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Theorem nmf2 20843
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 2462 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
31, 2grpidcl 15874 . . . . 5  |-  ( W  e.  Grp  ->  ( 0g `  W )  e.  X )
4 metcl 20565 . . . . . 6  |-  ( ( E  e.  ( Met `  X )  /\  x  e.  X  /\  ( 0g `  W )  e.  X )  ->  (
x E ( 0g
`  W ) )  e.  RR )
543comr 1199 . . . . 5  |-  ( ( ( 0g `  W
)  e.  X  /\  E  e.  ( Met `  X )  /\  x  e.  X )  ->  (
x E ( 0g
`  W ) )  e.  RR )
63, 5syl3an1 1256 . . . 4  |-  ( ( W  e.  Grp  /\  E  e.  ( Met `  X )  /\  x  e.  X )  ->  (
x E ( 0g
`  W ) )  e.  RR )
763expa 1191 . . 3  |-  ( ( ( W  e.  Grp  /\  E  e.  ( Met `  X ) )  /\  x  e.  X )  ->  ( x E ( 0g `  W ) )  e.  RR )
8 eqid 2462 . . 3  |-  ( x  e.  X  |->  ( x E ( 0g `  W ) ) )  =  ( x  e.  X  |->  ( x E ( 0g `  W
) ) )
97, 8fmptd 6038 . 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 20841 . . . 4  |-  ( W  e.  Grp  ->  N  =  ( x  e.  X  |->  ( x E ( 0g `  W
) ) ) )
1413adantr 465 . . 3  |-  ( ( W  e.  Grp  /\  E  e.  ( Met `  X ) )  ->  N  =  ( x  e.  X  |->  ( x E ( 0g `  W ) ) ) )
1514feq1d 5710 . 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 369    = wceq 1374    e. wcel 1762    |-> cmpt 4500    X. cxp 4992    |` cres 4996   -->wf 5577   ` cfv 5581  (class class class)co 6277   RRcr 9482   Basecbs 14481   distcds 14555   0gc0g 14686   Grpcgrp 15718   Metcme 18170   normcnm 20827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7414  df-0g 14688  df-mnd 15723  df-grp 15853  df-met 18179  df-nm 20833
This theorem is referenced by:  isngp2  20847  isngp3  20848  nmf  20864
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