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Theorem asclfn 19317
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 6663 . 2 (𝑥( ·𝑠𝑊)(1r𝑊)) ∈ V
2 asclfn.a . . 3 𝐴 = (algSc‘𝑊)
3 asclfn.f . . 3 𝐹 = (Scalar‘𝑊)
4 asclfn.k . . 3 𝐾 = (Base‘𝐹)
5 eqid 2620 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
6 eqid 2620 . . 3 (1r𝑊) = (1r𝑊)
72, 3, 4, 5, 6asclfval 19315 . 2 𝐴 = (𝑥𝐾 ↦ (𝑥( ·𝑠𝑊)(1r𝑊)))
81, 7fnmpti 6009 1 𝐴 Fn 𝐾
Colors of variables: wff setvar class
Syntax hints:   = wceq 1481   Fn wfn 5871  cfv 5876  (class class class)co 6635  Basecbs 15838  Scalarcsca 15925   ·𝑠 cvsca 15926  1rcur 18482  algSccascl 19292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-slot 15842  df-base 15844  df-ascl 19295
This theorem is referenced by:  issubassa2  19326  subrgascl  19479
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