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Theorem dflinc2 41461
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 41457 . 2 linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖)))))
2 elmapfn 7825 . . . . . . . 8 (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) → 𝑠 Fn 𝑣)
32adantr 481 . . . . . . 7 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → 𝑠 Fn 𝑣)
4 fnresi 5968 . . . . . . . 8 ( I ↾ 𝑣) Fn 𝑣
54a1i 11 . . . . . . 7 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → ( I ↾ 𝑣) Fn 𝑣)
6 vex 3194 . . . . . . . 8 𝑣 ∈ V
76a1i 11 . . . . . . 7 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → 𝑣 ∈ V)
8 inidm 3805 . . . . . . 7 (𝑣𝑣) = 𝑣
9 eqidd 2627 . . . . . . 7 (((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) ∧ 𝑖𝑣) → (𝑠𝑖) = (𝑠𝑖))
10 fvresi 6394 . . . . . . . 8 (𝑖𝑣 → (( I ↾ 𝑣)‘𝑖) = 𝑖)
1110adantl 482 . . . . . . 7 (((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) ∧ 𝑖𝑣) → (( I ↾ 𝑣)‘𝑖) = 𝑖)
123, 5, 7, 7, 8, 9, 11offval 6858 . . . . . 6 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → (𝑠𝑓 ( ·𝑠𝑚)( I ↾ 𝑣)) = (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖)))
1312eqcomd 2632 . . . . 5 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖)) = (𝑠𝑓 ( ·𝑠𝑚)( I ↾ 𝑣)))
1413oveq2d 6621 . . . 4 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → (𝑚 Σg (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖))) = (𝑚 Σg (𝑠𝑓 ( ·𝑠𝑚)( I ↾ 𝑣))))
1514mpt2eq3ia 6674 . . 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 4706 . 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 2648 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 384   = wceq 1480  wcel 1992  Vcvv 3191  𝒫 cpw 4135  cmpt 4678   I cid 4989  cres 5081   Fn wfn 5845  cfv 5850  (class class class)co 6605  cmpt2 6607  𝑓 cof 6849  𝑚 cmap 7803  Basecbs 15776  Scalarcsca 15860   ·𝑠 cvsca 15861   Σg cgsu 16017   linC clinc 41455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851  df-1st 7116  df-2nd 7117  df-map 7805  df-linc 41457
This theorem is referenced by: (None)
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