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Theorem nmpropd 18633
Description: Weak property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
nmpropd.1  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
nmpropd.2  |-  ( ph  ->  ( +g  `  K
)  =  ( +g  `  L ) )
nmpropd.3  |-  ( ph  ->  ( dist `  K
)  =  ( dist `  L ) )
Assertion
Ref Expression
nmpropd  |-  ( ph  ->  ( norm `  K
)  =  ( norm `  L ) )

Proof of Theorem nmpropd
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmpropd.1 . . 3  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
2 nmpropd.3 . . . 4  |-  ( ph  ->  ( dist `  K
)  =  ( dist `  L ) )
3 eqidd 2436 . . . 4  |-  ( ph  ->  x  =  x )
4 eqidd 2436 . . . . 5  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  K ) )
5 nmpropd.2 . . . . . 6  |-  ( ph  ->  ( +g  `  K
)  =  ( +g  `  L ) )
65proplem3 13908 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
) )  ->  (
x ( +g  `  K
) y )  =  ( x ( +g  `  L ) y ) )
74, 1, 6grpidpropd 14714 . . . 4  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
82, 3, 7oveq123d 6094 . . 3  |-  ( ph  ->  ( x ( dist `  K ) ( 0g
`  K ) )  =  ( x (
dist `  L )
( 0g `  L
) ) )
91, 8mpteq12dv 4279 . 2  |-  ( ph  ->  ( x  e.  (
Base `  K )  |->  ( x ( dist `  K ) ( 0g
`  K ) ) )  =  ( x  e.  ( Base `  L
)  |->  ( x (
dist `  L )
( 0g `  L
) ) ) )
10 eqid 2435 . . 3  |-  ( norm `  K )  =  (
norm `  K )
11 eqid 2435 . . 3  |-  ( Base `  K )  =  (
Base `  K )
12 eqid 2435 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
13 eqid 2435 . . 3  |-  ( dist `  K )  =  (
dist `  K )
1410, 11, 12, 13nmfval 18628 . 2  |-  ( norm `  K )  =  ( x  e.  ( Base `  K )  |->  ( x ( dist `  K
) ( 0g `  K ) ) )
15 eqid 2435 . . 3  |-  ( norm `  L )  =  (
norm `  L )
16 eqid 2435 . . 3  |-  ( Base `  L )  =  (
Base `  L )
17 eqid 2435 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
18 eqid 2435 . . 3  |-  ( dist `  L )  =  (
dist `  L )
1915, 16, 17, 18nmfval 18628 . 2  |-  ( norm `  L )  =  ( x  e.  ( Base `  L )  |->  ( x ( dist `  L
) ( 0g `  L ) ) )
209, 14, 193eqtr4g 2492 1  |-  ( ph  ->  ( norm `  K
)  =  ( norm `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   distcds 13530   0gc0g 13715   normcnm 18616
This theorem is referenced by:  sranlm  18712  zlmnm  24342
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
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-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-0g 13719  df-nm 18622
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