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Theorem asclfval 16074
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 5525 . . . . . . . 8  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
3 asclfval.f . . . . . . . 8  |-  F  =  (Scalar `  W )
42, 3syl6eqr 2333 . . . . . . 7  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
54fveq2d 5529 . . . . . 6  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  F )
)
6 asclfval.k . . . . . 6  |-  K  =  ( Base `  F
)
75, 6syl6eqr 2333 . . . . 5  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  K )
8 fveq2 5525 . . . . . . 7  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
9 asclfval.s . . . . . . 7  |-  .x.  =  ( .s `  W )
108, 9syl6eqr 2333 . . . . . 6  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
11 eqidd 2284 . . . . . 6  |-  ( w  =  W  ->  x  =  x )
12 fveq2 5525 . . . . . . 7  |-  ( w  =  W  ->  ( 1r `  w )  =  ( 1r `  W
) )
13 asclfval.o . . . . . . 7  |-  .1.  =  ( 1r `  W )
1412, 13syl6eqr 2333 . . . . . 6  |-  ( w  =  W  ->  ( 1r `  w )  =  .1.  )
1510, 11, 14oveq123d 5879 . . . . 5  |-  ( w  =  W  ->  (
x ( .s `  w ) ( 1r
`  w ) )  =  ( x  .x.  .1.  ) )
167, 15mpteq12dv 4098 . . . 4  |-  ( w  =  W  ->  (
x  e.  ( Base `  (Scalar `  w )
)  |->  ( x ( .s `  w ) ( 1r `  w
) ) )  =  ( x  e.  K  |->  ( x  .x.  .1.  ) ) )
17 df-ascl 16055 . . . 4  |- algSc  =  ( w  e.  _V  |->  ( x  e.  ( Base `  (Scalar `  w )
)  |->  ( x ( .s `  w ) ( 1r `  w
) ) ) )
183fveq2i 5528 . . . . . . 7  |-  ( Base `  F )  =  (
Base `  (Scalar `  W
) )
196, 18eqtri 2303 . . . . . 6  |-  K  =  ( Base `  (Scalar `  W ) )
20 fvex 5539 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  e.  _V
2119, 20eqeltri 2353 . . . . 5  |-  K  e. 
_V
2221mptex 5746 . . . 4  |-  ( x  e.  K  |->  ( x 
.x.  .1.  ) )  e.  _V
2316, 17, 22fvmpt 5602 . . 3  |-  ( W  e.  _V  ->  (algSc `  W )  =  ( x  e.  K  |->  ( x  .x.  .1.  )
) )
24 fvprc 5519 . . . . 5  |-  ( -.  W  e.  _V  ->  (algSc `  W )  =  (/) )
25 mpt0 5371 . . . . 5  |-  ( x  e.  (/)  |->  ( x  .x.  .1.  ) )  =  (/)
2624, 25syl6eqr 2333 . . . 4  |-  ( -.  W  e.  _V  ->  (algSc `  W )  =  ( x  e.  (/)  |->  ( x 
.x.  .1.  ) )
)
27 fvprc 5519 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  (Scalar `  W )  =  (/) )
283, 27syl5eq 2327 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  F  =  (/) )
2928fveq2d 5529 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (
Base `  F )  =  ( Base `  (/) ) )
30 base0 13185 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
3129, 30syl6eqr 2333 . . . . . 6  |-  ( -.  W  e.  _V  ->  (
Base `  F )  =  (/) )
326, 31syl5eq 2327 . . . . 5  |-  ( -.  W  e.  _V  ->  K  =  (/) )
33 mpteq1 4100 . . . . 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 2318 . . 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 2303 1  |-  A  =  ( x  e.  K  |->  ( x  .x.  .1.  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   1rcur 15339  algSccascl 16052
This theorem is referenced by:  asclval  16075  asclfn  16076  asclf  16077  rnascl  16082  ressascl  16083  asclpropd  16085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-slot 13152  df-base 13153  df-ascl 16055
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