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Theorem ssps 24273
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 2451 . . . . . . . . . . 11  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 ssps.s . . . . . . . . . . 11  |-  S  =  ( .sOLD `  U )
31, 2nvsf 24142 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  S : ( CC  X.  ( BaseSet `  U ) ) --> (
BaseSet `  U ) )
4 ffun 5662 . . . . . . . . . 10  |-  ( S : ( CC  X.  ( BaseSet `  U )
) --> ( BaseSet `  U
)  ->  Fun  S )
53, 4syl 16 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  Fun  S )
6 funres 5558 . . . . . . . . 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 24269 . . . . . . . . 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 24142 . . . . . . . . 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 5660 . . . . . . . 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 5621 . . . . . . . . 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 2451 . . . . . . . . . . . 12  |-  ( +v
`  U )  =  ( +v `  U
)
20 eqid 2451 . . . . . . . . . . . 12  |-  ( +v
`  W )  =  ( +v `  W
)
21 eqid 2451 . . . . . . . . . . . 12  |-  ( normCV `  U )  =  (
normCV
`  U )
22 eqid 2451 . . . . . . . . . . . 12  |-  ( normCV `  W )  =  (
normCV
`  W )
2319, 20, 2, 12, 21, 22, 9isssp 24267 . . . . . . . . . . 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 1001 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  R  C_  S )
26 ssres 5237 . . . . . . . . 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 3492 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  R  C_  ( S  |`  ( CC  X.  Y ) ) )
298, 16, 283jca 1168 . . . . . 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 6335 . . . . . 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 2459 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( x  e.  CC  /\  y  e.  Y ) )  ->  ( x R y )  =  ( x ( S  |`  ( CC  X.  Y
) ) y ) )
3332ralrimivva 2907 . . 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 2451 . . 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 5660 . . . . . 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 3476 . . . . 5  |-  CC  C_  CC
401, 11, 9sspba 24270 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  ( BaseSet `  U )
)
41 xpss12 5046 . . . . 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 5625 . . . 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 6299 . . 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 965    = wceq 1370    e. wcel 1758   A.wral 2795    C_ wss 3429    X. cxp 4939    |` cres 4943   Fun wfun 5513    Fn wfn 5514   -->wf 5515   ` cfv 5519  (class class class)co 6193   CCcc 9384   NrmCVeccnv 24107   +vcpv 24108   BaseSetcba 24109   .sOLDcns 24110   normCVcnmcv 24113   SubSpcss 24264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-1st 6680  df-2nd 6681  df-vc 24069  df-nv 24115  df-va 24118  df-ba 24119  df-sm 24120  df-0v 24121  df-nmcv 24123  df-ssp 24265
This theorem is referenced by:  sspsval  24274
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