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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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