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Theorem dflinc2 48327
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 48323 . 2 linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖)))))
2 elmapfn 8905 . . . . . . . 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 6697 . . . . . . . 8 ( I ↾ 𝑣) Fn 𝑣
54a1i 11 . . . . . . 7 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → ( I ↾ 𝑣) Fn 𝑣)
6 vex 3484 . . . . . . . 8 𝑣 ∈ V
76a1i 11 . . . . . . 7 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → 𝑣 ∈ V)
8 inidm 4227 . . . . . . 7 (𝑣𝑣) = 𝑣
9 eqidd 2738 . . . . . . 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 2743 . . . . 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 5247 . 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 2765 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 1540  wcel 2108  Vcvv 3480  𝒫 cpw 4600  cmpt 5225   I cid 5577  cres 5687   Fn wfn 6556  cfv 6561  (class class class)co 7431  cmpo 7433  f cof 7695  m cmap 8866  Basecbs 17247  Scalarcsca 17300   ·𝑠 cvsca 17301   Σg cgsu 17485   linC clinc 48321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-1st 8014  df-2nd 8015  df-map 8868  df-linc 48323
This theorem is referenced by: (None)
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