MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  istgp Unicode version

Theorem istgp 17708
Description: The predicate "is a topological group". Definition of [BourbakiTop1] p. III.1 (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
istgp.1  |-  J  =  ( TopOpen `  G )
istgp.2  |-  I  =  ( inv g `  G )
Assertion
Ref Expression
istgp  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e. TopMnd  /\  I  e.  ( J  Cn  J
) ) )

Proof of Theorem istgp
StepHypRef Expression
1 elin 3319 . . 3  |-  ( G  e.  ( Grp  i^i TopMnd )  <-> 
( G  e.  Grp  /\  G  e. TopMnd ) )
21anbi1i 679 . 2  |-  ( ( G  e.  ( Grp 
i^i TopMnd )  /\  I  e.  ( J  Cn  J
) )  <->  ( ( G  e.  Grp  /\  G  e. TopMnd )  /\  I  e.  ( J  Cn  J
) ) )
3 fvex 5458 . . . . 5  |-  ( TopOpen `  f )  e.  _V
43a1i 12 . . . 4  |-  ( f  =  G  ->  ( TopOpen
`  f )  e. 
_V )
5 simpl 445 . . . . . . 7  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
f  =  G )
65fveq2d 5448 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( inv g `  f )  =  ( inv g `  G
) )
7 istgp.2 . . . . . 6  |-  I  =  ( inv g `  G )
86, 7syl6eqr 2306 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( inv g `  f )  =  I )
9 id 21 . . . . . . 7  |-  ( j  =  ( TopOpen `  f
)  ->  j  =  ( TopOpen `  f )
)
10 fveq2 5444 . . . . . . . 8  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  ( TopOpen `  G )
)
11 istgp.1 . . . . . . . 8  |-  J  =  ( TopOpen `  G )
1210, 11syl6eqr 2306 . . . . . . 7  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  J )
139, 12sylan9eqr 2310 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
j  =  J )
1413, 13oveq12d 5796 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( j  Cn  j
)  =  ( J  Cn  J ) )
158, 14eleq12d 2324 . . . 4  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( ( inv g `  f )  e.  ( j  Cn  j )  <-> 
I  e.  ( J  Cn  J ) ) )
164, 15sbcied 2988 . . 3  |-  ( f  =  G  ->  ( [. ( TopOpen `  f )  /  j ]. ( inv g `  f )  e.  ( j  Cn  j )  <->  I  e.  ( J  Cn  J
) ) )
17 df-tgp 17704 . . 3  |-  TopGrp  =  {
f  e.  ( Grp 
i^i TopMnd )  |  [. ( TopOpen
`  f )  / 
j ]. ( inv g `  f )  e.  ( j  Cn  j ) }
1816, 17elrab2 2893 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  ( Grp  i^i TopMnd )  /\  I  e.  ( J  Cn  J
) ) )
19 df-3an 941 . 2  |-  ( ( G  e.  Grp  /\  G  e. TopMnd  /\  I  e.  ( J  Cn  J
) )  <->  ( ( G  e.  Grp  /\  G  e. TopMnd )  /\  I  e.  ( J  Cn  J
) ) )
202, 18, 193bitr4i 270 1  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e. TopMnd  /\  I  e.  ( J  Cn  J
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   _Vcvv 2757   [.wsbc 2952    i^i cin 3112   ` cfv 4659  (class class class)co 5778   TopOpenctopn 13274   Grpcgrp 14310   inv gcminusg 14311    Cn ccn 16902  TopMndctmd 17701   TopGrpctgp 17702
This theorem is referenced by:  tgpgrp  17709  tgptmd  17710  tgpinv  17716  istgp2  17722  oppgtgp  17729  symgtgp  17732  subgtgp  17736  prdstgpd  17755  tlmtgp  17826  nrgtdrg  18151
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-xp 4661  df-cnv 4663  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fv 4675  df-ov 5781  df-tgp 17704
  Copyright terms: Public domain W3C validator