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Theorem pjfval 18136
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  =  ( proj1 `  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 5696 . . . . . . 7  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
3 pjfval.l . . . . . . 7  |-  L  =  ( LSubSp `  W )
42, 3syl6eqr 2493 . . . . . 6  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  L )
5 fveq2 5696 . . . . . . . 8  |-  ( w  =  W  ->  ( proj1 `  w )  =  ( proj1 `  W ) )
6 pjfval.p . . . . . . . 8  |-  P  =  ( proj1 `  W )
75, 6syl6eqr 2493 . . . . . . 7  |-  ( w  =  W  ->  ( proj1 `  w )  =  P )
8 eqidd 2444 . . . . . . 7  |-  ( w  =  W  ->  x  =  x )
9 fveq2 5696 . . . . . . . . 9  |-  ( w  =  W  ->  ( ocv `  w )  =  ( ocv `  W
) )
10 pjfval.o . . . . . . . . 9  |-  ._|_  =  ( ocv `  W )
119, 10syl6eqr 2493 . . . . . . . 8  |-  ( w  =  W  ->  ( ocv `  w )  = 
._|_  )
1211fveq1d 5698 . . . . . . 7  |-  ( w  =  W  ->  (
( ocv `  w
) `  x )  =  (  ._|_  `  x
) )
137, 8, 12oveq123d 6117 . . . . . 6  |-  ( w  =  W  ->  (
x ( proj1 `  w ) ( ( ocv `  w ) `
 x ) )  =  ( x P (  ._|_  `  x ) ) )
144, 13mpteq12dv 4375 . . . . 5  |-  ( w  =  W  ->  (
x  e.  ( LSubSp `  w )  |->  ( x ( proj1 `  w ) ( ( ocv `  w ) `
 x ) ) )  =  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) ) )
15 fveq2 5696 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
16 pjfval.v . . . . . . . 8  |-  V  =  ( Base `  W
)
1715, 16syl6eqr 2493 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  V )
1817, 17oveq12d 6114 . . . . . 6  |-  ( w  =  W  ->  (
( Base `  w )  ^m  ( Base `  w
) )  =  ( V  ^m  V ) )
1918xpeq2d 4869 . . . . 5  |-  ( w  =  W  ->  ( _V  X.  ( ( Base `  w )  ^m  ( Base `  w ) ) )  =  ( _V 
X.  ( V  ^m  V ) ) )
2014, 19ineq12d 3558 . . . 4  |-  ( w  =  W  ->  (
( x  e.  (
LSubSp `  w )  |->  ( x ( proj1 `  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 18133 . . . 4  |-  proj  =  ( w  e.  _V  |->  ( ( x  e.  ( LSubSp `  w )  |->  ( x ( proj1 `  w )
( ( ocv `  w
) `  x )
) )  i^i  ( _V  X.  ( ( Base `  w )  ^m  ( Base `  w ) ) ) ) )
22 fvex 5706 . . . . . . . 8  |-  ( LSubSp `  W )  e.  _V
233, 22eqeltri 2513 . . . . . . 7  |-  L  e. 
_V
2423inex1 4438 . . . . . 6  |-  ( L  i^i  _V )  e. 
_V
25 ovex 6121 . . . . . . 7  |-  ( V  ^m  V )  e. 
_V
2625inex2 4439 . . . . . 6  |-  ( _V 
i^i  ( V  ^m  V ) )  e. 
_V
2724, 26xpex 6513 . . . . 5  |-  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )  e.  _V
28 eqid 2443 . . . . . . . 8  |-  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  =  ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )
29 ovex 6121 . . . . . . . . 9  |-  ( x P (  ._|_  `  x
) )  e.  _V
3029a1i 11 . . . . . . . 8  |-  ( x  e.  L  ->  (
x P (  ._|_  `  x ) )  e. 
_V )
3128, 30fmpti 5871 . . . . . . 7  |-  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) ) : L --> _V
32 fssxp 5575 . . . . . . 7  |-  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) ) : L --> _V  ->  ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) ) 
C_  ( L  X.  _V ) )
33 ssrin 3580 . . . . . . 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 4977 . . . . . 6  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )
3634, 35sseqtri 3393 . . . . 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 4442 . . . 4  |-  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )  e.  _V
3820, 21, 37fvmpt 5779 . . 3  |-  ( W  e.  _V  ->  ( proj `  W )  =  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) ) )
39 fvprc 5690 . . . 4  |-  ( -.  W  e.  _V  ->  (
proj `  W )  =  (/) )
40 inss1 3575 . . . . 5  |-  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )  C_  (
x  e.  L  |->  ( x P (  ._|_  `  x ) ) )
41 fvprc 5690 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  (
LSubSp `  W )  =  (/) )
423, 41syl5eq 2487 . . . . . . 7  |-  ( -.  W  e.  _V  ->  L  =  (/) )
4342mpteq1d 4378 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  =  ( x  e.  (/)  |->  ( x P (  ._|_  `  x ) ) ) )
44 mpt0 5543 . . . . . 6  |-  ( x  e.  (/)  |->  ( x P (  ._|_  `  x ) ) )  =  (/)
4543, 44syl6eq 2491 . . . . 5  |-  ( -.  W  e.  _V  ->  ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  =  (/) )
46 sseq0 3674 . . . . 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 663 . . . 4  |-  ( -.  W  e.  _V  ->  ( ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  i^i  ( _V 
X.  ( V  ^m  V ) ) )  =  (/) )
4839, 47eqtr4d 2478 . . 3  |-  ( -.  W  e.  _V  ->  (
proj `  W )  =  ( ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) ) )
4938, 48pm2.61i 164 . 2  |-  ( proj `  W )  =  ( ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  i^i  ( _V 
X.  ( V  ^m  V ) ) )
501, 49eqtri 2463 1  |-  K  =  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1369    e. wcel 1756   _Vcvv 2977    i^i cin 3332    C_ wss 3333   (/)c0 3642    e. cmpt 4355    X. cxp 4843   -->wf 5419   ` cfv 5423  (class class class)co 6096    ^m cmap 7219   Basecbs 14179   proj1cpj1 16139   LSubSpclss 17018   ocvcocv 18090   projcpj 18130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-ov 6099  df-pj 18133
This theorem is referenced by:  pjdm  18137  pjpm  18138  pjfval2  18139
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