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Theorem nmpropd2 21409
Description: Strong property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
nmpropd2.1  |-  ( ph  ->  B  =  ( Base `  K ) )
nmpropd2.2  |-  ( ph  ->  B  =  ( Base `  L ) )
nmpropd2.3  |-  ( ph  ->  K  e.  Grp )
nmpropd2.4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
nmpropd2.5  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
Assertion
Ref Expression
nmpropd2  |-  ( ph  ->  ( norm `  K
)  =  ( norm `  L ) )
Distinct variable groups:    x, y, B    x, K, y    x, L, y    ph, x, y

Proof of Theorem nmpropd2
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 nmpropd2.1 . . . 4  |-  ( ph  ->  B  =  ( Base `  K ) )
2 nmpropd2.2 . . . 4  |-  ( ph  ->  B  =  ( Base `  L ) )
31, 2eqtr3d 2447 . . 3  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
4 nmpropd2.5 . . . . . 6  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
51sqxpeqd 4851 . . . . . . 7  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
65reseq2d 5096 . . . . . 6  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
74, 6eqtr3d 2447 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
82sqxpeqd 4851 . . . . . 6  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
98reseq2d 5096 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
107, 9eqtr3d 2447 . . . 4  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
11 eqidd 2405 . . . 4  |-  ( ph  ->  a  =  a )
12 nmpropd2.4 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
131, 2, 12grpidpropd 16214 . . . 4  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
1410, 11, 13oveq123d 6301 . . 3  |-  ( ph  ->  ( a ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ( 0g `  K
) )  =  ( a ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) ( 0g `  L ) ) )
153, 14mpteq12dv 4475 . 2  |-  ( ph  ->  ( a  e.  (
Base `  K )  |->  ( a ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ( 0g `  K
) ) )  =  ( a  e.  (
Base `  L )  |->  ( a ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) ( 0g `  L
) ) ) )
16 nmpropd2.3 . . 3  |-  ( ph  ->  K  e.  Grp )
17 eqid 2404 . . . 4  |-  ( norm `  K )  =  (
norm `  K )
18 eqid 2404 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
19 eqid 2404 . . . 4  |-  ( 0g
`  K )  =  ( 0g `  K
)
20 eqid 2404 . . . 4  |-  ( dist `  K )  =  (
dist `  K )
21 eqid 2404 . . . 4  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
2217, 18, 19, 20, 21nmfval2 21405 . . 3  |-  ( K  e.  Grp  ->  ( norm `  K )  =  ( a  e.  (
Base `  K )  |->  ( a ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ( 0g `  K
) ) ) )
2316, 22syl 17 . 2  |-  ( ph  ->  ( norm `  K
)  =  ( a  e.  ( Base `  K
)  |->  ( a ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) ( 0g `  K ) ) ) )
241, 2, 12grppropd 16394 . . . 4  |-  ( ph  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
2516, 24mpbid 212 . . 3  |-  ( ph  ->  L  e.  Grp )
26 eqid 2404 . . . 4  |-  ( norm `  L )  =  (
norm `  L )
27 eqid 2404 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
28 eqid 2404 . . . 4  |-  ( 0g
`  L )  =  ( 0g `  L
)
29 eqid 2404 . . . 4  |-  ( dist `  L )  =  (
dist `  L )
30 eqid 2404 . . . 4  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
3126, 27, 28, 29, 30nmfval2 21405 . . 3  |-  ( L  e.  Grp  ->  ( norm `  L )  =  ( a  e.  (
Base `  L )  |->  ( a ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) ( 0g `  L
) ) ) )
3225, 31syl 17 . 2  |-  ( ph  ->  ( norm `  L
)  =  ( a  e.  ( Base `  L
)  |->  ( a ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) ( 0g `  L ) ) ) )
3315, 23, 323eqtr4d 2455 1  |-  ( ph  ->  ( norm `  K
)  =  ( norm `  L ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844    |-> cmpt 4455    X. cxp 4823    |` cres 4827   ` cfv 5571  (class class class)co 6280   Basecbs 14843   +g cplusg 14911   distcds 14920   0gc0g 15056   Grpcgrp 16379   normcnm 21391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-fv 5579  df-riota 6242  df-ov 6283  df-0g 15058  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-grp 16383  df-nm 21397
This theorem is referenced by:  ngppropd  21445
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