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Theorem nmpropd 20304
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 2452 . . . 4  |-  ( ph  ->  x  =  x )
4 eqidd 2452 . . . . 5  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  K ) )
5 nmpropd.2 . . . . . 6  |-  ( ph  ->  ( +g  `  K
)  =  ( +g  `  L ) )
65proplem3 14733 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
) )  ->  (
x ( +g  `  K
) y )  =  ( x ( +g  `  L ) y ) )
74, 1, 6grpidpropd 15551 . . . 4  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
82, 3, 7oveq123d 6213 . . 3  |-  ( ph  ->  ( x ( dist `  K ) ( 0g
`  K ) )  =  ( x (
dist `  L )
( 0g `  L
) ) )
91, 8mpteq12dv 4470 . 2  |-  ( ph  ->  ( x  e.  (
Base `  K )  |->  ( x ( dist `  K ) ( 0g
`  K ) ) )  =  ( x  e.  ( Base `  L
)  |->  ( x (
dist `  L )
( 0g `  L
) ) ) )
10 eqid 2451 . . 3  |-  ( norm `  K )  =  (
norm `  K )
11 eqid 2451 . . 3  |-  ( Base `  K )  =  (
Base `  K )
12 eqid 2451 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
13 eqid 2451 . . 3  |-  ( dist `  K )  =  (
dist `  K )
1410, 11, 12, 13nmfval 20299 . 2  |-  ( norm `  K )  =  ( x  e.  ( Base `  K )  |->  ( x ( dist `  K
) ( 0g `  K ) ) )
15 eqid 2451 . . 3  |-  ( norm `  L )  =  (
norm `  L )
16 eqid 2451 . . 3  |-  ( Base `  L )  =  (
Base `  L )
17 eqid 2451 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
18 eqid 2451 . . 3  |-  ( dist `  L )  =  (
dist `  L )
1915, 16, 17, 18nmfval 20299 . 2  |-  ( norm `  L )  =  ( x  e.  ( Base `  L )  |->  ( x ( dist `  L
) ( 0g `  L ) ) )
209, 14, 193eqtr4g 2517 1  |-  ( ph  ->  ( norm `  K
)  =  ( norm `  L ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    |-> cmpt 4450   ` cfv 5518  (class class class)co 6192   Basecbs 14278   +g cplusg 14342   distcds 14351   0gc0g 14482   normcnm 20287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526  df-ov 6195  df-0g 14484  df-nm 20293
This theorem is referenced by:  sranlm  20383  zlmnm  26531
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