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Theorem asclfn 19659
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 6910 . 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 2799 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
6 eqid 2799 . . 3 (1r𝑊) = (1r𝑊)
72, 3, 4, 5, 6asclfval 19657 . 2 𝐴 = (𝑥𝐾 ↦ (𝑥( ·𝑠𝑊)(1r𝑊)))
81, 7fnmpti 6233 1 𝐴 Fn 𝐾
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653   Fn wfn 6096  cfv 6101  (class class class)co 6878  Basecbs 16184  Scalarcsca 16270   ·𝑠 cvsca 16271  1rcur 18817  algSccascl 19634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-slot 16188  df-base 16190  df-ascl 19637
This theorem is referenced by:  issubassa2  19668  subrgascl  19820
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