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Theorem nmpropd 20987
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 2444 . . . 4  |-  ( ph  ->  x  =  x )
4 eqidd 2444 . . . . 5  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  K ) )
5 nmpropd.2 . . . . . 6  |-  ( ph  ->  ( +g  `  K
)  =  ( +g  `  L ) )
65oveqdr 6305 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
) )  ->  (
x ( +g  `  K
) y )  =  ( x ( +g  `  L ) y ) )
74, 1, 6grpidpropd 15762 . . . 4  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
82, 3, 7oveq123d 6302 . . 3  |-  ( ph  ->  ( x ( dist `  K ) ( 0g
`  K ) )  =  ( x (
dist `  L )
( 0g `  L
) ) )
91, 8mpteq12dv 4515 . 2  |-  ( ph  ->  ( x  e.  (
Base `  K )  |->  ( x ( dist `  K ) ( 0g
`  K ) ) )  =  ( x  e.  ( Base `  L
)  |->  ( x (
dist `  L )
( 0g `  L
) ) ) )
10 eqid 2443 . . 3  |-  ( norm `  K )  =  (
norm `  K )
11 eqid 2443 . . 3  |-  ( Base `  K )  =  (
Base `  K )
12 eqid 2443 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
13 eqid 2443 . . 3  |-  ( dist `  K )  =  (
dist `  K )
1410, 11, 12, 13nmfval 20982 . 2  |-  ( norm `  K )  =  ( x  e.  ( Base `  K )  |->  ( x ( dist `  K
) ( 0g `  K ) ) )
15 eqid 2443 . . 3  |-  ( norm `  L )  =  (
norm `  L )
16 eqid 2443 . . 3  |-  ( Base `  L )  =  (
Base `  L )
17 eqid 2443 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
18 eqid 2443 . . 3  |-  ( dist `  L )  =  (
dist `  L )
1915, 16, 17, 18nmfval 20982 . 2  |-  ( norm `  L )  =  ( x  e.  ( Base `  L )  |->  ( x ( dist `  L
) ( 0g `  L ) ) )
209, 14, 193eqtr4g 2509 1  |-  ( ph  ->  ( norm `  K
)  =  ( norm `  L ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804    |-> cmpt 4495   ` cfv 5578  (class class class)co 6281   Basecbs 14509   +g cplusg 14574   distcds 14583   0gc0g 14714   normcnm 20970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284  df-0g 14716  df-nm 20976
This theorem is referenced by:  sranlm  21066  zlmnm  27820
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