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Theorem asclfval 16090
Description: Function value of the algebraic scalars function. (Contributed by Mario Carneiro, 8-Mar-2015.)
Hypotheses
Ref Expression
asclfval.a  |-  A  =  (algSc `  W )
asclfval.f  |-  F  =  (Scalar `  W )
asclfval.k  |-  K  =  ( Base `  F
)
asclfval.s  |-  .x.  =  ( .s `  W )
asclfval.o  |-  .1.  =  ( 1r `  W )
Assertion
Ref Expression
asclfval  |-  A  =  ( x  e.  K  |->  ( x  .x.  .1.  ) )
Distinct variable groups:    x, K    x,  .1.    x,  .x.    x, W
Allowed substitution hints:    A( x)    F( x)

Proof of Theorem asclfval
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 asclfval.a . 2  |-  A  =  (algSc `  W )
2 fveq2 5541 . . . . . . . 8  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
3 asclfval.f . . . . . . . 8  |-  F  =  (Scalar `  W )
42, 3syl6eqr 2346 . . . . . . 7  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
54fveq2d 5545 . . . . . 6  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  F )
)
6 asclfval.k . . . . . 6  |-  K  =  ( Base `  F
)
75, 6syl6eqr 2346 . . . . 5  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  K )
8 fveq2 5541 . . . . . . 7  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
9 asclfval.s . . . . . . 7  |-  .x.  =  ( .s `  W )
108, 9syl6eqr 2346 . . . . . 6  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
11 eqidd 2297 . . . . . 6  |-  ( w  =  W  ->  x  =  x )
12 fveq2 5541 . . . . . . 7  |-  ( w  =  W  ->  ( 1r `  w )  =  ( 1r `  W
) )
13 asclfval.o . . . . . . 7  |-  .1.  =  ( 1r `  W )
1412, 13syl6eqr 2346 . . . . . 6  |-  ( w  =  W  ->  ( 1r `  w )  =  .1.  )
1510, 11, 14oveq123d 5895 . . . . 5  |-  ( w  =  W  ->  (
x ( .s `  w ) ( 1r
`  w ) )  =  ( x  .x.  .1.  ) )
167, 15mpteq12dv 4114 . . . 4  |-  ( w  =  W  ->  (
x  e.  ( Base `  (Scalar `  w )
)  |->  ( x ( .s `  w ) ( 1r `  w
) ) )  =  ( x  e.  K  |->  ( x  .x.  .1.  ) ) )
17 df-ascl 16071 . . . 4  |- algSc  =  ( w  e.  _V  |->  ( x  e.  ( Base `  (Scalar `  w )
)  |->  ( x ( .s `  w ) ( 1r `  w
) ) ) )
183fveq2i 5544 . . . . . . 7  |-  ( Base `  F )  =  (
Base `  (Scalar `  W
) )
196, 18eqtri 2316 . . . . . 6  |-  K  =  ( Base `  (Scalar `  W ) )
20 fvex 5555 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  e.  _V
2119, 20eqeltri 2366 . . . . 5  |-  K  e. 
_V
2221mptex 5762 . . . 4  |-  ( x  e.  K  |->  ( x 
.x.  .1.  ) )  e.  _V
2316, 17, 22fvmpt 5618 . . 3  |-  ( W  e.  _V  ->  (algSc `  W )  =  ( x  e.  K  |->  ( x  .x.  .1.  )
) )
24 fvprc 5535 . . . . 5  |-  ( -.  W  e.  _V  ->  (algSc `  W )  =  (/) )
25 mpt0 5387 . . . . 5  |-  ( x  e.  (/)  |->  ( x  .x.  .1.  ) )  =  (/)
2624, 25syl6eqr 2346 . . . 4  |-  ( -.  W  e.  _V  ->  (algSc `  W )  =  ( x  e.  (/)  |->  ( x 
.x.  .1.  ) )
)
27 fvprc 5535 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  (Scalar `  W )  =  (/) )
283, 27syl5eq 2340 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  F  =  (/) )
2928fveq2d 5545 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (
Base `  F )  =  ( Base `  (/) ) )
30 base0 13201 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
3129, 30syl6eqr 2346 . . . . . 6  |-  ( -.  W  e.  _V  ->  (
Base `  F )  =  (/) )
326, 31syl5eq 2340 . . . . 5  |-  ( -.  W  e.  _V  ->  K  =  (/) )
33 mpteq1 4116 . . . . 5  |-  ( K  =  (/)  ->  ( x  e.  K  |->  ( x 
.x.  .1.  ) )  =  ( x  e.  (/)  |->  ( x  .x.  .1.  ) ) )
3432, 33syl 15 . . . 4  |-  ( -.  W  e.  _V  ->  ( x  e.  K  |->  ( x  .x.  .1.  )
)  =  ( x  e.  (/)  |->  ( x  .x.  .1.  ) ) )
3526, 34eqtr4d 2331 . . 3  |-  ( -.  W  e.  _V  ->  (algSc `  W )  =  ( x  e.  K  |->  ( x  .x.  .1.  )
) )
3623, 35pm2.61i 156 . 2  |-  (algSc `  W )  =  ( x  e.  K  |->  ( x  .x.  .1.  )
)
371, 36eqtri 2316 1  |-  A  =  ( x  e.  K  |->  ( x  .x.  .1.  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .scvsca 13228   1rcur 15355  algSccascl 16068
This theorem is referenced by:  asclval  16091  asclfn  16092  asclf  16093  rnascl  16098  ressascl  16099  asclpropd  16101
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  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-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-slot 13168  df-base 13169  df-ascl 16071
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