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Theorem isngp 18643
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 3530 . . 3  |-  ( G  e.  ( Grp  i^i  MetSp
)  <->  ( G  e. 
Grp  /\  G  e.  MetSp
) )
21anbi1i 677 . 2  |-  ( ( G  e.  ( Grp 
i^i  MetSp )  /\  ( N  o.  .-  )  C_  D )  <->  ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D ) )
3 fveq2 5728 . . . . . 6  |-  ( g  =  G  ->  ( norm `  g )  =  ( norm `  G
) )
4 isngp.n . . . . . 6  |-  N  =  ( norm `  G
)
53, 4syl6eqr 2486 . . . . 5  |-  ( g  =  G  ->  ( norm `  g )  =  N )
6 fveq2 5728 . . . . . 6  |-  ( g  =  G  ->  ( -g `  g )  =  ( -g `  G
) )
7 isngp.z . . . . . 6  |-  .-  =  ( -g `  G )
86, 7syl6eqr 2486 . . . . 5  |-  ( g  =  G  ->  ( -g `  g )  = 
.-  )
95, 8coeq12d 5037 . . . 4  |-  ( g  =  G  ->  (
( norm `  g )  o.  ( -g `  g
) )  =  ( N  o.  .-  )
)
10 fveq2 5728 . . . . 5  |-  ( g  =  G  ->  ( dist `  g )  =  ( dist `  G
) )
11 isngp.d . . . . 5  |-  D  =  ( dist `  G
)
1210, 11syl6eqr 2486 . . . 4  |-  ( g  =  G  ->  ( dist `  g )  =  D )
139, 12sseq12d 3377 . . 3  |-  ( g  =  G  ->  (
( ( norm `  g
)  o.  ( -g `  g ) )  C_  ( dist `  g )  <->  ( N  o.  .-  )  C_  D ) )
14 df-ngp 18631 . . 3  |- NrmGrp  =  {
g  e.  ( Grp 
i^i  MetSp )  |  ( ( norm `  g
)  o.  ( -g `  g ) )  C_  ( dist `  g ) }
1513, 14elrab2 3094 . 2  |-  ( G  e. NrmGrp 
<->  ( G  e.  ( Grp  i^i  MetSp )  /\  ( N  o.  .-  )  C_  D ) )
16 df-3an 938 . 2  |-  ( ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  C_  D )  <->  ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D ) )
172, 15, 163bitr4i 269 1  |-  ( G  e. NrmGrp 
<->  ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  C_  D ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    i^i cin 3319    C_ wss 3320    o. ccom 4882   ` cfv 5454   distcds 13538   Grpcgrp 14685   -gcsg 14688   MetSpcmt 18348   normcnm 18624  NrmGrpcngp 18625
This theorem is referenced by:  isngp2  18644  ngpgrp  18646  ngpms  18647  tngngp2  18693  cnngp  18814  zhmnrg  24351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-co 4887  df-iota 5418  df-fv 5462  df-ngp 18631
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