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Theorem scafval 19582
Description: The scalar multiplication operation as 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)
scaffval.s · = ( ·𝑠𝑊)
Assertion
Ref Expression
scafval ((𝑋𝐾𝑌𝐵) → (𝑋 𝑌) = (𝑋 · 𝑌))

Proof of Theorem scafval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 7154 . 2 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥 · 𝑦) = (𝑋 · 𝑌))
2 scaffval.b . . 3 𝐵 = (Base‘𝑊)
3 scaffval.f . . 3 𝐹 = (Scalar‘𝑊)
4 scaffval.k . . 3 𝐾 = (Base‘𝐹)
5 scaffval.a . . 3 = ( ·sf𝑊)
6 scaffval.s . . 3 · = ( ·𝑠𝑊)
72, 3, 4, 5, 6scaffval 19581 . 2 = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
8 ovex 7178 . 2 (𝑋 · 𝑌) ∈ V
91, 7, 8ovmpoa 7294 1 ((𝑋𝐾𝑌𝐵) → (𝑋 𝑌) = (𝑋 · 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145  Basecbs 16471  Scalarcsca 16556   ·𝑠 cvsca 16557   ·sf cscaf 19564
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-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
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-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-pw 4537  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-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-scaf 19566
This theorem is referenced by:  lmodfopne  19601  cnmpt1vsca  22729  cnmpt2vsca  22730  nlmvscn  23223
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