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Theorem nmpropd2 20303
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 2494 . . 3  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
4 nmpropd2.5 . . . . . 6  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
51, 1xpeq12d 4963 . . . . . . 7  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
65reseq2d 5208 . . . . . 6  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
74, 6eqtr3d 2494 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
82, 2xpeq12d 4963 . . . . . 6  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
98reseq2d 5208 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
107, 9eqtr3d 2494 . . . 4  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
11 eqidd 2452 . . . 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 15549 . . . 4  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
1410, 11, 13oveq123d 6211 . . 3  |-  ( ph  ->  ( a ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ( 0g `  K
) )  =  ( a ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) ( 0g `  L ) ) )
153, 14mpteq12dv 4468 . 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 2451 . . . 4  |-  ( norm `  K )  =  (
norm `  K )
18 eqid 2451 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
19 eqid 2451 . . . 4  |-  ( 0g
`  K )  =  ( 0g `  K
)
20 eqid 2451 . . . 4  |-  ( dist `  K )  =  (
dist `  K )
21 eqid 2451 . . . 4  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
2217, 18, 19, 20, 21nmfval2 20299 . . 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 15658 . . . 4  |-  ( ph  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
2516, 24mpbid 210 . . 3  |-  ( ph  ->  L  e.  Grp )
26 eqid 2451 . . . 4  |-  ( norm `  L )  =  (
norm `  L )
27 eqid 2451 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
28 eqid 2451 . . . 4  |-  ( 0g
`  L )  =  ( 0g `  L
)
29 eqid 2451 . . . 4  |-  ( dist `  L )  =  (
dist `  L )
30 eqid 2451 . . . 4  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
3126, 27, 28, 29, 30nmfval2 20299 . . 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 2502 1  |-  ( ph  ->  ( norm `  K
)  =  ( norm `  L ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    |-> cmpt 4448    X. cxp 4936    |` cres 4940   ` cfv 5516  (class class class)co 6190   Basecbs 14276   +g cplusg 14340   distcds 14349   0gc0g 14480   Grpcgrp 15512   normcnm 20285
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 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
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-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-fv 5524  df-riota 6151  df-ov 6193  df-0g 14482  df-mnd 15517  df-grp 15647  df-nm 20291
This theorem is referenced by:  ngppropd  20339
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