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Theorem nmpropd 20842
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 2461 . . . 4  |-  ( ph  ->  x  =  x )
4 eqidd 2461 . . . . 5  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  K ) )
5 nmpropd.2 . . . . . 6  |-  ( ph  ->  ( +g  `  K
)  =  ( +g  `  L ) )
65proplem3 14935 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
) )  ->  (
x ( +g  `  K
) y )  =  ( x ( +g  `  L ) y ) )
74, 1, 6grpidpropd 15753 . . . 4  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
82, 3, 7oveq123d 6296 . . 3  |-  ( ph  ->  ( x ( dist `  K ) ( 0g
`  K ) )  =  ( x (
dist `  L )
( 0g `  L
) ) )
91, 8mpteq12dv 4518 . 2  |-  ( ph  ->  ( x  e.  (
Base `  K )  |->  ( x ( dist `  K ) ( 0g
`  K ) ) )  =  ( x  e.  ( Base `  L
)  |->  ( x (
dist `  L )
( 0g `  L
) ) ) )
10 eqid 2460 . . 3  |-  ( norm `  K )  =  (
norm `  K )
11 eqid 2460 . . 3  |-  ( Base `  K )  =  (
Base `  K )
12 eqid 2460 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
13 eqid 2460 . . 3  |-  ( dist `  K )  =  (
dist `  K )
1410, 11, 12, 13nmfval 20837 . 2  |-  ( norm `  K )  =  ( x  e.  ( Base `  K )  |->  ( x ( dist `  K
) ( 0g `  K ) ) )
15 eqid 2460 . . 3  |-  ( norm `  L )  =  (
norm `  L )
16 eqid 2460 . . 3  |-  ( Base `  L )  =  (
Base `  L )
17 eqid 2460 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
18 eqid 2460 . . 3  |-  ( dist `  L )  =  (
dist `  L )
1915, 16, 17, 18nmfval 20837 . 2  |-  ( norm `  L )  =  ( x  e.  ( Base `  L )  |->  ( x ( dist `  L
) ( 0g `  L ) ) )
209, 14, 193eqtr4g 2526 1  |-  ( ph  ->  ( norm `  K
)  =  ( norm `  L ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   distcds 14553   0gc0g 14684   normcnm 20825
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 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
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 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-0g 14686  df-nm 20831
This theorem is referenced by:  sranlm  20921  zlmnm  27433
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