Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lincval Structured version   Visualization version   GIF version

Theorem lincval 41992
 Description: The value of a linear combination. (Contributed by AV, 30-Mar-2019.)
Assertion
Ref Expression
lincval ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑆( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))))
Distinct variable groups:   𝑥,𝑀   𝑥,𝑆   𝑥,𝑉
Allowed substitution hint:   𝑋(𝑥)

Proof of Theorem lincval
Dummy variables 𝑠 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lincop 41991 . . . 4 (𝑀𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))))
213ad2ant1 1075 . . 3 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))))
32oveqd 6566 . 2 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑆( linC ‘𝑀)𝑉) = (𝑆(𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))))𝑉))
4 simp2 1055 . . 3 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
5 simp3 1056 . . 3 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
6 ovex 6577 . . . 4 (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))) ∈ V
76a1i 11 . . 3 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))) ∈ V)
8 simpr 476 . . . . . 6 ((𝑠 = 𝑆𝑣 = 𝑉) → 𝑣 = 𝑉)
9 fveq1 6102 . . . . . . . 8 (𝑠 = 𝑆 → (𝑠𝑥) = (𝑆𝑥))
109oveq1d 6564 . . . . . . 7 (𝑠 = 𝑆 → ((𝑠𝑥)( ·𝑠𝑀)𝑥) = ((𝑆𝑥)( ·𝑠𝑀)𝑥))
1110adantr 480 . . . . . 6 ((𝑠 = 𝑆𝑣 = 𝑉) → ((𝑠𝑥)( ·𝑠𝑀)𝑥) = ((𝑆𝑥)( ·𝑠𝑀)𝑥))
128, 11mpteq12dv 4663 . . . . 5 ((𝑠 = 𝑆𝑣 = 𝑉) → (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)) = (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥)))
1312oveq2d 6565 . . . 4 ((𝑠 = 𝑆𝑣 = 𝑉) → (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))) = (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))))
14 oveq2 6557 . . . 4 (𝑣 = 𝑉 → ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣) = ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
15 eqid 2610 . . . 4 (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))))
1613, 14, 15ovmpt2x2 41912 . . 3 ((𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))) ∈ V) → (𝑆(𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))))𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))))
174, 5, 7, 16syl3anc 1318 . 2 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑆(𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))))𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))))
183, 17eqtrd 2644 1 ((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑆( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  𝒫 cpw 4108   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ↑𝑚 cmap 7744  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772   Σg cgsu 15924   linC clinc 41987 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-linc 41989 This theorem is referenced by:  lincfsuppcl  41996  linccl  41997  lincval0  41998  lincvalsng  41999  lincvalpr  42001  lincvalsc0  42004  linc0scn0  42006  lincdifsn  42007  linc1  42008  lincellss  42009  lincsum  42012  lincscm  42013  lindslinindimp2lem4  42044  lindslinindsimp2lem5  42045  snlindsntor  42054  lincresunit3lem2  42063  lincresunit3  42064  zlmodzxzldeplem3  42085
 Copyright terms: Public domain W3C validator