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Theorem nmpropd 18116
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 2284 . . . 4  |-  ( ph  ->  x  =  x )
4 eqidd 2284 . . . . 5  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  K ) )
5 nmpropd.2 . . . . . 6  |-  ( ph  ->  ( +g  `  K
)  =  ( +g  `  L ) )
65proplem3 13593 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
) )  ->  (
x ( +g  `  K
) y )  =  ( x ( +g  `  L ) y ) )
74, 1, 6grpidpropd 14399 . . . 4  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
82, 3, 7oveq123d 5879 . . 3  |-  ( ph  ->  ( x ( dist `  K ) ( 0g
`  K ) )  =  ( x (
dist `  L )
( 0g `  L
) ) )
91, 8mpteq12dv 4098 . 2  |-  ( ph  ->  ( x  e.  (
Base `  K )  |->  ( x ( dist `  K ) ( 0g
`  K ) ) )  =  ( x  e.  ( Base `  L
)  |->  ( x (
dist `  L )
( 0g `  L
) ) ) )
10 eqid 2283 . . 3  |-  ( norm `  K )  =  (
norm `  K )
11 eqid 2283 . . 3  |-  ( Base `  K )  =  (
Base `  K )
12 eqid 2283 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
13 eqid 2283 . . 3  |-  ( dist `  K )  =  (
dist `  K )
1410, 11, 12, 13nmfval 18111 . 2  |-  ( norm `  K )  =  ( x  e.  ( Base `  K )  |->  ( x ( dist `  K
) ( 0g `  K ) ) )
15 eqid 2283 . . 3  |-  ( norm `  L )  =  (
norm `  L )
16 eqid 2283 . . 3  |-  ( Base `  L )  =  (
Base `  L )
17 eqid 2283 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
18 eqid 2283 . . 3  |-  ( dist `  L )  =  (
dist `  L )
1915, 16, 17, 18nmfval 18111 . 2  |-  ( norm `  L )  =  ( x  e.  ( Base `  L )  |->  ( x ( dist `  L
) ( 0g `  L ) ) )
209, 14, 193eqtr4g 2340 1  |-  ( ph  ->  ( norm `  K
)  =  ( norm `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   distcds 13217   0gc0g 13400   normcnm 18099
This theorem is referenced by:  sranlm  18195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-0g 13404  df-nm 18105
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