MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssps Structured version   Unicode version

Theorem ssps 25770
Description: Scalar multiplication on a subspace is a restriction of scalar multiplication on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
ssps.y  |-  Y  =  ( BaseSet `  W )
ssps.s  |-  S  =  ( .sOLD `  U )
ssps.r  |-  R  =  ( .sOLD `  W )
ssps.h  |-  H  =  ( SubSp `  U )
Assertion
Ref Expression
ssps  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  R  =  ( S  |`  ( CC  X.  Y
) ) )

Proof of Theorem ssps
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2457 . . . . . . . . . . 11  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 ssps.s . . . . . . . . . . 11  |-  S  =  ( .sOLD `  U )
31, 2nvsf 25639 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  S : ( CC  X.  ( BaseSet `  U ) ) --> (
BaseSet `  U ) )
4 ffun 5739 . . . . . . . . . 10  |-  ( S : ( CC  X.  ( BaseSet `  U )
) --> ( BaseSet `  U
)  ->  Fun  S )
53, 4syl 16 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  Fun  S )
6 funres 5633 . . . . . . . . 9  |-  ( Fun 
S  ->  Fun  ( S  |`  ( CC  X.  Y
) ) )
75, 6syl 16 . . . . . . . 8  |-  ( U  e.  NrmCVec  ->  Fun  ( S  |`  ( CC  X.  Y
) ) )
87adantr 465 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Fun  ( S  |`  ( CC 
X.  Y ) ) )
9 ssps.h . . . . . . . . . 10  |-  H  =  ( SubSp `  U )
109sspnv 25766 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  W  e.  NrmCVec )
11 ssps.y . . . . . . . . . 10  |-  Y  =  ( BaseSet `  W )
12 ssps.r . . . . . . . . . 10  |-  R  =  ( .sOLD `  W )
1311, 12nvsf 25639 . . . . . . . . 9  |-  ( W  e.  NrmCVec  ->  R : ( CC  X.  Y ) --> Y )
1410, 13syl 16 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  R : ( CC  X.  Y ) --> Y )
15 ffn 5737 . . . . . . . 8  |-  ( R : ( CC  X.  Y ) --> Y  ->  R  Fn  ( CC  X.  Y ) )
1614, 15syl 16 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  R  Fn  ( CC  X.  Y
) )
17 fnresdm 5696 . . . . . . . . 9  |-  ( R  Fn  ( CC  X.  Y )  ->  ( R  |`  ( CC  X.  Y ) )  =  R )
1816, 17syl 16 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( R  |`  ( CC  X.  Y ) )  =  R )
19 eqid 2457 . . . . . . . . . . . 12  |-  ( +v
`  U )  =  ( +v `  U
)
20 eqid 2457 . . . . . . . . . . . 12  |-  ( +v
`  W )  =  ( +v `  W
)
21 eqid 2457 . . . . . . . . . . . 12  |-  ( normCV `  U )  =  (
normCV
`  U )
22 eqid 2457 . . . . . . . . . . . 12  |-  ( normCV `  W )  =  (
normCV
`  W )
2319, 20, 2, 12, 21, 22, 9isssp 25764 . . . . . . . . . . 11  |-  ( U  e.  NrmCVec  ->  ( W  e.  H  <->  ( W  e.  NrmCVec 
/\  ( ( +v
`  W )  C_  ( +v `  U )  /\  R  C_  S  /\  ( normCV `  W )  C_  ( normCV `  U ) ) ) ) )
2423simplbda 624 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  (
( +v `  W
)  C_  ( +v `  U )  /\  R  C_  S  /\  ( normCV `  W )  C_  ( normCV `  U ) ) )
2524simp2d 1009 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  R  C_  S )
26 ssres 5309 . . . . . . . . 9  |-  ( R 
C_  S  ->  ( R  |`  ( CC  X.  Y ) )  C_  ( S  |`  ( CC 
X.  Y ) ) )
2725, 26syl 16 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( R  |`  ( CC  X.  Y ) )  C_  ( S  |`  ( CC 
X.  Y ) ) )
2818, 27eqsstr3d 3534 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  R  C_  ( S  |`  ( CC  X.  Y ) ) )
298, 16, 283jca 1176 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( Fun  ( S  |`  ( CC  X.  Y ) )  /\  R  Fn  ( CC  X.  Y )  /\  R  C_  ( S  |`  ( CC  X.  Y
) ) ) )
30 oprssov 6443 . . . . . 6  |-  ( ( ( Fun  ( S  |`  ( CC  X.  Y
) )  /\  R  Fn  ( CC  X.  Y
)  /\  R  C_  ( S  |`  ( CC  X.  Y ) ) )  /\  ( x  e.  CC  /\  y  e.  Y ) )  -> 
( x ( S  |`  ( CC  X.  Y
) ) y )  =  ( x R y ) )
3129, 30sylan 471 . . . . 5  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( x  e.  CC  /\  y  e.  Y ) )  ->  ( x
( S  |`  ( CC  X.  Y ) ) y )  =  ( x R y ) )
3231eqcomd 2465 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( x  e.  CC  /\  y  e.  Y ) )  ->  ( x R y )  =  ( x ( S  |`  ( CC  X.  Y
) ) y ) )
3332ralrimivva 2878 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  A. x  e.  CC  A. y  e.  Y  ( x R y )  =  ( x ( S  |`  ( CC  X.  Y
) ) y ) )
34 eqid 2457 . . 3  |-  ( CC 
X.  Y )  =  ( CC  X.  Y
)
3533, 34jctil 537 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  (
( CC  X.  Y
)  =  ( CC 
X.  Y )  /\  A. x  e.  CC  A. y  e.  Y  (
x R y )  =  ( x ( S  |`  ( CC  X.  Y ) ) y ) ) )
36 ffn 5737 . . . . . 6  |-  ( S : ( CC  X.  ( BaseSet `  U )
) --> ( BaseSet `  U
)  ->  S  Fn  ( CC  X.  ( BaseSet
`  U ) ) )
373, 36syl 16 . . . . 5  |-  ( U  e.  NrmCVec  ->  S  Fn  ( CC  X.  ( BaseSet `  U
) ) )
3837adantr 465 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  S  Fn  ( CC  X.  ( BaseSet
`  U ) ) )
39 ssid 3518 . . . . 5  |-  CC  C_  CC
401, 11, 9sspba 25767 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  ( BaseSet `  U )
)
41 xpss12 5117 . . . . 5  |-  ( ( CC  C_  CC  /\  Y  C_  ( BaseSet `  U )
)  ->  ( CC  X.  Y )  C_  ( CC  X.  ( BaseSet `  U
) ) )
4239, 40, 41sylancr 663 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( CC  X.  Y )  C_  ( CC  X.  ( BaseSet
`  U ) ) )
43 fnssres 5700 . . . 4  |-  ( ( S  Fn  ( CC 
X.  ( BaseSet `  U
) )  /\  ( CC  X.  Y )  C_  ( CC  X.  ( BaseSet
`  U ) ) )  ->  ( S  |`  ( CC  X.  Y
) )  Fn  ( CC  X.  Y ) )
4438, 42, 43syl2anc 661 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( S  |`  ( CC  X.  Y ) )  Fn  ( CC  X.  Y
) )
45 eqfnov 6407 . . 3  |-  ( ( R  Fn  ( CC 
X.  Y )  /\  ( S  |`  ( CC 
X.  Y ) )  Fn  ( CC  X.  Y ) )  -> 
( R  =  ( S  |`  ( CC  X.  Y ) )  <->  ( ( CC  X.  Y )  =  ( CC  X.  Y
)  /\  A. x  e.  CC  A. y  e.  Y  ( x R y )  =  ( x ( S  |`  ( CC  X.  Y
) ) y ) ) ) )
4616, 44, 45syl2anc 661 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( R  =  ( S  |`  ( CC  X.  Y
) )  <->  ( ( CC  X.  Y )  =  ( CC  X.  Y
)  /\  A. x  e.  CC  A. y  e.  Y  ( x R y )  =  ( x ( S  |`  ( CC  X.  Y
) ) y ) ) ) )
4735, 46mpbird 232 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  R  =  ( S  |`  ( CC  X.  Y
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807    C_ wss 3471    X. cxp 5006    |` cres 5010   Fun wfun 5588    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   NrmCVeccnv 25604   +vcpv 25605   BaseSetcba 25606   .sOLDcns 25607   normCVcnmcv 25610   SubSpcss 25761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-1st 6799  df-2nd 6800  df-vc 25566  df-nv 25612  df-va 25615  df-ba 25616  df-sm 25617  df-0v 25618  df-nmcv 25620  df-ssp 25762
This theorem is referenced by:  sspsval  25771
  Copyright terms: Public domain W3C validator