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Theorem nmpropd2 18133
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 2330 . . 3  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
4 nmpropd2.5 . . . . . 6  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
51, 1xpeq12d 4730 . . . . . . 7  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
65reseq2d 4971 . . . . . 6  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
74, 6eqtr3d 2330 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
82, 2xpeq12d 4730 . . . . . 6  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
98reseq2d 4971 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
107, 9eqtr3d 2330 . . . 4  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
11 eqidd 2297 . . . 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 14415 . . . 4  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
1410, 11, 13oveq123d 5895 . . 3  |-  ( ph  ->  ( a ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ( 0g `  K
) )  =  ( a ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) ( 0g `  L ) ) )
153, 14mpteq12dv 4114 . 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 2296 . . . 4  |-  ( norm `  K )  =  (
norm `  K )
18 eqid 2296 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
19 eqid 2296 . . . 4  |-  ( 0g
`  K )  =  ( 0g `  K
)
20 eqid 2296 . . . 4  |-  ( dist `  K )  =  (
dist `  K )
21 eqid 2296 . . . 4  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
2217, 18, 19, 20, 21nmfval2 18129 . . 3  |-  ( K  e.  Grp  ->  ( norm `  K )  =  ( a  e.  (
Base `  K )  |->  ( a ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ( 0g `  K
) ) ) )
2316, 22syl 15 . 2  |-  ( ph  ->  ( norm `  K
)  =  ( a  e.  ( Base `  K
)  |->  ( a ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) ( 0g `  K ) ) ) )
241, 2, 12grppropd 14516 . . . 4  |-  ( ph  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
2516, 24mpbid 201 . . 3  |-  ( ph  ->  L  e.  Grp )
26 eqid 2296 . . . 4  |-  ( norm `  L )  =  (
norm `  L )
27 eqid 2296 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
28 eqid 2296 . . . 4  |-  ( 0g
`  L )  =  ( 0g `  L
)
29 eqid 2296 . . . 4  |-  ( dist `  L )  =  (
dist `  L )
30 eqid 2296 . . . 4  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
3126, 27, 28, 29, 30nmfval2 18129 . . 3  |-  ( L  e.  Grp  ->  ( norm `  L )  =  ( a  e.  (
Base `  L )  |->  ( a ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) ( 0g `  L
) ) ) )
3225, 31syl 15 . 2  |-  ( ph  ->  ( norm `  L
)  =  ( a  e.  ( Base `  L
)  |->  ( a ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) ( 0g `  L ) ) ) )
3315, 23, 323eqtr4d 2338 1  |-  ( ph  ->  ( norm `  K
)  =  ( norm `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    e. cmpt 4093    X. cxp 4703    |` cres 4707   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   distcds 13233   0gc0g 13416   Grpcgrp 14378   normcnm 18115
This theorem is referenced by:  ngppropd  18169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-riota 6320  df-0g 13420  df-mnd 14383  df-grp 14505  df-nm 18121
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