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Theorem asclfval 17961
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 5856 . . . . . . . 8  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
3 asclfval.f . . . . . . . 8  |-  F  =  (Scalar `  W )
42, 3syl6eqr 2502 . . . . . . 7  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
54fveq2d 5860 . . . . . 6  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  F )
)
6 asclfval.k . . . . . 6  |-  K  =  ( Base `  F
)
75, 6syl6eqr 2502 . . . . 5  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  K )
8 fveq2 5856 . . . . . . 7  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
9 asclfval.s . . . . . . 7  |-  .x.  =  ( .s `  W )
108, 9syl6eqr 2502 . . . . . 6  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
11 eqidd 2444 . . . . . 6  |-  ( w  =  W  ->  x  =  x )
12 fveq2 5856 . . . . . . 7  |-  ( w  =  W  ->  ( 1r `  w )  =  ( 1r `  W
) )
13 asclfval.o . . . . . . 7  |-  .1.  =  ( 1r `  W )
1412, 13syl6eqr 2502 . . . . . 6  |-  ( w  =  W  ->  ( 1r `  w )  =  .1.  )
1510, 11, 14oveq123d 6302 . . . . 5  |-  ( w  =  W  ->  (
x ( .s `  w ) ( 1r
`  w ) )  =  ( x  .x.  .1.  ) )
167, 15mpteq12dv 4515 . . . 4  |-  ( w  =  W  ->  (
x  e.  ( Base `  (Scalar `  w )
)  |->  ( x ( .s `  w ) ( 1r `  w
) ) )  =  ( x  e.  K  |->  ( x  .x.  .1.  ) ) )
17 df-ascl 17941 . . . 4  |- algSc  =  ( w  e.  _V  |->  ( x  e.  ( Base `  (Scalar `  w )
)  |->  ( x ( .s `  w ) ( 1r `  w
) ) ) )
183fveq2i 5859 . . . . . . 7  |-  ( Base `  F )  =  (
Base `  (Scalar `  W
) )
196, 18eqtri 2472 . . . . . 6  |-  K  =  ( Base `  (Scalar `  W ) )
20 fvex 5866 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  e.  _V
2119, 20eqeltri 2527 . . . . 5  |-  K  e. 
_V
2221mptex 6128 . . . 4  |-  ( x  e.  K  |->  ( x 
.x.  .1.  ) )  e.  _V
2316, 17, 22fvmpt 5941 . . 3  |-  ( W  e.  _V  ->  (algSc `  W )  =  ( x  e.  K  |->  ( x  .x.  .1.  )
) )
24 fvprc 5850 . . . . 5  |-  ( -.  W  e.  _V  ->  (algSc `  W )  =  (/) )
25 mpt0 5698 . . . . 5  |-  ( x  e.  (/)  |->  ( x  .x.  .1.  ) )  =  (/)
2624, 25syl6eqr 2502 . . . 4  |-  ( -.  W  e.  _V  ->  (algSc `  W )  =  ( x  e.  (/)  |->  ( x 
.x.  .1.  ) )
)
27 fvprc 5850 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  (Scalar `  W )  =  (/) )
283, 27syl5eq 2496 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  F  =  (/) )
2928fveq2d 5860 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (
Base `  F )  =  ( Base `  (/) ) )
30 base0 14652 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
3129, 30syl6eqr 2502 . . . . . 6  |-  ( -.  W  e.  _V  ->  (
Base `  F )  =  (/) )
326, 31syl5eq 2496 . . . . 5  |-  ( -.  W  e.  _V  ->  K  =  (/) )
3332mpteq1d 4518 . . . 4  |-  ( -.  W  e.  _V  ->  ( x  e.  K  |->  ( x  .x.  .1.  )
)  =  ( x  e.  (/)  |->  ( x  .x.  .1.  ) ) )
3426, 33eqtr4d 2487 . . 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 2472 1  |-  A  =  ( x  e.  K  |->  ( x  .x.  .1.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1383    e. wcel 1804   _Vcvv 3095   (/)c0 3770    |-> cmpt 4495   ` cfv 5578  (class class class)co 6281   Basecbs 14613  Scalarcsca 14681   .scvsca 14682   1rcur 17131  algSccascl 17938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-slot 14617  df-base 14618  df-ascl 17941
This theorem is referenced by:  asclval  17962  asclfn  17963  asclf  17964  rnascl  17970  ressascl  17971  asclpropd  17973
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