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Theorem asclfn 16125
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 5925 . 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 2316 . . 3  |-  ( .s
`  W )  =  ( .s `  W
)
6 eqid 2316 . . 3  |-  ( 1r
`  W )  =  ( 1r `  W
)
72, 3, 4, 5, 6asclfval 16123 . 2  |-  A  =  ( x  e.  K  |->  ( x ( .s
`  W ) ( 1r `  W ) ) )
81, 7fnmpti 5409 1  |-  A  Fn  K
Colors of variables: wff set class
Syntax hints:    = wceq 1633    Fn wfn 5287   ` cfv 5292  (class class class)co 5900   Basecbs 13195  Scalarcsca 13258   .scvsca 13259   1rcur 15388  algSccascl 16101
This theorem is referenced by:  issubassa2  16133  subrgascl  16288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-slot 13199  df-base 13200  df-ascl 16104
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