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Theorem sspn 24255
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 24245 . . . 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 24167 . . . 4  |-  ( W  e.  NrmCVec  ->  M : Y --> RR )
62, 5syl 16 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M : Y --> RR )
7 ffn 5643 . . 3  |-  ( M : Y --> RR  ->  M  Fn  Y )
86, 7syl 16 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  Fn  Y )
9 eqid 2450 . . . . . 6  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
10 sspn.n . . . . . 6  |-  N  =  ( normCV `  U )
119, 10nvf 24167 . . . . 5  |-  ( U  e.  NrmCVec  ->  N : (
BaseSet `  U ) --> RR )
12 ffn 5643 . . . . 5  |-  ( N : ( BaseSet `  U
) --> RR  ->  N  Fn  ( BaseSet `  U )
)
1311, 12syl 16 . . . 4  |-  ( U  e.  NrmCVec  ->  N  Fn  ( BaseSet
`  U ) )
1413adantr 465 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  N  Fn  ( BaseSet `  U )
)
159, 3, 1sspba 24246 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  ( BaseSet `  U )
)
16 fnssres 5608 . . 3  |-  ( ( N  Fn  ( BaseSet `  U )  /\  Y  C_  ( BaseSet `  U )
)  ->  ( N  |`  Y )  Fn  Y
)
1714, 15, 16syl2anc 661 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( N  |`  Y )  Fn  Y )
18 ffun 5645 . . . . . . 7  |-  ( N : ( BaseSet `  U
) --> RR  ->  Fun  N )
1911, 18syl 16 . . . . . 6  |-  ( U  e.  NrmCVec  ->  Fun  N )
20 funres 5541 . . . . . 6  |-  ( Fun 
N  ->  Fun  ( N  |`  Y ) )
2119, 20syl 16 . . . . 5  |-  ( U  e.  NrmCVec  ->  Fun  ( N  |`  Y ) )
2221ad2antrr 725 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  Fun  ( N  |`  Y ) )
23 fnresdm 5604 . . . . . . 7  |-  ( M  Fn  Y  ->  ( M  |`  Y )  =  M )
248, 23syl 16 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( M  |`  Y )  =  M )
25 eqid 2450 . . . . . . . . . 10  |-  ( +v
`  U )  =  ( +v `  U
)
26 eqid 2450 . . . . . . . . . 10  |-  ( +v
`  W )  =  ( +v `  W
)
27 eqid 2450 . . . . . . . . . 10  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
28 eqid 2450 . . . . . . . . . 10  |-  ( .sOLD `  W )  =  ( .sOLD `  W )
2925, 26, 27, 28, 10, 4, 1isssp 24243 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  ( W  e.  H  <->  ( W  e.  NrmCVec 
/\  ( ( +v
`  W )  C_  ( +v `  U )  /\  ( .sOLD `  W )  C_  ( .sOLD `  U )  /\  M  C_  N
) ) ) )
3029simplbda 624 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  (
( +v `  W
)  C_  ( +v `  U )  /\  ( .sOLD `  W ) 
C_  ( .sOLD `  U )  /\  M  C_  N ) )
3130simp3d 1002 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  C_  N )
32 ssres 5220 . . . . . . 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 3475 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  C_  ( N  |`  Y ) )
3534adantr 465 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  M  C_  ( N  |`  Y ) )
36 fdm 5647 . . . . . . . 8  |-  ( M : Y --> RR  ->  dom 
M  =  Y )
375, 36syl 16 . . . . . . 7  |-  ( W  e.  NrmCVec  ->  dom  M  =  Y )
3837eleq2d 2519 . . . . . 6  |-  ( W  e.  NrmCVec  ->  ( x  e. 
dom  M  <->  x  e.  Y
) )
3938biimpar 485 . . . . 5  |-  ( ( W  e.  NrmCVec  /\  x  e.  Y )  ->  x  e.  dom  M )
402, 39sylan 471 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  x  e.  dom  M )
41 funssfv 5790 . . . 4  |-  ( ( Fun  ( N  |`  Y )  /\  M  C_  ( N  |`  Y )  /\  x  e.  dom  M )  ->  ( ( N  |`  Y ) `  x )  =  ( M `  x ) )
4222, 35, 40, 41syl3anc 1219 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  ( ( N  |`  Y ) `  x )  =  ( M `  x ) )
4342eqcomd 2457 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  x  e.  Y
)  ->  ( M `  x )  =  ( ( N  |`  Y ) `
 x ) )
448, 17, 43eqfnfvd 5885 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  =  ( N  |`  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757    C_ wss 3412   dom cdm 4924    |` cres 4926   Fun wfun 5496    Fn wfn 5497   -->wf 5498   ` cfv 5502   RRcr 9368   NrmCVeccnv 24083   +vcpv 24084   BaseSetcba 24085   .sOLDcns 24086   normCVcnmcv 24089   SubSpcss 24240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-ov 6179  df-oprab 6180  df-1st 6663  df-2nd 6664  df-vc 24045  df-nv 24091  df-va 24094  df-ba 24095  df-sm 24096  df-0v 24097  df-nmcv 24099  df-ssp 24241
This theorem is referenced by:  sspnval  24256
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