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Theorem ssps 22229
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  =  ( .s OLD `  U
)
ssps.r  |-  R  =  ( .s OLD `  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 2436 . . . . . . . . . . 11  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 ssps.s . . . . . . . . . . 11  |-  S  =  ( .s OLD `  U
)
31, 2nvsf 22098 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  S : ( CC  X.  ( BaseSet `  U ) ) --> (
BaseSet `  U ) )
4 ffun 5593 . . . . . . . . . 10  |-  ( S : ( CC  X.  ( BaseSet `  U )
) --> ( BaseSet `  U
)  ->  Fun  S )
53, 4syl 16 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  Fun  S )
6 funres 5492 . . . . . . . . 9  |-  ( Fun 
S  ->  Fun  ( S  |`  ( CC  X.  Y
) ) )
75, 6syl 16 . . . . . . . 8  |-  ( U  e.  NrmCVec  ->  Fun  ( S  |`  ( CC  X.  Y
) ) )
87adantr 452 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Fun  ( S  |`  ( CC 
X.  Y ) ) )
9 ssps.h . . . . . . . . . 10  |-  H  =  ( SubSp `  U )
109sspnv 22225 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  W  e.  NrmCVec )
11 ssps.y . . . . . . . . . 10  |-  Y  =  ( BaseSet `  W )
12 ssps.r . . . . . . . . . 10  |-  R  =  ( .s OLD `  W
)
1311, 12nvsf 22098 . . . . . . . . 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 5591 . . . . . . . 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 5554 . . . . . . . . 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 2436 . . . . . . . . . . . 12  |-  ( +v
`  U )  =  ( +v `  U
)
20 eqid 2436 . . . . . . . . . . . 12  |-  ( +v
`  W )  =  ( +v `  W
)
21 eqid 2436 . . . . . . . . . . . 12  |-  ( normCV `  U )  =  (
normCV
`  U )
22 eqid 2436 . . . . . . . . . . . 12  |-  ( normCV `  W )  =  (
normCV
`  W )
2319, 20, 2, 12, 21, 22, 9isssp 22223 . . . . . . . . . . 11  |-  ( U  e.  NrmCVec  ->  ( W  e.  H  <->  ( W  e.  NrmCVec 
/\  ( ( +v
`  W )  C_  ( +v `  U )  /\  R  C_  S  /\  ( normCV `  W )  C_  ( normCV `  U ) ) ) ) )
2423simplbda 608 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  (
( +v `  W
)  C_  ( +v `  U )  /\  R  C_  S  /\  ( normCV `  W )  C_  ( normCV `  U ) ) )
2524simp2d 970 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  R  C_  S )
26 ssres 5172 . . . . . . . . 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 3383 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  R  C_  ( S  |`  ( CC  X.  Y ) ) )
298, 16, 283jca 1134 . . . . . 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 6215 . . . . . 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 458 . . . . 5  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( x  e.  CC  /\  y  e.  Y ) )  ->  ( x
( S  |`  ( CC  X.  Y ) ) y )  =  ( x R y ) )
3231eqcomd 2441 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( x  e.  CC  /\  y  e.  Y ) )  ->  ( x R y )  =  ( x ( S  |`  ( CC  X.  Y
) ) y ) )
3332ralrimivva 2798 . . 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 2436 . . 3  |-  ( CC 
X.  Y )  =  ( CC  X.  Y
)
3533, 34jctil 524 . 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 5591 . . . . . 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 452 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  S  Fn  ( CC  X.  ( BaseSet
`  U ) ) )
39 ssid 3367 . . . . 5  |-  CC  C_  CC
401, 11, 9sspba 22226 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  ( BaseSet `  U )
)
41 xpss12 4981 . . . . 5  |-  ( ( CC  C_  CC  /\  Y  C_  ( BaseSet `  U )
)  ->  ( CC  X.  Y )  C_  ( CC  X.  ( BaseSet `  U
) ) )
4239, 40, 41sylancr 645 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( CC  X.  Y )  C_  ( CC  X.  ( BaseSet
`  U ) ) )
43 fnssres 5558 . . . 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 643 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( S  |`  ( CC  X.  Y ) )  Fn  ( CC  X.  Y
) )
45 eqfnov 6176 . . 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 643 . 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 224 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  R  =  ( S  |`  ( CC  X.  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320    X. cxp 4876    |` cres 4880   Fun wfun 5448    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988   NrmCVeccnv 22063   +vcpv 22064   BaseSetcba 22065   .s
OLDcns 22066   normCVcnmcv 22069   SubSpcss 22220
This theorem is referenced by:  sspsval  22230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-1st 6349  df-2nd 6350  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-0v 22077  df-nmcv 22079  df-ssp 22221
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