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Theorem pjfval 16934
Description: The value of the projection function. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
pjfval.v  |-  V  =  ( Base `  W
)
pjfval.l  |-  L  =  ( LSubSp `  W )
pjfval.o  |-  ._|_  =  ( ocv `  W )
pjfval.p  |-  P  =  ( proj 1 `  W )
pjfval.k  |-  K  =  ( proj `  W
)
Assertion
Ref Expression
pjfval  |-  K  =  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
Distinct variable groups:    x,  ._|_    x, L    x, P    x, V    x, W
Allowed substitution hint:    K( x)

Proof of Theorem pjfval
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 pjfval.k . 2  |-  K  =  ( proj `  W
)
2 fveq2 5729 . . . . . . 7  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
3 pjfval.l . . . . . . 7  |-  L  =  ( LSubSp `  W )
42, 3syl6eqr 2487 . . . . . 6  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  L )
5 fveq2 5729 . . . . . . . 8  |-  ( w  =  W  ->  ( proj 1 `  w )  =  ( proj 1 `  W ) )
6 pjfval.p . . . . . . . 8  |-  P  =  ( proj 1 `  W )
75, 6syl6eqr 2487 . . . . . . 7  |-  ( w  =  W  ->  ( proj 1 `  w )  =  P )
8 eqidd 2438 . . . . . . 7  |-  ( w  =  W  ->  x  =  x )
9 fveq2 5729 . . . . . . . . 9  |-  ( w  =  W  ->  ( ocv `  w )  =  ( ocv `  W
) )
10 pjfval.o . . . . . . . . 9  |-  ._|_  =  ( ocv `  W )
119, 10syl6eqr 2487 . . . . . . . 8  |-  ( w  =  W  ->  ( ocv `  w )  = 
._|_  )
1211fveq1d 5731 . . . . . . 7  |-  ( w  =  W  ->  (
( ocv `  w
) `  x )  =  (  ._|_  `  x
) )
137, 8, 12oveq123d 6103 . . . . . 6  |-  ( w  =  W  ->  (
x ( proj 1 `  w ) ( ( ocv `  w ) `
 x ) )  =  ( x P (  ._|_  `  x ) ) )
144, 13mpteq12dv 4288 . . . . 5  |-  ( w  =  W  ->  (
x  e.  ( LSubSp `  w )  |->  ( x ( proj 1 `  w ) ( ( ocv `  w ) `
 x ) ) )  =  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) ) )
15 fveq2 5729 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
16 pjfval.v . . . . . . . 8  |-  V  =  ( Base `  W
)
1715, 16syl6eqr 2487 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  V )
1817, 17oveq12d 6100 . . . . . 6  |-  ( w  =  W  ->  (
( Base `  w )  ^m  ( Base `  w
) )  =  ( V  ^m  V ) )
1918xpeq2d 4903 . . . . 5  |-  ( w  =  W  ->  ( _V  X.  ( ( Base `  w )  ^m  ( Base `  w ) ) )  =  ( _V 
X.  ( V  ^m  V ) ) )
2014, 19ineq12d 3544 . . . 4  |-  ( w  =  W  ->  (
( x  e.  (
LSubSp `  w )  |->  ( x ( proj 1 `  w ) ( ( ocv `  w ) `
 x ) ) )  i^i  ( _V 
X.  ( ( Base `  w )  ^m  ( Base `  w ) ) ) )  =  ( ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  i^i  ( _V 
X.  ( V  ^m  V ) ) ) )
21 df-pj 16931 . . . 4  |-  proj  =  ( w  e.  _V  |->  ( ( x  e.  ( LSubSp `  w )  |->  ( x ( proj
1 `  w )
( ( ocv `  w
) `  x )
) )  i^i  ( _V  X.  ( ( Base `  w )  ^m  ( Base `  w ) ) ) ) )
22 fvex 5743 . . . . . . . 8  |-  ( LSubSp `  W )  e.  _V
233, 22eqeltri 2507 . . . . . . 7  |-  L  e. 
_V
2423inex1 4345 . . . . . 6  |-  ( L  i^i  _V )  e. 
_V
25 ovex 6107 . . . . . . 7  |-  ( V  ^m  V )  e. 
_V
2625inex2 4346 . . . . . 6  |-  ( _V 
i^i  ( V  ^m  V ) )  e. 
_V
2724, 26xpex 4991 . . . . 5  |-  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )  e.  _V
28 eqid 2437 . . . . . . . 8  |-  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  =  ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )
29 ovex 6107 . . . . . . . . 9  |-  ( x P (  ._|_  `  x
) )  e.  _V
3029a1i 11 . . . . . . . 8  |-  ( x  e.  L  ->  (
x P (  ._|_  `  x ) )  e. 
_V )
3128, 30fmpti 5893 . . . . . . 7  |-  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) ) : L --> _V
32 fssxp 5603 . . . . . . 7  |-  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) ) : L --> _V  ->  ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) ) 
C_  ( L  X.  _V ) )
33 ssrin 3567 . . . . . . 7  |-  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) ) 
C_  ( L  X.  _V )  ->  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )  C_  (
( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) ) )
3431, 32, 33mp2b 10 . . . . . 6  |-  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )  C_  (
( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )
35 inxp 5008 . . . . . 6  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )
3634, 35sseqtri 3381 . . . . 5  |-  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )  C_  (
( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )
3727, 36ssexi 4349 . . . 4  |-  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )  e.  _V
3820, 21, 37fvmpt 5807 . . 3  |-  ( W  e.  _V  ->  ( proj `  W )  =  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) ) )
39 fvprc 5723 . . . 4  |-  ( -.  W  e.  _V  ->  (
proj `  W )  =  (/) )
40 inss1 3562 . . . . 5  |-  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )  C_  (
x  e.  L  |->  ( x P (  ._|_  `  x ) ) )
41 fvprc 5723 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  (
LSubSp `  W )  =  (/) )
423, 41syl5eq 2481 . . . . . . 7  |-  ( -.  W  e.  _V  ->  L  =  (/) )
4342mpteq1d 4291 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  =  ( x  e.  (/)  |->  ( x P (  ._|_  `  x ) ) ) )
44 mpt0 5573 . . . . . 6  |-  ( x  e.  (/)  |->  ( x P (  ._|_  `  x ) ) )  =  (/)
4543, 44syl6eq 2485 . . . . 5  |-  ( -.  W  e.  _V  ->  ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  =  (/) )
46 sseq0 3660 . . . . 5  |-  ( ( ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) ) 
C_  ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  /\  (
x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  =  (/) )  ->  (
( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  i^i  ( _V 
X.  ( V  ^m  V ) ) )  =  (/) )
4740, 45, 46sylancr 646 . . . 4  |-  ( -.  W  e.  _V  ->  ( ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  i^i  ( _V 
X.  ( V  ^m  V ) ) )  =  (/) )
4839, 47eqtr4d 2472 . . 3  |-  ( -.  W  e.  _V  ->  (
proj `  W )  =  ( ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) ) )
4938, 48pm2.61i 159 . 2  |-  ( proj `  W )  =  ( ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  i^i  ( _V 
X.  ( V  ^m  V ) ) )
501, 49eqtri 2457 1  |-  K  =  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   _Vcvv 2957    i^i cin 3320    C_ wss 3321   (/)c0 3629    e. cmpt 4267    X. cxp 4877   -->wf 5451   ` cfv 5455  (class class class)co 6082    ^m cmap 7019   Basecbs 13470   proj 1cpj1 15270   LSubSpclss 16009   ocvcocv 16888   projcpj 16928
This theorem is referenced by:  pjdm  16935  pjpm  16936  pjfval2  16937
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-ov 6085  df-pj 16931
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