
This theorem is referenced by:  tmdmnd  17758  tmdtps  17759  tmdcn  17766  istgp2  17774  oppgtmd  17780  symgtgp  17784  submtmd  17787  prdstmdd  17806  nrgtrg  18200  mhmhmeotmd  23300  xrge0tmdALT  23327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-tmd 17755
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