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Theorem sspn 21312
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 21302 . . . 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 21224 . . . 4  |-  ( W  e.  NrmCVec  ->  M : Y --> RR )
62, 5syl 15 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M : Y --> RR )
7 ffn 5389 . . 3  |-  ( M : Y --> RR  ->  M  Fn  Y )
86, 7syl 15 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  Fn  Y )
9 eqid 2283 . . . . . 6  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
10 sspn.n . . . . . 6  |-  N  =  ( normCV `  U )
119, 10nvf 21224 . . . . 5  |-  ( U  e.  NrmCVec  ->  N : (
BaseSet `  U ) --> RR )
12 ffn 5389 . . . . 5  |-  ( N : ( BaseSet `  U
) --> RR  ->  N  Fn  ( BaseSet `  U )
)
1311, 12syl 15 . . . 4  |-  ( U  e.  NrmCVec  ->  N  Fn  ( BaseSet
`  U ) )
1413adantr 451 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  N  Fn  ( BaseSet `  U )
)
159, 3, 1sspba 21303 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  ( BaseSet `  U )
)
16 fnssres 5357 . . 3  |-  ( ( N  Fn  ( BaseSet `  U )  /\  Y  C_  ( BaseSet `  U )
)  ->  ( N  |`  Y )  Fn  Y
)
1714, 15, 16syl2anc 642 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( N  |`  Y )  Fn  Y )
18 ffun 5391 . . . . . . 7  |-  ( N : ( BaseSet `  U
) --> RR  ->  Fun  N )
1911, 18syl 15 . . . . . 6  |-  ( U  e.  NrmCVec  ->  Fun  N )
20 funres 5293 . . . . . 6  |-  ( Fun 
N  ->  Fun  ( N  |`  Y ) )
2119, 20syl 15 . . . . 5  |-  ( U  e.  NrmCVec  ->  Fun  ( N  |`  Y ) )
2221ad2antrr 706 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  Fun  ( N  |`  Y ) )
23 fnresdm 5353 . . . . . . 7  |-  ( M  Fn  Y  ->  ( M  |`  Y )  =  M )
248, 23syl 15 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( M  |`  Y )  =  M )
25 eqid 2283 . . . . . . . . . 10  |-  ( +v
`  U )  =  ( +v `  U
)
26 eqid 2283 . . . . . . . . . 10  |-  ( +v
`  W )  =  ( +v `  W
)
27 eqid 2283 . . . . . . . . . 10  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
28 eqid 2283 . . . . . . . . . 10  |-  ( .s
OLD `  W )  =  ( .s OLD `  W )
2925, 26, 27, 28, 10, 4, 1isssp 21300 . . . . . . . . 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 607 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  (
( +v `  W
)  C_  ( +v `  U )  /\  ( .s OLD `  W ) 
C_  ( .s OLD `  U )  /\  M  C_  N ) )
3130simp3d 969 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  C_  N )
32 ssres 4981 . . . . . . 7  |-  ( M 
C_  N  ->  ( M  |`  Y )  C_  ( N  |`  Y ) )
3331, 32syl 15 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( M  |`  Y )  C_  ( N  |`  Y ) )
3424, 33eqsstr3d 3213 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  C_  ( N  |`  Y ) )
3534adantr 451 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  M  C_  ( N  |`  Y ) )
36 fdm 5393 . . . . . . . 8  |-  ( M : Y --> RR  ->  dom 
M  =  Y )
375, 36syl 15 . . . . . . 7  |-  ( W  e.  NrmCVec  ->  dom  M  =  Y )
3837eleq2d 2350 . . . . . 6  |-  ( W  e.  NrmCVec  ->  ( x  e. 
dom  M  <->  x  e.  Y
) )
3938biimpar 471 . . . . 5  |-  ( ( W  e.  NrmCVec  /\  x  e.  Y )  ->  x  e.  dom  M )
402, 39sylan 457 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  x  e.  dom  M )
41 funssfv 5543 . . . 4  |-  ( ( Fun  ( N  |`  Y )  /\  M  C_  ( N  |`  Y )  /\  x  e.  dom  M )  ->  ( ( N  |`  Y ) `  x )  =  ( M `  x ) )
4222, 35, 40, 41syl3anc 1182 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  ( ( N  |`  Y ) `  x )  =  ( M `  x ) )
4342eqcomd 2288 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  ( M `  x )  =  ( ( N  |`  Y ) `
 x ) )
448, 17, 43eqfnfvd 5625 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  =  ( N  |`  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   dom cdm 4689    |` cres 4691   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255   RRcr 8736   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143   normCVcnmcv 21146   SubSpcss 21297
This theorem is referenced by:  sspnval  21313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-1st 6122  df-2nd 6123  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156  df-ssp 21298
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