Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dflinc2 Structured version   Visualization version   GIF version

Theorem dflinc2 48256
Description: Alternative definition of linear combinations using the function operation. (Contributed by AV, 1-Apr-2019.)
Assertion
Ref Expression
dflinc2 linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑠f ( ·𝑠𝑚)( I ↾ 𝑣)))))
Distinct variable group:   𝑚,𝑠,𝑣

Proof of Theorem dflinc2
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 df-linc 48252 . 2 linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖)))))
2 elmapfn 8904 . . . . . . . 8 (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) → 𝑠 Fn 𝑣)
32adantr 480 . . . . . . 7 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → 𝑠 Fn 𝑣)
4 fnresi 6698 . . . . . . . 8 ( I ↾ 𝑣) Fn 𝑣
54a1i 11 . . . . . . 7 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → ( I ↾ 𝑣) Fn 𝑣)
6 vex 3482 . . . . . . . 8 𝑣 ∈ V
76a1i 11 . . . . . . 7 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → 𝑣 ∈ V)
8 inidm 4235 . . . . . . 7 (𝑣𝑣) = 𝑣
9 eqidd 2736 . . . . . . 7 (((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) ∧ 𝑖𝑣) → (𝑠𝑖) = (𝑠𝑖))
10 fvresi 7193 . . . . . . . 8 (𝑖𝑣 → (( I ↾ 𝑣)‘𝑖) = 𝑖)
1110adantl 481 . . . . . . 7 (((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) ∧ 𝑖𝑣) → (( I ↾ 𝑣)‘𝑖) = 𝑖)
123, 5, 7, 7, 8, 9, 11offval 7706 . . . . . 6 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → (𝑠f ( ·𝑠𝑚)( I ↾ 𝑣)) = (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖)))
1312eqcomd 2741 . . . . 5 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖)) = (𝑠f ( ·𝑠𝑚)( I ↾ 𝑣)))
1413oveq2d 7447 . . . 4 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → (𝑚 Σg (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖))) = (𝑚 Σg (𝑠f ( ·𝑠𝑚)( I ↾ 𝑣))))
1514mpoeq3ia 7511 . . 3 (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖)))) = (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑠f ( ·𝑠𝑚)( I ↾ 𝑣))))
1615mpteq2i 5253 . 2 (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖))))) = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑠f ( ·𝑠𝑚)( I ↾ 𝑣)))))
171, 16eqtri 2763 1 linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑠f ( ·𝑠𝑚)( I ↾ 𝑣)))))
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  𝒫 cpw 4605  cmpt 5231   I cid 5582  cres 5691   Fn wfn 6558  cfv 6563  (class class class)co 7431  cmpo 7433  f cof 7695  m cmap 8865  Basecbs 17245  Scalarcsca 17301   ·𝑠 cvsca 17302   Σg cgsu 17487   linC clinc 48250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-1st 8013  df-2nd 8014  df-map 8867  df-linc 48252
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator