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Theorem pjfval 19346
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 5879 . . . . . . 7  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
3 pjfval.l . . . . . . 7  |-  L  =  ( LSubSp `  W )
42, 3syl6eqr 2523 . . . . . 6  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  L )
5 fveq2 5879 . . . . . . . 8  |-  ( w  =  W  ->  ( proj1 `  w )  =  ( proj1 `  W ) )
6 pjfval.p . . . . . . . 8  |-  P  =  ( proj1 `  W )
75, 6syl6eqr 2523 . . . . . . 7  |-  ( w  =  W  ->  ( proj1 `  w )  =  P )
8 eqidd 2472 . . . . . . 7  |-  ( w  =  W  ->  x  =  x )
9 fveq2 5879 . . . . . . . . 9  |-  ( w  =  W  ->  ( ocv `  w )  =  ( ocv `  W
) )
10 pjfval.o . . . . . . . . 9  |-  ._|_  =  ( ocv `  W )
119, 10syl6eqr 2523 . . . . . . . 8  |-  ( w  =  W  ->  ( ocv `  w )  = 
._|_  )
1211fveq1d 5881 . . . . . . 7  |-  ( w  =  W  ->  (
( ocv `  w
) `  x )  =  (  ._|_  `  x
) )
137, 8, 12oveq123d 6329 . . . . . 6  |-  ( w  =  W  ->  (
x ( proj1 `  w ) ( ( ocv `  w ) `
 x ) )  =  ( x P (  ._|_  `  x ) ) )
144, 13mpteq12dv 4474 . . . . 5  |-  ( w  =  W  ->  (
x  e.  ( LSubSp `  w )  |->  ( x ( proj1 `  w ) ( ( ocv `  w ) `
 x ) ) )  =  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) ) )
15 fveq2 5879 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
16 pjfval.v . . . . . . . 8  |-  V  =  ( Base `  W
)
1715, 16syl6eqr 2523 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  V )
1817, 17oveq12d 6326 . . . . . 6  |-  ( w  =  W  ->  (
( Base `  w )  ^m  ( Base `  w
) )  =  ( V  ^m  V ) )
1918xpeq2d 4863 . . . . 5  |-  ( w  =  W  ->  ( _V  X.  ( ( Base `  w )  ^m  ( Base `  w ) ) )  =  ( _V 
X.  ( V  ^m  V ) ) )
2014, 19ineq12d 3626 . . . 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 19343 . . . 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 5889 . . . . . . . 8  |-  ( LSubSp `  W )  e.  _V
233, 22eqeltri 2545 . . . . . . 7  |-  L  e. 
_V
2423inex1 4537 . . . . . 6  |-  ( L  i^i  _V )  e. 
_V
25 ovex 6336 . . . . . . 7  |-  ( V  ^m  V )  e. 
_V
2625inex2 4538 . . . . . 6  |-  ( _V 
i^i  ( V  ^m  V ) )  e. 
_V
2724, 26xpex 6614 . . . . 5  |-  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )  e.  _V
28 eqid 2471 . . . . . . . 8  |-  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  =  ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )
29 ovex 6336 . . . . . . . . 9  |-  ( x P (  ._|_  `  x
) )  e.  _V
3029a1i 11 . . . . . . . 8  |-  ( x  e.  L  ->  (
x P (  ._|_  `  x ) )  e. 
_V )
3128, 30fmpti 6060 . . . . . . 7  |-  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) ) : L --> _V
32 fssxp 5753 . . . . . . 7  |-  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) ) : L --> _V  ->  ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) ) 
C_  ( L  X.  _V ) )
33 ssrin 3648 . . . . . . 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 4972 . . . . . 6  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )
3634, 35sseqtri 3450 . . . . 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 4541 . . . 4  |-  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )  e.  _V
3820, 21, 37fvmpt 5963 . . 3  |-  ( W  e.  _V  ->  ( proj `  W )  =  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) ) )
39 fvprc 5873 . . . 4  |-  ( -.  W  e.  _V  ->  (
proj `  W )  =  (/) )
40 inss1 3643 . . . . 5  |-  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )  C_  (
x  e.  L  |->  ( x P (  ._|_  `  x ) ) )
41 fvprc 5873 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  (
LSubSp `  W )  =  (/) )
423, 41syl5eq 2517 . . . . . . 7  |-  ( -.  W  e.  _V  ->  L  =  (/) )
4342mpteq1d 4477 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  =  ( x  e.  (/)  |->  ( x P (  ._|_  `  x ) ) ) )
44 mpt0 5715 . . . . . 6  |-  ( x  e.  (/)  |->  ( x P (  ._|_  `  x ) ) )  =  (/)
4543, 44syl6eq 2521 . . . . 5  |-  ( -.  W  e.  _V  ->  ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  =  (/) )
46 sseq0 3769 . . . . 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 676 . . . 4  |-  ( -.  W  e.  _V  ->  ( ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  i^i  ( _V 
X.  ( V  ^m  V ) ) )  =  (/) )
4839, 47eqtr4d 2508 . . 3  |-  ( -.  W  e.  _V  ->  (
proj `  W )  =  ( ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) ) )
4938, 48pm2.61i 169 . 2  |-  ( proj `  W )  =  ( ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  i^i  ( _V 
X.  ( V  ^m  V ) ) )
501, 49eqtri 2493 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 1452    e. wcel 1904   _Vcvv 3031    i^i cin 3389    C_ wss 3390   (/)c0 3722    |-> cmpt 4454    X. cxp 4837   -->wf 5585   ` cfv 5589  (class class class)co 6308    ^m cmap 7490   Basecbs 15199   proj1cpj1 17365   LSubSpclss 18233   ocvcocv 19300   projcpj 19340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-ov 6311  df-pj 19343
This theorem is referenced by:  pjdm  19347  pjpm  19348  pjfval2  19349
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