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Theorem nmpropd2 20850
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 2510 . . 3  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
4 nmpropd2.5 . . . . . 6  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
51, 1xpeq12d 5024 . . . . . . 7  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
65reseq2d 5271 . . . . . 6  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
74, 6eqtr3d 2510 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
82, 2xpeq12d 5024 . . . . . 6  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
98reseq2d 5271 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
107, 9eqtr3d 2510 . . . 4  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
11 eqidd 2468 . . . 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 15760 . . . 4  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
1410, 11, 13oveq123d 6303 . . 3  |-  ( ph  ->  ( a ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ( 0g `  K
) )  =  ( a ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) ( 0g `  L ) ) )
153, 14mpteq12dv 4525 . 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 2467 . . . 4  |-  ( norm `  K )  =  (
norm `  K )
18 eqid 2467 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
19 eqid 2467 . . . 4  |-  ( 0g
`  K )  =  ( 0g `  K
)
20 eqid 2467 . . . 4  |-  ( dist `  K )  =  (
dist `  K )
21 eqid 2467 . . . 4  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
2217, 18, 19, 20, 21nmfval2 20846 . . 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 15869 . . . 4  |-  ( ph  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
2516, 24mpbid 210 . . 3  |-  ( ph  ->  L  e.  Grp )
26 eqid 2467 . . . 4  |-  ( norm `  L )  =  (
norm `  L )
27 eqid 2467 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
28 eqid 2467 . . . 4  |-  ( 0g
`  L )  =  ( 0g `  L
)
29 eqid 2467 . . . 4  |-  ( dist `  L )  =  (
dist `  L )
30 eqid 2467 . . . 4  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
3126, 27, 28, 29, 30nmfval2 20846 . . 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 2518 1  |-  ( ph  ->  ( norm `  K
)  =  ( norm `  L ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    |-> cmpt 4505    X. cxp 4997    |` cres 5001   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   distcds 14560   0gc0g 14691   Grpcgrp 15723   normcnm 20832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-riota 6243  df-ov 6285  df-0g 14693  df-mnd 15728  df-grp 15858  df-nm 20838
This theorem is referenced by:  ngppropd  20886
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