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Theorem scaffn 18653
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 2609 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
61, 2, 3, 4, 5scaffval 18650 . 2 = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥( ·𝑠𝑊)𝑦))
7 ovex 6555 . 2 (𝑥( ·𝑠𝑊)𝑦) ∈ V
86, 7fnmpt2i 7105 1 Fn (𝐾 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474   × cxp 5026   Fn wfn 5785  cfv 5790  (class class class)co 6527  Basecbs 15641  Scalarcsca 15717   ·𝑠 cvsca 15718   ·sf cscaf 18633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-slot 15645  df-base 15646  df-scaf 18635
This theorem is referenced by: (None)
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