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Theorem dflinc2 42594
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 42590 . 2 linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖)))))
2 elmapfn 7965 . . . . . . . 8 (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) → 𝑠 Fn 𝑣)
32adantr 472 . . . . . . 7 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → 𝑠 Fn 𝑣)
4 fnresi 6089 . . . . . . . 8 ( I ↾ 𝑣) Fn 𝑣
54a1i 11 . . . . . . 7 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → ( I ↾ 𝑣) Fn 𝑣)
6 vex 3275 . . . . . . . 8 𝑣 ∈ V
76a1i 11 . . . . . . 7 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → 𝑣 ∈ V)
8 inidm 3898 . . . . . . 7 (𝑣𝑣) = 𝑣
9 eqidd 2693 . . . . . . 7 (((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) ∧ 𝑖𝑣) → (𝑠𝑖) = (𝑠𝑖))
10 fvresi 6523 . . . . . . . 8 (𝑖𝑣 → (( I ↾ 𝑣)‘𝑖) = 𝑖)
1110adantl 473 . . . . . . 7 (((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) ∧ 𝑖𝑣) → (( I ↾ 𝑣)‘𝑖) = 𝑖)
123, 5, 7, 7, 8, 9, 11offval 6989 . . . . . 6 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → (𝑠𝑓 ( ·𝑠𝑚)( I ↾ 𝑣)) = (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖)))
1312eqcomd 2698 . . . . 5 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖)) = (𝑠𝑓 ( ·𝑠𝑚)( I ↾ 𝑣)))
1413oveq2d 6749 . . . 4 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → (𝑚 Σg (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖))) = (𝑚 Σg (𝑠𝑓 ( ·𝑠𝑚)( I ↾ 𝑣))))
1514mpt2eq3ia 6805 . . 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 4817 . 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 2714 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 383   = wceq 1564  wcel 2071  Vcvv 3272  𝒫 cpw 4234  cmpt 4805   I cid 5095  cres 5188   Fn wfn 5964  cfv 5969  (class class class)co 6733  cmpt2 6735  𝑓 cof 6980  𝑚 cmap 7942  Basecbs 15948  Scalarcsca 16035   ·𝑠 cvsca 16036   Σg cgsu 16192   linC clinc 42588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-8 2073  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-rep 4847  ax-sep 4857  ax-nul 4865  ax-pow 4916  ax-pr 4979  ax-un 7034
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-rex 2988  df-reu 2989  df-rab 2991  df-v 3274  df-sbc 3510  df-csb 3608  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-nul 3992  df-if 4163  df-pw 4236  df-sn 4254  df-pr 4256  df-op 4260  df-uni 4513  df-iun 4598  df-br 4729  df-opab 4789  df-mpt 4806  df-id 5096  df-xp 5192  df-rel 5193  df-cnv 5194  df-co 5195  df-dm 5196  df-rn 5197  df-res 5198  df-ima 5199  df-iota 5932  df-fun 5971  df-fn 5972  df-f 5973  df-f1 5974  df-fo 5975  df-f1o 5976  df-fv 5977  df-ov 6736  df-oprab 6737  df-mpt2 6738  df-of 6982  df-1st 7253  df-2nd 7254  df-map 7944  df-linc 42590
This theorem is referenced by: (None)
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