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Theorem nmf2 18513
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 2389 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
31, 2grpidcl 14762 . . . . 5  |-  ( W  e.  Grp  ->  ( 0g `  W )  e.  X )
4 metcl 18273 . . . . . 6  |-  ( ( E  e.  ( Met `  X )  /\  x  e.  X  /\  ( 0g `  W )  e.  X )  ->  (
x E ( 0g
`  W ) )  e.  RR )
543comr 1161 . . . . 5  |-  ( ( ( 0g `  W
)  e.  X  /\  E  e.  ( Met `  X )  /\  x  e.  X )  ->  (
x E ( 0g
`  W ) )  e.  RR )
63, 5syl3an1 1217 . . . 4  |-  ( ( W  e.  Grp  /\  E  e.  ( Met `  X )  /\  x  e.  X )  ->  (
x E ( 0g
`  W ) )  e.  RR )
763expa 1153 . . 3  |-  ( ( ( W  e.  Grp  /\  E  e.  ( Met `  X ) )  /\  x  e.  X )  ->  ( x E ( 0g `  W ) )  e.  RR )
8 eqid 2389 . . 3  |-  ( x  e.  X  |->  ( x E ( 0g `  W ) ) )  =  ( x  e.  X  |->  ( x E ( 0g `  W
) ) )
97, 8fmptd 5834 . 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 18511 . . . 4  |-  ( W  e.  Grp  ->  N  =  ( x  e.  X  |->  ( x E ( 0g `  W
) ) ) )
1413adantr 452 . . 3  |-  ( ( W  e.  Grp  /\  E  e.  ( Met `  X ) )  ->  N  =  ( x  e.  X  |->  ( x E ( 0g `  W ) ) ) )
1514feq1d 5522 . 2  |-  ( ( W  e.  Grp  /\  E  e.  ( Met `  X ) )  -> 
( N : X --> RR 
<->  ( x  e.  X  |->  ( x E ( 0g `  W ) ) ) : X --> RR ) )
169, 15mpbird 224 1  |-  ( ( W  e.  Grp  /\  E  e.  ( Met `  X ) )  ->  N : X --> RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    e. cmpt 4209    X. cxp 4818    |` cres 4822   -->wf 5392   ` cfv 5396  (class class class)co 6022   RRcr 8924   Basecbs 13398   distcds 13467   0gc0g 13652   Grpcgrp 14614   Metcme 16615   normcnm 18497
This theorem is referenced by:  isngp2  18517  isngp3  18518  nmf  18534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-map 6958  df-0g 13656  df-mnd 14619  df-grp 14741  df-met 16622  df-nm 18503
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