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Theorem isngp 18220
Description: The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
isngp.n  |-  N  =  ( norm `  G
)
isngp.z  |-  .-  =  ( -g `  G )
isngp.d  |-  D  =  ( dist `  G
)
Assertion
Ref Expression
isngp  |-  ( G  e. NrmGrp 
<->  ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  C_  D ) )

Proof of Theorem isngp
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 elin 3434 . . 3  |-  ( G  e.  ( Grp  i^i  MetSp
)  <->  ( G  e. 
Grp  /\  G  e.  MetSp
) )
21anbi1i 676 . 2  |-  ( ( G  e.  ( Grp 
i^i  MetSp )  /\  ( N  o.  .-  )  C_  D )  <->  ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D ) )
3 fveq2 5608 . . . . . 6  |-  ( g  =  G  ->  ( norm `  g )  =  ( norm `  G
) )
4 isngp.n . . . . . 6  |-  N  =  ( norm `  G
)
53, 4syl6eqr 2408 . . . . 5  |-  ( g  =  G  ->  ( norm `  g )  =  N )
6 fveq2 5608 . . . . . 6  |-  ( g  =  G  ->  ( -g `  g )  =  ( -g `  G
) )
7 isngp.z . . . . . 6  |-  .-  =  ( -g `  G )
86, 7syl6eqr 2408 . . . . 5  |-  ( g  =  G  ->  ( -g `  g )  = 
.-  )
95, 8coeq12d 4930 . . . 4  |-  ( g  =  G  ->  (
( norm `  g )  o.  ( -g `  g
) )  =  ( N  o.  .-  )
)
10 fveq2 5608 . . . . 5  |-  ( g  =  G  ->  ( dist `  g )  =  ( dist `  G
) )
11 isngp.d . . . . 5  |-  D  =  ( dist `  G
)
1210, 11syl6eqr 2408 . . . 4  |-  ( g  =  G  ->  ( dist `  g )  =  D )
139, 12sseq12d 3283 . . 3  |-  ( g  =  G  ->  (
( ( norm `  g
)  o.  ( -g `  g ) )  C_  ( dist `  g )  <->  ( N  o.  .-  )  C_  D ) )
14 df-ngp 18208 . . 3  |- NrmGrp  =  {
g  e.  ( Grp 
i^i  MetSp )  |  ( ( norm `  g
)  o.  ( -g `  g ) )  C_  ( dist `  g ) }
1513, 14elrab2 3001 . 2  |-  ( G  e. NrmGrp 
<->  ( G  e.  ( Grp  i^i  MetSp )  /\  ( N  o.  .-  )  C_  D ) )
16 df-3an 936 . 2  |-  ( ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  C_  D )  <->  ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D ) )
172, 15, 163bitr4i 268 1  |-  ( G  e. NrmGrp 
<->  ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  C_  D ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    i^i cin 3227    C_ wss 3228    o. ccom 4775   ` cfv 5337   distcds 13314   Grpcgrp 14461   -gcsg 14464   MetSpcmt 17985   normcnm 18201  NrmGrpcngp 18202
This theorem is referenced by:  isngp2  18221  ngpgrp  18223  ngpms  18224  tngngp2  18270  cnngp  18391  zhmnrg  23626
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-co 4780  df-iota 5301  df-fv 5345  df-ngp 18208
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