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Theorem nmpropd2 18644
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 2472 . . 3  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
4 nmpropd2.5 . . . . . 6  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
51, 1xpeq12d 4905 . . . . . . 7  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
65reseq2d 5148 . . . . . 6  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
74, 6eqtr3d 2472 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
82, 2xpeq12d 4905 . . . . . 6  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
98reseq2d 5148 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
107, 9eqtr3d 2472 . . . 4  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
11 eqidd 2439 . . . 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 14724 . . . 4  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
1410, 11, 13oveq123d 6104 . . 3  |-  ( ph  ->  ( a ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ( 0g `  K
) )  =  ( a ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) ( 0g `  L ) ) )
153, 14mpteq12dv 4289 . 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 2438 . . . 4  |-  ( norm `  K )  =  (
norm `  K )
18 eqid 2438 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
19 eqid 2438 . . . 4  |-  ( 0g
`  K )  =  ( 0g `  K
)
20 eqid 2438 . . . 4  |-  ( dist `  K )  =  (
dist `  K )
21 eqid 2438 . . . 4  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
2217, 18, 19, 20, 21nmfval2 18640 . . 3  |-  ( K  e.  Grp  ->  ( norm `  K )  =  ( a  e.  (
Base `  K )  |->  ( a ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ( 0g `  K
) ) ) )
2316, 22syl 16 . 2  |-  ( ph  ->  ( norm `  K
)  =  ( a  e.  ( Base `  K
)  |->  ( a ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) ( 0g `  K ) ) ) )
241, 2, 12grppropd 14825 . . . 4  |-  ( ph  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
2516, 24mpbid 203 . . 3  |-  ( ph  ->  L  e.  Grp )
26 eqid 2438 . . . 4  |-  ( norm `  L )  =  (
norm `  L )
27 eqid 2438 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
28 eqid 2438 . . . 4  |-  ( 0g
`  L )  =  ( 0g `  L
)
29 eqid 2438 . . . 4  |-  ( dist `  L )  =  (
dist `  L )
30 eqid 2438 . . . 4  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
3126, 27, 28, 29, 30nmfval2 18640 . . 3  |-  ( L  e.  Grp  ->  ( norm `  L )  =  ( a  e.  (
Base `  L )  |->  ( a ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) ( 0g `  L
) ) ) )
3225, 31syl 16 . 2  |-  ( ph  ->  ( norm `  L
)  =  ( a  e.  ( Base `  L
)  |->  ( a ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) ( 0g `  L ) ) ) )
3315, 23, 323eqtr4d 2480 1  |-  ( ph  ->  ( norm `  K
)  =  ( norm `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    e. cmpt 4268    X. cxp 4878    |` cres 4882   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   distcds 13540   0gc0g 13725   Grpcgrp 14687   normcnm 18626
This theorem is referenced by:  ngppropd  18680
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-riota 6551  df-0g 13729  df-mnd 14692  df-grp 14814  df-nm 18632
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