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Theorem istmd 19645
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 3539 . . 3  |-  ( G  e.  ( Mnd  i^i  TopSp
)  <->  ( G  e. 
Mnd  /\  G  e.  TopSp
) )
21anbi1i 695 . 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 5701 . . . . 5  |-  ( TopOpen `  f )  e.  _V
43a1i 11 . . . 4  |-  ( f  =  G  ->  ( TopOpen
`  f )  e. 
_V )
5 simpl 457 . . . . . . 7  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
f  =  G )
65fveq2d 5695 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( +f `  f )  =  ( +f `  G
) )
7 istmd.1 . . . . . 6  |-  F  =  ( +f `  G )
86, 7syl6eqr 2493 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( +f `  f )  =  F )
9 id 22 . . . . . . . 8  |-  ( j  =  ( TopOpen `  f
)  ->  j  =  ( TopOpen `  f )
)
10 fveq2 5691 . . . . . . . . 9  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  ( TopOpen `  G )
)
11 istmd.2 . . . . . . . . 9  |-  J  =  ( TopOpen `  G )
1210, 11syl6eqr 2493 . . . . . . . 8  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  J )
139, 12sylan9eqr 2497 . . . . . . 7  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
j  =  J )
1413, 13oveq12d 6109 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( j  tX  j
)  =  ( J 
tX  J ) )
1514, 13oveq12d 6109 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( ( j  tX  j )  Cn  j
)  =  ( ( J  tX  J )  Cn  J ) )
168, 15eleq12d 2511 . . . 4  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( ( +f `  f )  e.  ( ( j  tX  j
)  Cn  j )  <-> 
F  e.  ( ( J  tX  J )  Cn  J ) ) )
174, 16sbcied 3223 . . 3  |-  ( f  =  G  ->  ( [. ( TopOpen `  f )  /  j ]. ( +f `  f
)  e.  ( ( j  tX  j )  Cn  j )  <->  F  e.  ( ( J  tX  J )  Cn  J
) ) )
18 df-tmd 19643 . . 3  |- TopMnd  =  {
f  e.  ( Mnd 
i^i  TopSp )  |  [. ( TopOpen `  f )  /  j ]. ( +f `  f
)  e.  ( ( j  tX  j )  Cn  j ) }
1917, 18elrab2 3119 . 2  |-  ( G  e. TopMnd 
<->  ( G  e.  ( Mnd  i^i  TopSp )  /\  F  e.  ( ( J  tX  J )  Cn  J ) ) )
20 df-3an 967 . 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 277 1  |-  ( G  e. TopMnd 
<->  ( G  e.  Mnd  /\  G  e.  TopSp  /\  F  e.  ( ( J  tX  J )  Cn  J
) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2972   [.wsbc 3186    i^i cin 3327   ` cfv 5418  (class class class)co 6091   TopOpenctopn 14360   Mndcmnd 15409   +fcplusf 15412   TopSpctps 18501    Cn ccn 18828    tX ctx 19133  TopMndctmd 19641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426  df-ov 6094  df-tmd 19643
This theorem is referenced by:  tmdmnd  19646  tmdtps  19647  tmdcn  19654  istgp2  19662  oppgtmd  19668  symgtgp  19672  submtmd  19675  prdstmdd  19694  nrgtrg  20270  mhmhmeotmd  26357  xrge0tmdOLD  26375
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