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Theorem asclfn 20656
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 7189 . 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 2758 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
6 eqid 2758 . . 3 (1r𝑊) = (1r𝑊)
72, 3, 4, 5, 6asclfval 20654 . 2 𝐴 = (𝑥𝐾 ↦ (𝑥( ·𝑠𝑊)(1r𝑊)))
81, 7fnmpti 6479 1 𝐴 Fn 𝐾
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538   Fn wfn 6335  cfv 6340  (class class class)co 7156  Basecbs 16554  Scalarcsca 16639   ·𝑠 cvsca 16640  1rcur 19332  algSccascl 20630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pr 5302
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-slot 16558  df-base 16560  df-ascl 20633
This theorem is referenced by:  issubassa2  20668  subrgascl  20840
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