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Theorem asclfval 18193
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 5803 . . . . . . . 8  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
3 asclfval.f . . . . . . . 8  |-  F  =  (Scalar `  W )
42, 3syl6eqr 2459 . . . . . . 7  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
54fveq2d 5807 . . . . . 6  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  F )
)
6 asclfval.k . . . . . 6  |-  K  =  ( Base `  F
)
75, 6syl6eqr 2459 . . . . 5  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  K )
8 fveq2 5803 . . . . . . 7  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
9 asclfval.s . . . . . . 7  |-  .x.  =  ( .s `  W )
108, 9syl6eqr 2459 . . . . . 6  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
11 eqidd 2401 . . . . . 6  |-  ( w  =  W  ->  x  =  x )
12 fveq2 5803 . . . . . . 7  |-  ( w  =  W  ->  ( 1r `  w )  =  ( 1r `  W
) )
13 asclfval.o . . . . . . 7  |-  .1.  =  ( 1r `  W )
1412, 13syl6eqr 2459 . . . . . 6  |-  ( w  =  W  ->  ( 1r `  w )  =  .1.  )
1510, 11, 14oveq123d 6253 . . . . 5  |-  ( w  =  W  ->  (
x ( .s `  w ) ( 1r
`  w ) )  =  ( x  .x.  .1.  ) )
167, 15mpteq12dv 4470 . . . 4  |-  ( w  =  W  ->  (
x  e.  ( Base `  (Scalar `  w )
)  |->  ( x ( .s `  w ) ( 1r `  w
) ) )  =  ( x  e.  K  |->  ( x  .x.  .1.  ) ) )
17 df-ascl 18173 . . . 4  |- algSc  =  ( w  e.  _V  |->  ( x  e.  ( Base `  (Scalar `  w )
)  |->  ( x ( .s `  w ) ( 1r `  w
) ) ) )
183fveq2i 5806 . . . . . . 7  |-  ( Base `  F )  =  (
Base `  (Scalar `  W
) )
196, 18eqtri 2429 . . . . . 6  |-  K  =  ( Base `  (Scalar `  W ) )
20 fvex 5813 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  e.  _V
2119, 20eqeltri 2484 . . . . 5  |-  K  e. 
_V
2221mptex 6078 . . . 4  |-  ( x  e.  K  |->  ( x 
.x.  .1.  ) )  e.  _V
2316, 17, 22fvmpt 5886 . . 3  |-  ( W  e.  _V  ->  (algSc `  W )  =  ( x  e.  K  |->  ( x  .x.  .1.  )
) )
24 fvprc 5797 . . . . 5  |-  ( -.  W  e.  _V  ->  (algSc `  W )  =  (/) )
25 mpt0 5645 . . . . 5  |-  ( x  e.  (/)  |->  ( x  .x.  .1.  ) )  =  (/)
2624, 25syl6eqr 2459 . . . 4  |-  ( -.  W  e.  _V  ->  (algSc `  W )  =  ( x  e.  (/)  |->  ( x 
.x.  .1.  ) )
)
27 fvprc 5797 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  (Scalar `  W )  =  (/) )
283, 27syl5eq 2453 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  F  =  (/) )
2928fveq2d 5807 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (
Base `  F )  =  ( Base `  (/) ) )
30 base0 14772 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
3129, 30syl6eqr 2459 . . . . . 6  |-  ( -.  W  e.  _V  ->  (
Base `  F )  =  (/) )
326, 31syl5eq 2453 . . . . 5  |-  ( -.  W  e.  _V  ->  K  =  (/) )
3332mpteq1d 4473 . . . 4  |-  ( -.  W  e.  _V  ->  ( x  e.  K  |->  ( x  .x.  .1.  )
)  =  ( x  e.  (/)  |->  ( x  .x.  .1.  ) ) )
3426, 33eqtr4d 2444 . . 3  |-  ( -.  W  e.  _V  ->  (algSc `  W )  =  ( x  e.  K  |->  ( x  .x.  .1.  )
) )
3523, 34pm2.61i 164 . 2  |-  (algSc `  W )  =  ( x  e.  K  |->  ( x  .x.  .1.  )
)
361, 35eqtri 2429 1  |-  A  =  ( x  e.  K  |->  ( x  .x.  .1.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1403    e. wcel 1840   _Vcvv 3056   (/)c0 3735    |-> cmpt 4450   ` cfv 5523  (class class class)co 6232   Basecbs 14731  Scalarcsca 14802   .scvsca 14803   1rcur 17363  algSccascl 18170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-slot 14735  df-base 14736  df-ascl 18173
This theorem is referenced by:  asclval  18194  asclfn  18195  asclf  18196  rnascl  18202  ressascl  18203  asclpropd  18205
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