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Theorem nmf2 18131
Description: The norm is a function from the base set into the reals. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
nmf2.n  |-  N  =  ( norm `  W
)
nmf2.x  |-  X  =  ( Base `  W
)
nmf2.d  |-  D  =  ( dist `  W
)
nmf2.e  |-  E  =  ( D  |`  ( X  X.  X ) )
Assertion
Ref Expression
nmf2  |-  ( ( W  e.  Grp  /\  E  e.  ( Met `  X ) )  ->  N : X --> RR )

Proof of Theorem nmf2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 nmf2.x . . . . . 6  |-  X  =  ( Base `  W
)
2 eqid 2296 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
31, 2grpidcl 14526 . . . . 5  |-  ( W  e.  Grp  ->  ( 0g `  W )  e.  X )
4 metcl 17913 . . . . . 6  |-  ( ( E  e.  ( Met `  X )  /\  x  e.  X  /\  ( 0g `  W )  e.  X )  ->  (
x E ( 0g
`  W ) )  e.  RR )
543comr 1159 . . . . 5  |-  ( ( ( 0g `  W
)  e.  X  /\  E  e.  ( Met `  X )  /\  x  e.  X )  ->  (
x E ( 0g
`  W ) )  e.  RR )
63, 5syl3an1 1215 . . . 4  |-  ( ( W  e.  Grp  /\  E  e.  ( Met `  X )  /\  x  e.  X )  ->  (
x E ( 0g
`  W ) )  e.  RR )
763expa 1151 . . 3  |-  ( ( ( W  e.  Grp  /\  E  e.  ( Met `  X ) )  /\  x  e.  X )  ->  ( x E ( 0g `  W ) )  e.  RR )
8 eqid 2296 . . 3  |-  ( x  e.  X  |->  ( x E ( 0g `  W ) ) )  =  ( x  e.  X  |->  ( x E ( 0g `  W
) ) )
97, 8fmptd 5700 . 2  |-  ( ( W  e.  Grp  /\  E  e.  ( Met `  X ) )  -> 
( x  e.  X  |->  ( x E ( 0g `  W ) ) ) : X --> RR )
10 nmf2.n . . . . 5  |-  N  =  ( norm `  W
)
11 nmf2.d . . . . 5  |-  D  =  ( dist `  W
)
12 nmf2.e . . . . 5  |-  E  =  ( D  |`  ( X  X.  X ) )
1310, 1, 2, 11, 12nmfval2 18129 . . . 4  |-  ( W  e.  Grp  ->  N  =  ( x  e.  X  |->  ( x E ( 0g `  W
) ) ) )
1413adantr 451 . . 3  |-  ( ( W  e.  Grp  /\  E  e.  ( Met `  X ) )  ->  N  =  ( x  e.  X  |->  ( x E ( 0g `  W ) ) ) )
1514feq1d 5395 . 2  |-  ( ( W  e.  Grp  /\  E  e.  ( Met `  X ) )  -> 
( N : X --> RR 
<->  ( x  e.  X  |->  ( x E ( 0g `  W ) ) ) : X --> RR ) )
169, 15mpbird 223 1  |-  ( ( W  e.  Grp  /\  E  e.  ( Met `  X ) )  ->  N : X --> RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    e. cmpt 4093    X. cxp 4703    |` cres 4707   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   Basecbs 13164   distcds 13233   0gc0g 13416   Grpcgrp 14378   Metcme 16386   normcnm 18115
This theorem is referenced by:  isngp2  18135  isngp3  18136  nmf  18152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-map 6790  df-0g 13420  df-mnd 14383  df-grp 14505  df-met 16390  df-nm 18121
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