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Theorem sspn 22083
Description: The norm on a subspace is a restriction of the norm on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
sspn.y  |-  Y  =  ( BaseSet `  W )
sspn.n  |-  N  =  ( normCV `  U )
sspn.m  |-  M  =  ( normCV `  W )
sspn.h  |-  H  =  ( SubSp `  U )
Assertion
Ref Expression
sspn  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  =  ( N  |`  Y ) )

Proof of Theorem sspn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sspn.h . . . . 5  |-  H  =  ( SubSp `  U )
21sspnv 22073 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  W  e.  NrmCVec )
3 sspn.y . . . . 5  |-  Y  =  ( BaseSet `  W )
4 sspn.m . . . . 5  |-  M  =  ( normCV `  W )
53, 4nvf 21995 . . . 4  |-  ( W  e.  NrmCVec  ->  M : Y --> RR )
62, 5syl 16 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M : Y --> RR )
7 ffn 5531 . . 3  |-  ( M : Y --> RR  ->  M  Fn  Y )
86, 7syl 16 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  Fn  Y )
9 eqid 2387 . . . . . 6  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
10 sspn.n . . . . . 6  |-  N  =  ( normCV `  U )
119, 10nvf 21995 . . . . 5  |-  ( U  e.  NrmCVec  ->  N : (
BaseSet `  U ) --> RR )
12 ffn 5531 . . . . 5  |-  ( N : ( BaseSet `  U
) --> RR  ->  N  Fn  ( BaseSet `  U )
)
1311, 12syl 16 . . . 4  |-  ( U  e.  NrmCVec  ->  N  Fn  ( BaseSet
`  U ) )
1413adantr 452 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  N  Fn  ( BaseSet `  U )
)
159, 3, 1sspba 22074 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  ( BaseSet `  U )
)
16 fnssres 5498 . . 3  |-  ( ( N  Fn  ( BaseSet `  U )  /\  Y  C_  ( BaseSet `  U )
)  ->  ( N  |`  Y )  Fn  Y
)
1714, 15, 16syl2anc 643 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( N  |`  Y )  Fn  Y )
18 ffun 5533 . . . . . . 7  |-  ( N : ( BaseSet `  U
) --> RR  ->  Fun  N )
1911, 18syl 16 . . . . . 6  |-  ( U  e.  NrmCVec  ->  Fun  N )
20 funres 5432 . . . . . 6  |-  ( Fun 
N  ->  Fun  ( N  |`  Y ) )
2119, 20syl 16 . . . . 5  |-  ( U  e.  NrmCVec  ->  Fun  ( N  |`  Y ) )
2221ad2antrr 707 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  Fun  ( N  |`  Y ) )
23 fnresdm 5494 . . . . . . 7  |-  ( M  Fn  Y  ->  ( M  |`  Y )  =  M )
248, 23syl 16 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( M  |`  Y )  =  M )
25 eqid 2387 . . . . . . . . . 10  |-  ( +v
`  U )  =  ( +v `  U
)
26 eqid 2387 . . . . . . . . . 10  |-  ( +v
`  W )  =  ( +v `  W
)
27 eqid 2387 . . . . . . . . . 10  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
28 eqid 2387 . . . . . . . . . 10  |-  ( .s
OLD `  W )  =  ( .s OLD `  W )
2925, 26, 27, 28, 10, 4, 1isssp 22071 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  ( W  e.  H  <->  ( W  e.  NrmCVec 
/\  ( ( +v
`  W )  C_  ( +v `  U )  /\  ( .s OLD `  W )  C_  ( .s OLD `  U )  /\  M  C_  N
) ) ) )
3029simplbda 608 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  (
( +v `  W
)  C_  ( +v `  U )  /\  ( .s OLD `  W ) 
C_  ( .s OLD `  U )  /\  M  C_  N ) )
3130simp3d 971 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  C_  N )
32 ssres 5112 . . . . . . 7  |-  ( M 
C_  N  ->  ( M  |`  Y )  C_  ( N  |`  Y ) )
3331, 32syl 16 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( M  |`  Y )  C_  ( N  |`  Y ) )
3424, 33eqsstr3d 3326 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  C_  ( N  |`  Y ) )
3534adantr 452 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  M  C_  ( N  |`  Y ) )
36 fdm 5535 . . . . . . . 8  |-  ( M : Y --> RR  ->  dom 
M  =  Y )
375, 36syl 16 . . . . . . 7  |-  ( W  e.  NrmCVec  ->  dom  M  =  Y )
3837eleq2d 2454 . . . . . 6  |-  ( W  e.  NrmCVec  ->  ( x  e. 
dom  M  <->  x  e.  Y
) )
3938biimpar 472 . . . . 5  |-  ( ( W  e.  NrmCVec  /\  x  e.  Y )  ->  x  e.  dom  M )
402, 39sylan 458 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  x  e.  dom  M )
41 funssfv 5686 . . . 4  |-  ( ( Fun  ( N  |`  Y )  /\  M  C_  ( N  |`  Y )  /\  x  e.  dom  M )  ->  ( ( N  |`  Y ) `  x )  =  ( M `  x ) )
4222, 35, 40, 41syl3anc 1184 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  ( ( N  |`  Y ) `  x )  =  ( M `  x ) )
4342eqcomd 2392 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  ( M `  x )  =  ( ( N  |`  Y ) `
 x ) )
448, 17, 43eqfnfvd 5769 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  =  ( N  |`  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    C_ wss 3263   dom cdm 4818    |` cres 4820   Fun wfun 5388    Fn wfn 5389   -->wf 5390   ` cfv 5394   RRcr 8922   NrmCVeccnv 21911   +vcpv 21912   BaseSetcba 21913   .s
OLDcns 21914   normCVcnmcv 21917   SubSpcss 22068
This theorem is referenced by:  sspnval  22084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-1st 6288  df-2nd 6289  df-vc 21873  df-nv 21919  df-va 21922  df-ba 21923  df-sm 21924  df-0v 21925  df-nmcv 21927  df-ssp 22069
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