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Theorem asclfval 17853
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 5872 . . . . . . . 8  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
3 asclfval.f . . . . . . . 8  |-  F  =  (Scalar `  W )
42, 3syl6eqr 2526 . . . . . . 7  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
54fveq2d 5876 . . . . . 6  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  F )
)
6 asclfval.k . . . . . 6  |-  K  =  ( Base `  F
)
75, 6syl6eqr 2526 . . . . 5  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  K )
8 fveq2 5872 . . . . . . 7  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
9 asclfval.s . . . . . . 7  |-  .x.  =  ( .s `  W )
108, 9syl6eqr 2526 . . . . . 6  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
11 eqidd 2468 . . . . . 6  |-  ( w  =  W  ->  x  =  x )
12 fveq2 5872 . . . . . . 7  |-  ( w  =  W  ->  ( 1r `  w )  =  ( 1r `  W
) )
13 asclfval.o . . . . . . 7  |-  .1.  =  ( 1r `  W )
1412, 13syl6eqr 2526 . . . . . 6  |-  ( w  =  W  ->  ( 1r `  w )  =  .1.  )
1510, 11, 14oveq123d 6316 . . . . 5  |-  ( w  =  W  ->  (
x ( .s `  w ) ( 1r
`  w ) )  =  ( x  .x.  .1.  ) )
167, 15mpteq12dv 4531 . . . 4  |-  ( w  =  W  ->  (
x  e.  ( Base `  (Scalar `  w )
)  |->  ( x ( .s `  w ) ( 1r `  w
) ) )  =  ( x  e.  K  |->  ( x  .x.  .1.  ) ) )
17 df-ascl 17833 . . . 4  |- algSc  =  ( w  e.  _V  |->  ( x  e.  ( Base `  (Scalar `  w )
)  |->  ( x ( .s `  w ) ( 1r `  w
) ) ) )
183fveq2i 5875 . . . . . . 7  |-  ( Base `  F )  =  (
Base `  (Scalar `  W
) )
196, 18eqtri 2496 . . . . . 6  |-  K  =  ( Base `  (Scalar `  W ) )
20 fvex 5882 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  e.  _V
2119, 20eqeltri 2551 . . . . 5  |-  K  e. 
_V
2221mptex 6142 . . . 4  |-  ( x  e.  K  |->  ( x 
.x.  .1.  ) )  e.  _V
2316, 17, 22fvmpt 5957 . . 3  |-  ( W  e.  _V  ->  (algSc `  W )  =  ( x  e.  K  |->  ( x  .x.  .1.  )
) )
24 fvprc 5866 . . . . 5  |-  ( -.  W  e.  _V  ->  (algSc `  W )  =  (/) )
25 mpt0 5714 . . . . 5  |-  ( x  e.  (/)  |->  ( x  .x.  .1.  ) )  =  (/)
2624, 25syl6eqr 2526 . . . 4  |-  ( -.  W  e.  _V  ->  (algSc `  W )  =  ( x  e.  (/)  |->  ( x 
.x.  .1.  ) )
)
27 fvprc 5866 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  (Scalar `  W )  =  (/) )
283, 27syl5eq 2520 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  F  =  (/) )
2928fveq2d 5876 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (
Base `  F )  =  ( Base `  (/) ) )
30 base0 14546 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
3129, 30syl6eqr 2526 . . . . . 6  |-  ( -.  W  e.  _V  ->  (
Base `  F )  =  (/) )
326, 31syl5eq 2520 . . . . 5  |-  ( -.  W  e.  _V  ->  K  =  (/) )
3332mpteq1d 4534 . . . 4  |-  ( -.  W  e.  _V  ->  ( x  e.  K  |->  ( x  .x.  .1.  )
)  =  ( x  e.  (/)  |->  ( x  .x.  .1.  ) ) )
3426, 33eqtr4d 2511 . . 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 2496 1  |-  A  =  ( x  e.  K  |->  ( x  .x.  .1.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   _Vcvv 3118   (/)c0 3790    |-> cmpt 4511   ` cfv 5594  (class class class)co 6295   Basecbs 14507  Scalarcsca 14575   .scvsca 14576   1rcur 17025  algSccascl 17830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-slot 14511  df-base 14512  df-ascl 17833
This theorem is referenced by:  asclval  17854  asclfn  17855  asclf  17856  rnascl  17862  ressascl  17863  asclpropd  17865
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