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Theorem scaffn 20442
Description: The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
scaffval.b 𝐵 = (Base‘𝑊)
scaffval.f 𝐹 = (Scalar‘𝑊)
scaffval.k 𝐾 = (Base‘𝐹)
scaffval.a = ( ·sf𝑊)
Assertion
Ref Expression
scaffn Fn (𝐾 × 𝐵)

Proof of Theorem scaffn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 scaffval.b . . 3 𝐵 = (Base‘𝑊)
2 scaffval.f . . 3 𝐹 = (Scalar‘𝑊)
3 scaffval.k . . 3 𝐾 = (Base‘𝐹)
4 scaffval.a . . 3 = ( ·sf𝑊)
5 eqid 2731 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
61, 2, 3, 4, 5scaffval 20439 . 2 = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥( ·𝑠𝑊)𝑦))
7 ovex 7426 . 2 (𝑥( ·𝑠𝑊)𝑦) ∈ V
86, 7fnmpoi 8038 1 Fn (𝐾 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541   × cxp 5667   Fn wfn 6527  cfv 6532  (class class class)co 7393  Basecbs 17126  Scalarcsca 17182   ·𝑠 cvsca 17183   ·sf cscaf 20421
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-1st 7957  df-2nd 7958  df-scaf 20423
This theorem is referenced by: (None)
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