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Theorem sspn 27591
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 27581 . . . 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 27515 . . . 4 |- ( W e. NrmCVec -> M : Y --> RR )
62, 5syl 17 . . 3 |- ( ( U e. NrmCVec /\ W e. H ) -> M : Y --> RR )
7 ffn 6045 . . 3 |- ( M : Y --> RR -> M Fn Y )
86, 7syl 17 . 2 |- ( ( U e. NrmCVec /\ W e. H ) -> M Fn Y )
9 eqid 2622 . . . . . 6 |- ( BaseSet ` U ) = ( BaseSet ` U )
10 sspn.n . . . . . 6 |- N = ( normCV ` U )
119, 10nvf 27515 . . . . 5 |- ( U e. NrmCVec -> N : ( BaseSet ` U ) --> RR )
12 ffn 6045 . . . . 5 |- ( N : ( BaseSet ` U ) --> RR -> N Fn ( BaseSet ` U ) )
1311, 12syl 17 . . . 4 |- ( U e. NrmCVec -> N Fn ( BaseSet ` U ) )
1413adantr 481 . . 3 |- ( ( U e. NrmCVec /\ W e. H ) -> N Fn ( BaseSet ` U ) )
159, 3, 1sspba 27582 . . 3 |- ( ( U e. NrmCVec /\ W e. H ) -> Y C_ ( BaseSet ` U ) )
16 fnssres 6004 . . 3 |- ( ( N Fn ( BaseSet ` U ) /\ Y C_ ( BaseSet ` U ) ) -> ( N |` Y ) Fn Y )
1714, 15, 16syl2anc 693 . 2 |- ( ( U e. NrmCVec /\ W e. H ) -> ( N |` Y ) Fn Y )
18 ffun 6048 . . . . . . 7 |- ( N : ( BaseSet ` U ) --> RR -> Fun N )
1911, 18syl 17 . . . . . 6 |- ( U e. NrmCVec -> Fun N )
20 funres 5929 . . . . . 6 |- ( Fun N -> Fun ( N |` Y ) )
2119, 20syl 17 . . . . 5 |- ( U e. NrmCVec -> Fun ( N |` Y ) )
2221ad2antrr 762 . . . 4 |- ( ( ( U e. NrmCVec /\ W e. H ) /\ x e. Y ) -> Fun ( N |` Y ) )
23 fnresdm 6000 . . . . . . 7 |- ( M Fn Y -> ( M |` Y ) = M )
248, 23syl 17 . . . . . 6 |- ( ( U e. NrmCVec /\ W e. H ) -> ( M |` Y ) = M )
25 eqid 2622 . . . . . . . . . 10 |- ( +v ` U ) = ( +v ` U )
26 eqid 2622 . . . . . . . . . 10 |- ( +v ` W ) = ( +v ` W )
27 eqid 2622 . . . . . . . . . 10 |- ( .sOLD ` U ) = ( .sOLD ` U )
28 eqid 2622 . . . . . . . . . 10 |- ( .sOLD ` W ) = ( .sOLD ` W )
2925, 26, 27, 28, 10, 4, 1isssp 27579 . . . . . . . . 9 |- ( U e. NrmCVec -> ( W e. H <-> ( W e. NrmCVec /\ ( ( +v ` W ) C_ ( +v ` U ) /\ ( .sOLD ` W ) C_ ( .sOLD ` U ) /\ M C_ N ) ) ) )
3029simplbda 654 . . . . . . . 8 |- ( ( U e. NrmCVec /\ W e. H ) -> ( ( +v ` W ) C_ ( +v ` U ) /\ ( .sOLD ` W ) C_ ( .sOLD ` U ) /\ M C_ N ) )
3130simp3d 1075 . . . . . . 7 |- ( ( U e. NrmCVec /\ W e. H ) -> M C_ N )
32 ssres 5424 . . . . . . 7 |- ( M C_ N -> ( M |` Y ) C_ ( N |` Y ) )
3331, 32syl 17 . . . . . 6 |- ( ( U e. NrmCVec /\ W e. H ) -> ( M |` Y ) C_ ( N |` Y ) )
3424, 33eqsstr3d 3640 . . . . 5 |- ( ( U e. NrmCVec /\ W e. H ) -> M C_ ( N |` Y ) )
3534adantr 481 . . . 4 |- ( ( ( U e. NrmCVec /\ W e. H ) /\ x e. Y ) -> M C_ ( N |` Y ) )
36 fdm 6051 . . . . . . . 8 |- ( M : Y --> RR -> dom M = Y )
375, 36syl 17 . . . . . . 7 |- ( W e. NrmCVec -> dom M = Y )
3837eleq2d 2687 . . . . . 6 |- ( W e. NrmCVec -> ( x e. dom M <-> x e. Y ) )
3938biimpar 502 . . . . 5 |- ( ( W e. NrmCVec /\ x e. Y ) -> x e. dom M )
402, 39sylan 488 . . . 4 |- ( ( ( U e. NrmCVec /\ W e. H ) /\ x e. Y ) -> x e. dom M )
41 funssfv 6209 . . . 4 |- ( ( Fun ( N |` Y ) /\ M C_ ( N |` Y ) /\ x e. dom M ) -> ( ( N |` Y ) ` x ) = ( M ` x ) )
4222, 35, 40, 41syl3anc 1326 . . 3 |- ( ( ( U e. NrmCVec /\ W e. H ) /\ x e. Y ) -> ( ( N |` Y ) ` x ) = ( M ` x ) )
4342eqcomd 2628 . 2 |- ( ( ( U e. NrmCVec /\ W e. H ) /\ x e. Y ) -> ( M ` x ) = ( ( N |` Y ) ` x ) )
448, 17, 43eqfnfvd 6314 1 |- ( ( U e. NrmCVec /\ W e. H ) -> M = ( N |` Y ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   dom cdm 5114   |` cres 5116   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888   RRcr 9935   NrmCVeccnv 27439   +vcpv 27440   BaseSetcba 27441   .sOLDcns 27442   normCVcnmcv 27445   SubSpcss 27576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-1st 7168  df-2nd 7169  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455  df-ssp 27577
This theorem is referenced by:  sspnval  27592
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