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Theorem asclfn 20038
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 7178 . 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 2818 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
6 eqid 2818 . . 3 (1r𝑊) = (1r𝑊)
72, 3, 4, 5, 6asclfval 20036 . 2 𝐴 = (𝑥𝐾 ↦ (𝑥( ·𝑠𝑊)(1r𝑊)))
81, 7fnmpti 6484 1 𝐴 Fn 𝐾
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528   Fn wfn 6343  cfv 6348  (class class class)co 7145  Basecbs 16471  Scalarcsca 16556   ·𝑠 cvsca 16557  1rcur 19180  algSccascl 20012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-slot 16475  df-base 16477  df-ascl 20015
This theorem is referenced by:  issubassa2  20049  subrgascl  20206
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