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Theorem pjfval 18910
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 5848 . . . . . . 7  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
3 pjfval.l . . . . . . 7  |-  L  =  ( LSubSp `  W )
42, 3syl6eqr 2513 . . . . . 6  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  L )
5 fveq2 5848 . . . . . . . 8  |-  ( w  =  W  ->  ( proj1 `  w )  =  ( proj1 `  W ) )
6 pjfval.p . . . . . . . 8  |-  P  =  ( proj1 `  W )
75, 6syl6eqr 2513 . . . . . . 7  |-  ( w  =  W  ->  ( proj1 `  w )  =  P )
8 eqidd 2455 . . . . . . 7  |-  ( w  =  W  ->  x  =  x )
9 fveq2 5848 . . . . . . . . 9  |-  ( w  =  W  ->  ( ocv `  w )  =  ( ocv `  W
) )
10 pjfval.o . . . . . . . . 9  |-  ._|_  =  ( ocv `  W )
119, 10syl6eqr 2513 . . . . . . . 8  |-  ( w  =  W  ->  ( ocv `  w )  = 
._|_  )
1211fveq1d 5850 . . . . . . 7  |-  ( w  =  W  ->  (
( ocv `  w
) `  x )  =  (  ._|_  `  x
) )
137, 8, 12oveq123d 6291 . . . . . 6  |-  ( w  =  W  ->  (
x ( proj1 `  w ) ( ( ocv `  w ) `
 x ) )  =  ( x P (  ._|_  `  x ) ) )
144, 13mpteq12dv 4517 . . . . 5  |-  ( w  =  W  ->  (
x  e.  ( LSubSp `  w )  |->  ( x ( proj1 `  w ) ( ( ocv `  w ) `
 x ) ) )  =  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) ) )
15 fveq2 5848 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
16 pjfval.v . . . . . . . 8  |-  V  =  ( Base `  W
)
1715, 16syl6eqr 2513 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  V )
1817, 17oveq12d 6288 . . . . . 6  |-  ( w  =  W  ->  (
( Base `  w )  ^m  ( Base `  w
) )  =  ( V  ^m  V ) )
1918xpeq2d 5012 . . . . 5  |-  ( w  =  W  ->  ( _V  X.  ( ( Base `  w )  ^m  ( Base `  w ) ) )  =  ( _V 
X.  ( V  ^m  V ) ) )
2014, 19ineq12d 3687 . . . 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 18907 . . . 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 5858 . . . . . . . 8  |-  ( LSubSp `  W )  e.  _V
233, 22eqeltri 2538 . . . . . . 7  |-  L  e. 
_V
2423inex1 4578 . . . . . 6  |-  ( L  i^i  _V )  e. 
_V
25 ovex 6298 . . . . . . 7  |-  ( V  ^m  V )  e. 
_V
2625inex2 4579 . . . . . 6  |-  ( _V 
i^i  ( V  ^m  V ) )  e. 
_V
2724, 26xpex 6577 . . . . 5  |-  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )  e.  _V
28 eqid 2454 . . . . . . . 8  |-  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  =  ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )
29 ovex 6298 . . . . . . . . 9  |-  ( x P (  ._|_  `  x
) )  e.  _V
3029a1i 11 . . . . . . . 8  |-  ( x  e.  L  ->  (
x P (  ._|_  `  x ) )  e. 
_V )
3128, 30fmpti 6030 . . . . . . 7  |-  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) ) : L --> _V
32 fssxp 5725 . . . . . . 7  |-  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) ) : L --> _V  ->  ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) ) 
C_  ( L  X.  _V ) )
33 ssrin 3709 . . . . . . 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 5124 . . . . . 6  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )
3634, 35sseqtri 3521 . . . . 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 4582 . . . 4  |-  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )  e.  _V
3820, 21, 37fvmpt 5931 . . 3  |-  ( W  e.  _V  ->  ( proj `  W )  =  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) ) )
39 fvprc 5842 . . . 4  |-  ( -.  W  e.  _V  ->  (
proj `  W )  =  (/) )
40 inss1 3704 . . . . 5  |-  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )  C_  (
x  e.  L  |->  ( x P (  ._|_  `  x ) ) )
41 fvprc 5842 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  (
LSubSp `  W )  =  (/) )
423, 41syl5eq 2507 . . . . . . 7  |-  ( -.  W  e.  _V  ->  L  =  (/) )
4342mpteq1d 4520 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  =  ( x  e.  (/)  |->  ( x P (  ._|_  `  x ) ) ) )
44 mpt0 5690 . . . . . 6  |-  ( x  e.  (/)  |->  ( x P (  ._|_  `  x ) ) )  =  (/)
4543, 44syl6eq 2511 . . . . 5  |-  ( -.  W  e.  _V  ->  ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  =  (/) )
46 sseq0 3816 . . . . 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 661 . . . 4  |-  ( -.  W  e.  _V  ->  ( ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  i^i  ( _V 
X.  ( V  ^m  V ) ) )  =  (/) )
4839, 47eqtr4d 2498 . . 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 2483 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 1398    e. wcel 1823   _Vcvv 3106    i^i cin 3460    C_ wss 3461   (/)c0 3783    |-> cmpt 4497    X. cxp 4986   -->wf 5566   ` cfv 5570  (class class class)co 6270    ^m cmap 7412   Basecbs 14716   proj1cpj1 16854   LSubSpclss 17773   ocvcocv 18864   projcpj 18904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-pj 18907
This theorem is referenced by:  pjdm  18911  pjpm  18912  pjfval2  18913
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