MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pjfval Unicode version

Theorem pjfval 16622
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 5541 . . . . . . 7  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
3 pjfval.l . . . . . . 7  |-  L  =  ( LSubSp `  W )
42, 3syl6eqr 2346 . . . . . 6  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  L )
5 fveq2 5541 . . . . . . . 8  |-  ( w  =  W  ->  ( proj 1 `  w )  =  ( proj 1 `  W ) )
6 pjfval.p . . . . . . . 8  |-  P  =  ( proj 1 `  W )
75, 6syl6eqr 2346 . . . . . . 7  |-  ( w  =  W  ->  ( proj 1 `  w )  =  P )
8 eqidd 2297 . . . . . . 7  |-  ( w  =  W  ->  x  =  x )
9 fveq2 5541 . . . . . . . . 9  |-  ( w  =  W  ->  ( ocv `  w )  =  ( ocv `  W
) )
10 pjfval.o . . . . . . . . 9  |-  ._|_  =  ( ocv `  W )
119, 10syl6eqr 2346 . . . . . . . 8  |-  ( w  =  W  ->  ( ocv `  w )  = 
._|_  )
1211fveq1d 5543 . . . . . . 7  |-  ( w  =  W  ->  (
( ocv `  w
) `  x )  =  (  ._|_  `  x
) )
137, 8, 12oveq123d 5895 . . . . . 6  |-  ( w  =  W  ->  (
x ( proj 1 `  w ) ( ( ocv `  w ) `
 x ) )  =  ( x P (  ._|_  `  x ) ) )
144, 13mpteq12dv 4114 . . . . 5  |-  ( w  =  W  ->  (
x  e.  ( LSubSp `  w )  |->  ( x ( proj 1 `  w ) ( ( ocv `  w ) `
 x ) ) )  =  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) ) )
15 fveq2 5541 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
16 pjfval.v . . . . . . . 8  |-  V  =  ( Base `  W
)
1715, 16syl6eqr 2346 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  V )
1817, 17oveq12d 5892 . . . . . 6  |-  ( w  =  W  ->  (
( Base `  w )  ^m  ( Base `  w
) )  =  ( V  ^m  V ) )
1918xpeq2d 4729 . . . . 5  |-  ( w  =  W  ->  ( _V  X.  ( ( Base `  w )  ^m  ( Base `  w ) ) )  =  ( _V 
X.  ( V  ^m  V ) ) )
2014, 19ineq12d 3384 . . . 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 16619 . . . 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 5555 . . . . . . . 8  |-  ( LSubSp `  W )  e.  _V
233, 22eqeltri 2366 . . . . . . 7  |-  L  e. 
_V
2423inex1 4171 . . . . . 6  |-  ( L  i^i  _V )  e. 
_V
25 ovex 5899 . . . . . . 7  |-  ( V  ^m  V )  e. 
_V
2625inex2 4172 . . . . . 6  |-  ( _V 
i^i  ( V  ^m  V ) )  e. 
_V
2724, 26xpex 4817 . . . . 5  |-  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )  e.  _V
28 eqid 2296 . . . . . . . 8  |-  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  =  ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )
29 ovex 5899 . . . . . . . . 9  |-  ( x P (  ._|_  `  x
) )  e.  _V
3029a1i 10 . . . . . . . 8  |-  ( x  e.  L  ->  (
x P (  ._|_  `  x ) )  e. 
_V )
3128, 30fmpti 5699 . . . . . . 7  |-  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) ) : L --> _V
32 fssxp 5416 . . . . . . 7  |-  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) ) : L --> _V  ->  ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) ) 
C_  ( L  X.  _V ) )
33 ssrin 3407 . . . . . . 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 9 . . . . . 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 4834 . . . . . 6  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )
3634, 35sseqtri 3223 . . . . 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 4175 . . . 4  |-  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )  e.  _V
3820, 21, 37fvmpt 5618 . . 3  |-  ( W  e.  _V  ->  ( proj `  W )  =  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) ) )
39 fvprc 5535 . . . 4  |-  ( -.  W  e.  _V  ->  (
proj `  W )  =  (/) )
40 inss1 3402 . . . . 5  |-  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )  C_  (
x  e.  L  |->  ( x P (  ._|_  `  x ) ) )
41 fvprc 5535 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  (
LSubSp `  W )  =  (/) )
423, 41syl5eq 2340 . . . . . . 7  |-  ( -.  W  e.  _V  ->  L  =  (/) )
43 mpteq1 4116 . . . . . . 7  |-  ( L  =  (/)  ->  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  =  ( x  e.  (/)  |->  ( x P ( 
._|_  `  x ) ) ) )
4442, 43syl 15 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  =  ( x  e.  (/)  |->  ( x P (  ._|_  `  x ) ) ) )
45 mpt0 5387 . . . . . 6  |-  ( x  e.  (/)  |->  ( x P (  ._|_  `  x ) ) )  =  (/)
4644, 45syl6eq 2344 . . . . 5  |-  ( -.  W  e.  _V  ->  ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  =  (/) )
47 sseq0 3499 . . . . 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 ) ) )  =  (/) )
4840, 46, 47sylancr 644 . . . 4  |-  ( -.  W  e.  _V  ->  ( ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  i^i  ( _V 
X.  ( V  ^m  V ) ) )  =  (/) )
4939, 48eqtr4d 2331 . . 3  |-  ( -.  W  e.  _V  ->  (
proj `  W )  =  ( ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) ) )
5038, 49pm2.61i 156 . 2  |-  ( proj `  W )  =  ( ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  i^i  ( _V 
X.  ( V  ^m  V ) ) )
511, 50eqtri 2316 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 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468    e. cmpt 4093    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Basecbs 13164   proj 1cpj1 14962   LSubSpclss 15705   ocvcocv 16576   projcpj 16616
This theorem is referenced by:  pjdm  16623  pjpm  16624  pjfval2  16625
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-pj 16619
  Copyright terms: Public domain W3C validator