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Theorem ssps 21322
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 2296 . . . . . . . . . . 11  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 ssps.s . . . . . . . . . . 11  |-  S  =  ( .s OLD `  U
)
31, 2nvsf 21191 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  S : ( CC  X.  ( BaseSet `  U ) ) --> (
BaseSet `  U ) )
4 ffun 5407 . . . . . . . . . 10  |-  ( S : ( CC  X.  ( BaseSet `  U )
) --> ( BaseSet `  U
)  ->  Fun  S )
53, 4syl 15 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  Fun  S )
6 funres 5309 . . . . . . . . 9  |-  ( Fun 
S  ->  Fun  ( S  |`  ( CC  X.  Y
) ) )
75, 6syl 15 . . . . . . . 8  |-  ( U  e.  NrmCVec  ->  Fun  ( S  |`  ( CC  X.  Y
) ) )
87adantr 451 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Fun  ( S  |`  ( CC 
X.  Y ) ) )
9 ssps.h . . . . . . . . . 10  |-  H  =  ( SubSp `  U )
109sspnv 21318 . . . . . . . . 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 21191 . . . . . . . . 9  |-  ( W  e.  NrmCVec  ->  R : ( CC  X.  Y ) --> Y )
1410, 13syl 15 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  R : ( CC  X.  Y ) --> Y )
15 ffn 5405 . . . . . . . 8  |-  ( R : ( CC  X.  Y ) --> Y  ->  R  Fn  ( CC  X.  Y ) )
1614, 15syl 15 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  R  Fn  ( CC  X.  Y
) )
17 fnresdm 5369 . . . . . . . . 9  |-  ( R  Fn  ( CC  X.  Y )  ->  ( R  |`  ( CC  X.  Y ) )  =  R )
1816, 17syl 15 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( R  |`  ( CC  X.  Y ) )  =  R )
19 eqid 2296 . . . . . . . . . . . 12  |-  ( +v
`  U )  =  ( +v `  U
)
20 eqid 2296 . . . . . . . . . . . 12  |-  ( +v
`  W )  =  ( +v `  W
)
21 eqid 2296 . . . . . . . . . . . 12  |-  ( normCV `  U )  =  (
normCV
`  U )
22 eqid 2296 . . . . . . . . . . . 12  |-  ( normCV `  W )  =  (
normCV
`  W )
2319, 20, 2, 12, 21, 22, 9isssp 21316 . . . . . . . . . . 11  |-  ( U  e.  NrmCVec  ->  ( W  e.  H  <->  ( W  e.  NrmCVec 
/\  ( ( +v
`  W )  C_  ( +v `  U )  /\  R  C_  S  /\  ( normCV `  W )  C_  ( normCV `  U ) ) ) ) )
2423simplbda 607 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  (
( +v `  W
)  C_  ( +v `  U )  /\  R  C_  S  /\  ( normCV `  W )  C_  ( normCV `  U ) ) )
2524simp2d 968 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  R  C_  S )
26 ssres 4997 . . . . . . . . 9  |-  ( R 
C_  S  ->  ( R  |`  ( CC  X.  Y ) )  C_  ( S  |`  ( CC 
X.  Y ) ) )
2725, 26syl 15 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( R  |`  ( CC  X.  Y ) )  C_  ( S  |`  ( CC 
X.  Y ) ) )
2818, 27eqsstr3d 3226 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  R  C_  ( S  |`  ( CC  X.  Y ) ) )
298, 16, 283jca 1132 . . . . . 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 6005 . . . . . 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 457 . . . . 5  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( x  e.  CC  /\  y  e.  Y ) )  ->  ( x
( S  |`  ( CC  X.  Y ) ) y )  =  ( x R y ) )
3231eqcomd 2301 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( x  e.  CC  /\  y  e.  Y ) )  ->  ( x R y )  =  ( x ( S  |`  ( CC  X.  Y
) ) y ) )
3332ralrimivva 2648 . . 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 2296 . . 3  |-  ( CC 
X.  Y )  =  ( CC  X.  Y
)
3533, 34jctil 523 . 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 5405 . . . . . 6  |-  ( S : ( CC  X.  ( BaseSet `  U )
) --> ( BaseSet `  U
)  ->  S  Fn  ( CC  X.  ( BaseSet
`  U ) ) )
373, 36syl 15 . . . . 5  |-  ( U  e.  NrmCVec  ->  S  Fn  ( CC  X.  ( BaseSet `  U
) ) )
3837adantr 451 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  S  Fn  ( CC  X.  ( BaseSet
`  U ) ) )
39 ssid 3210 . . . . 5  |-  CC  C_  CC
401, 11, 9sspba 21319 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  ( BaseSet `  U )
)
41 xpss12 4808 . . . . 5  |-  ( ( CC  C_  CC  /\  Y  C_  ( BaseSet `  U )
)  ->  ( CC  X.  Y )  C_  ( CC  X.  ( BaseSet `  U
) ) )
4239, 40, 41sylancr 644 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( CC  X.  Y )  C_  ( CC  X.  ( BaseSet
`  U ) ) )
43 fnssres 5373 . . . 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 642 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( S  |`  ( CC  X.  Y ) )  Fn  ( CC  X.  Y
) )
45 eqfnov 5966 . . 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 642 . 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 223 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  R  =  ( S  |`  ( CC  X.  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165    X. cxp 4703    |` cres 4707   Fun wfun 5265    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   NrmCVeccnv 21156   +vcpv 21157   BaseSetcba 21158   .s
OLDcns 21159   normCVcnmcv 21162   SubSpcss 21313
This theorem is referenced by:  sspsval  21323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-1st 6138  df-2nd 6139  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-nmcv 21172  df-ssp 21314
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