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Theorem nmf2 21403
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 2402 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
31, 2grpidcl 16400 . . . . 5  |-  ( W  e.  Grp  ->  ( 0g `  W )  e.  X )
4 metcl 21125 . . . . . 6  |-  ( ( E  e.  ( Met `  X )  /\  x  e.  X  /\  ( 0g `  W )  e.  X )  ->  (
x E ( 0g
`  W ) )  e.  RR )
543comr 1205 . . . . 5  |-  ( ( ( 0g `  W
)  e.  X  /\  E  e.  ( Met `  X )  /\  x  e.  X )  ->  (
x E ( 0g
`  W ) )  e.  RR )
63, 5syl3an1 1263 . . . 4  |-  ( ( W  e.  Grp  /\  E  e.  ( Met `  X )  /\  x  e.  X )  ->  (
x E ( 0g
`  W ) )  e.  RR )
763expa 1197 . . 3  |-  ( ( ( W  e.  Grp  /\  E  e.  ( Met `  X ) )  /\  x  e.  X )  ->  ( x E ( 0g `  W ) )  e.  RR )
8 eqid 2402 . . 3  |-  ( x  e.  X  |->  ( x E ( 0g `  W ) ) )  =  ( x  e.  X  |->  ( x E ( 0g `  W
) ) )
97, 8fmptd 6032 . 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 21401 . . . 4  |-  ( W  e.  Grp  ->  N  =  ( x  e.  X  |->  ( x E ( 0g `  W
) ) ) )
1413adantr 463 . . 3  |-  ( ( W  e.  Grp  /\  E  e.  ( Met `  X ) )  ->  N  =  ( x  e.  X  |->  ( x E ( 0g `  W ) ) ) )
1514feq1d 5699 . 2  |-  ( ( W  e.  Grp  /\  E  e.  ( Met `  X ) )  -> 
( N : X --> RR 
<->  ( x  e.  X  |->  ( x E ( 0g `  W ) ) ) : X --> RR ) )
169, 15mpbird 232 1  |-  ( ( W  e.  Grp  /\  E  e.  ( Met `  X ) )  ->  N : X --> RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    |-> cmpt 4452    X. cxp 4820    |` cres 4824   -->wf 5564   ` cfv 5568  (class class class)co 6277   RRcr 9520   Basecbs 14839   distcds 14916   0gc0g 15052   Grpcgrp 16375   Metcme 18722   normcnm 21387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7458  df-0g 15054  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-grp 16379  df-met 18731  df-nm 21393
This theorem is referenced by:  isngp2  21407  isngp3  21408  nmf  21424
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