Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  scafval Structured version   Visualization version   GIF version

Theorem scafval 19650
 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 7145 . 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 19649 . 2 = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
8 ovex 7169 . 2 (𝑋 · 𝑌) ∈ V
91, 7, 8ovmpoa 7286 1 ((𝑋𝐾𝑌𝐵) → (𝑋 𝑌) = (𝑋 · 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ‘cfv 6325  (class class class)co 7136  Basecbs 16478  Scalarcsca 16563   ·𝑠 cvsca 16564   ·sf cscaf 19632 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 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-1st 7674  df-2nd 7675  df-scaf 19634 This theorem is referenced by:  lmodfopne  19669  cnmpt1vsca  22809  cnmpt2vsca  22810  nlmvscn  23303
 Copyright terms: Public domain W3C validator