MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  asclfn Structured version   Unicode version

Theorem asclfn 17525
Description: Unconditional functionality of the algebra scalars function. (Contributed by Mario Carneiro, 9-Mar-2015.)
Hypotheses
Ref Expression
asclfn.a  |-  A  =  (algSc `  W )
asclfn.f  |-  F  =  (Scalar `  W )
asclfn.k  |-  K  =  ( Base `  F
)
Assertion
Ref Expression
asclfn  |-  A  Fn  K

Proof of Theorem asclfn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ovex 6220 . 2  |-  ( x ( .s `  W
) ( 1r `  W ) )  e. 
_V
2 asclfn.a . . 3  |-  A  =  (algSc `  W )
3 asclfn.f . . 3  |-  F  =  (Scalar `  W )
4 asclfn.k . . 3  |-  K  =  ( Base `  F
)
5 eqid 2452 . . 3  |-  ( .s
`  W )  =  ( .s `  W
)
6 eqid 2452 . . 3  |-  ( 1r
`  W )  =  ( 1r `  W
)
72, 3, 4, 5, 6asclfval 17523 . 2  |-  A  =  ( x  e.  K  |->  ( x ( .s
`  W ) ( 1r `  W ) ) )
81, 7fnmpti 5642 1  |-  A  Fn  K
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    Fn wfn 5516   ` cfv 5521  (class class class)co 6195   Basecbs 14287  Scalarcsca 14355   .scvsca 14356   1rcur 16720  algSccascl 17501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-slot 14291  df-base 14292  df-ascl 17504
This theorem is referenced by:  issubassa2  17533  subrgascl  17699
  Copyright terms: Public domain W3C validator