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Theorem istmd 18065
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 3498 . . 3  |-  ( G  e.  ( Mnd  i^i  TopSp
)  <->  ( G  e. 
Mnd  /\  G  e.  TopSp
) )
21anbi1i 677 . 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 5709 . . . . 5  |-  ( TopOpen `  f )  e.  _V
43a1i 11 . . . 4  |-  ( f  =  G  ->  ( TopOpen
`  f )  e. 
_V )
5 simpl 444 . . . . . . 7  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
f  =  G )
65fveq2d 5699 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( + f `  f )  =  ( + f `  G
) )
7 istmd.1 . . . . . 6  |-  F  =  ( + f `  G )
86, 7syl6eqr 2462 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( + f `  f )  =  F )
9 id 20 . . . . . . . 8  |-  ( j  =  ( TopOpen `  f
)  ->  j  =  ( TopOpen `  f )
)
10 fveq2 5695 . . . . . . . . 9  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  ( TopOpen `  G )
)
11 istmd.2 . . . . . . . . 9  |-  J  =  ( TopOpen `  G )
1210, 11syl6eqr 2462 . . . . . . . 8  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  J )
139, 12sylan9eqr 2466 . . . . . . 7  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
j  =  J )
1413, 13oveq12d 6066 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( j  tX  j
)  =  ( J 
tX  J ) )
1514, 13oveq12d 6066 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( ( j  tX  j )  Cn  j
)  =  ( ( J  tX  J )  Cn  J ) )
168, 15eleq12d 2480 . . . 4  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( ( + f `  f )  e.  ( ( j  tX  j
)  Cn  j )  <-> 
F  e.  ( ( J  tX  J )  Cn  J ) ) )
174, 16sbcied 3165 . . 3  |-  ( f  =  G  ->  ( [. ( TopOpen `  f )  /  j ]. ( + f `  f )  e.  ( ( j 
tX  j )  Cn  j )  <->  F  e.  ( ( J  tX  J )  Cn  J
) ) )
18 df-tmd 18063 . . 3  |- TopMnd  =  {
f  e.  ( Mnd 
i^i  TopSp )  |  [. ( TopOpen `  f )  /  j ]. ( + f `  f )  e.  ( ( j 
tX  j )  Cn  j ) }
1917, 18elrab2 3062 . 2  |-  ( G  e. TopMnd 
<->  ( G  e.  ( Mnd  i^i  TopSp )  /\  F  e.  ( ( J  tX  J )  Cn  J ) ) )
20 df-3an 938 . 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 269 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2924   [.wsbc 3129    i^i cin 3287   ` cfv 5421  (class class class)co 6048   TopOpenctopn 13612   Mndcmnd 14647   + fcplusf 14650   TopSpctps 16924    Cn ccn 17250    tX ctx 17553  TopMndctmd 18061
This theorem is referenced by:  tmdmnd  18066  tmdtps  18067  tmdcn  18074  istgp2  18082  oppgtmd  18088  symgtgp  18092  submtmd  18095  prdstmdd  18114  nrgtrg  18686  mhmhmeotmd  24274  xrge0tmdOLD  24292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-nul 4306
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-ov 6051  df-tmd 18063
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