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Theorem nmf2 18632
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 2435 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
31, 2grpidcl 14825 . . . . 5  |-  ( W  e.  Grp  ->  ( 0g `  W )  e.  X )
4 metcl 18354 . . . . . 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 2435 . . 3  |-  ( x  e.  X  |->  ( x E ( 0g `  W ) ) )  =  ( x  e.  X  |->  ( x E ( 0g `  W
) ) )
97, 8fmptd 5885 . 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 18630 . . . 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 5572 . 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 1652    e. wcel 1725    e. cmpt 4258    X. cxp 4868    |` cres 4872   -->wf 5442   ` cfv 5446  (class class class)co 6073   RRcr 8981   Basecbs 13461   distcds 13530   0gc0g 13715   Grpcgrp 14677   Metcme 16679   normcnm 18616
This theorem is referenced by:  isngp2  18636  isngp3  18637  nmf  18653
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039
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-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-map 7012  df-0g 13719  df-mnd 14682  df-grp 14804  df-met 16688  df-nm 18622
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