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Theorem sspn 25766
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 25756 . . . 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 25678 . . . 4  |-  ( W  e.  NrmCVec  ->  M : Y --> RR )
62, 5syl 16 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M : Y --> RR )
7 ffn 5639 . . 3  |-  ( M : Y --> RR  ->  M  Fn  Y )
86, 7syl 16 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  Fn  Y )
9 eqid 2382 . . . . . 6  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
10 sspn.n . . . . . 6  |-  N  =  ( normCV `  U )
119, 10nvf 25678 . . . . 5  |-  ( U  e.  NrmCVec  ->  N : (
BaseSet `  U ) --> RR )
12 ffn 5639 . . . . 5  |-  ( N : ( BaseSet `  U
) --> RR  ->  N  Fn  ( BaseSet `  U )
)
1311, 12syl 16 . . . 4  |-  ( U  e.  NrmCVec  ->  N  Fn  ( BaseSet
`  U ) )
1413adantr 463 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  N  Fn  ( BaseSet `  U )
)
159, 3, 1sspba 25757 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  ( BaseSet `  U )
)
16 fnssres 5602 . . 3  |-  ( ( N  Fn  ( BaseSet `  U )  /\  Y  C_  ( BaseSet `  U )
)  ->  ( N  |`  Y )  Fn  Y
)
1714, 15, 16syl2anc 659 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( N  |`  Y )  Fn  Y )
18 ffun 5641 . . . . . . 7  |-  ( N : ( BaseSet `  U
) --> RR  ->  Fun  N )
1911, 18syl 16 . . . . . 6  |-  ( U  e.  NrmCVec  ->  Fun  N )
20 funres 5535 . . . . . 6  |-  ( Fun 
N  ->  Fun  ( N  |`  Y ) )
2119, 20syl 16 . . . . 5  |-  ( U  e.  NrmCVec  ->  Fun  ( N  |`  Y ) )
2221ad2antrr 723 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  Fun  ( N  |`  Y ) )
23 fnresdm 5598 . . . . . . 7  |-  ( M  Fn  Y  ->  ( M  |`  Y )  =  M )
248, 23syl 16 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( M  |`  Y )  =  M )
25 eqid 2382 . . . . . . . . . 10  |-  ( +v
`  U )  =  ( +v `  U
)
26 eqid 2382 . . . . . . . . . 10  |-  ( +v
`  W )  =  ( +v `  W
)
27 eqid 2382 . . . . . . . . . 10  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
28 eqid 2382 . . . . . . . . . 10  |-  ( .sOLD `  W )  =  ( .sOLD `  W )
2925, 26, 27, 28, 10, 4, 1isssp 25754 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  ( W  e.  H  <->  ( W  e.  NrmCVec 
/\  ( ( +v
`  W )  C_  ( +v `  U )  /\  ( .sOLD `  W )  C_  ( .sOLD `  U )  /\  M  C_  N
) ) ) )
3029simplbda 622 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  (
( +v `  W
)  C_  ( +v `  U )  /\  ( .sOLD `  W ) 
C_  ( .sOLD `  U )  /\  M  C_  N ) )
3130simp3d 1008 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  C_  N )
32 ssres 5211 . . . . . . 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 3452 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  C_  ( N  |`  Y ) )
3534adantr 463 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  M  C_  ( N  |`  Y ) )
36 fdm 5643 . . . . . . . 8  |-  ( M : Y --> RR  ->  dom 
M  =  Y )
375, 36syl 16 . . . . . . 7  |-  ( W  e.  NrmCVec  ->  dom  M  =  Y )
3837eleq2d 2452 . . . . . 6  |-  ( W  e.  NrmCVec  ->  ( x  e. 
dom  M  <->  x  e.  Y
) )
3938biimpar 483 . . . . 5  |-  ( ( W  e.  NrmCVec  /\  x  e.  Y )  ->  x  e.  dom  M )
402, 39sylan 469 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  x  e.  dom  M )
41 funssfv 5789 . . . 4  |-  ( ( Fun  ( N  |`  Y )  /\  M  C_  ( N  |`  Y )  /\  x  e.  dom  M )  ->  ( ( N  |`  Y ) `  x )  =  ( M `  x ) )
4222, 35, 40, 41syl3anc 1226 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  ( ( N  |`  Y ) `  x )  =  ( M `  x ) )
4342eqcomd 2390 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  ( M `  x )  =  ( ( N  |`  Y ) `
 x ) )
448, 17, 43eqfnfvd 5886 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  =  ( N  |`  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    C_ wss 3389   dom cdm 4913    |` cres 4915   Fun wfun 5490    Fn wfn 5491   -->wf 5492   ` cfv 5496   RRcr 9402   NrmCVeccnv 25594   +vcpv 25595   BaseSetcba 25596   .sOLDcns 25597   normCVcnmcv 25600   SubSpcss 25751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-1st 6699  df-2nd 6700  df-vc 25556  df-nv 25602  df-va 25605  df-ba 25606  df-sm 25607  df-0v 25608  df-nmcv 25610  df-ssp 25752
This theorem is referenced by:  sspnval  25767
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