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Theorem sspn 22077
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 22067 . . . 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 21989 . . . 4  |-  ( W  e.  NrmCVec  ->  M : Y --> RR )
62, 5syl 16 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M : Y --> RR )
7 ffn 5525 . . 3  |-  ( M : Y --> RR  ->  M  Fn  Y )
86, 7syl 16 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  Fn  Y )
9 eqid 2381 . . . . . 6  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
10 sspn.n . . . . . 6  |-  N  =  ( normCV `  U )
119, 10nvf 21989 . . . . 5  |-  ( U  e.  NrmCVec  ->  N : (
BaseSet `  U ) --> RR )
12 ffn 5525 . . . . 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 22068 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  ( BaseSet `  U )
)
16 fnssres 5492 . . 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 5527 . . . . . . 7  |-  ( N : ( BaseSet `  U
) --> RR  ->  Fun  N )
1911, 18syl 16 . . . . . 6  |-  ( U  e.  NrmCVec  ->  Fun  N )
20 funres 5426 . . . . . 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 5488 . . . . . . 7  |-  ( M  Fn  Y  ->  ( M  |`  Y )  =  M )
248, 23syl 16 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( M  |`  Y )  =  M )
25 eqid 2381 . . . . . . . . . 10  |-  ( +v
`  U )  =  ( +v `  U
)
26 eqid 2381 . . . . . . . . . 10  |-  ( +v
`  W )  =  ( +v `  W
)
27 eqid 2381 . . . . . . . . . 10  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
28 eqid 2381 . . . . . . . . . 10  |-  ( .s
OLD `  W )  =  ( .s OLD `  W )
2925, 26, 27, 28, 10, 4, 1isssp 22065 . . . . . . . . 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 5106 . . . . . . 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 3320 . . . . 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 5529 . . . . . . . 8  |-  ( M : Y --> RR  ->  dom 
M  =  Y )
375, 36syl 16 . . . . . . 7  |-  ( W  e.  NrmCVec  ->  dom  M  =  Y )
3837eleq2d 2448 . . . . . 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 5680 . . . 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 2386 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  ( M `  x )  =  ( ( N  |`  Y ) `
 x ) )
448, 17, 43eqfnfvd 5763 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 3257   dom cdm 4812    |` cres 4814   Fun wfun 5382    Fn wfn 5383   -->wf 5384   ` cfv 5388   RRcr 8916   NrmCVeccnv 21905   +vcpv 21906   BaseSetcba 21907   .s
OLDcns 21908   normCVcnmcv 21911   SubSpcss 22062
This theorem is referenced by:  sspnval  22078
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 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635
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 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-reu 2650  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-op 3760  df-uni 3952  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-id 4433  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-1st 6282  df-2nd 6283  df-vc 21867  df-nv 21913  df-va 21916  df-ba 21917  df-sm 21918  df-0v 21919  df-nmcv 21921  df-ssp 22063
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