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Theorem dflinc2 43039
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 43035 . 2 linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖)))))
2 elmapfn 8145 . . . . . . . 8 (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) → 𝑠 Fn 𝑣)
32adantr 474 . . . . . . 7 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → 𝑠 Fn 𝑣)
4 fnresi 6241 . . . . . . . 8 ( I ↾ 𝑣) Fn 𝑣
54a1i 11 . . . . . . 7 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → ( I ↾ 𝑣) Fn 𝑣)
6 vex 3417 . . . . . . . 8 𝑣 ∈ V
76a1i 11 . . . . . . 7 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → 𝑣 ∈ V)
8 inidm 4047 . . . . . . 7 (𝑣𝑣) = 𝑣
9 eqidd 2826 . . . . . . 7 (((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) ∧ 𝑖𝑣) → (𝑠𝑖) = (𝑠𝑖))
10 fvresi 6691 . . . . . . . 8 (𝑖𝑣 → (( I ↾ 𝑣)‘𝑖) = 𝑖)
1110adantl 475 . . . . . . 7 (((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) ∧ 𝑖𝑣) → (( I ↾ 𝑣)‘𝑖) = 𝑖)
123, 5, 7, 7, 8, 9, 11offval 7164 . . . . . 6 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → (𝑠𝑓 ( ·𝑠𝑚)( I ↾ 𝑣)) = (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖)))
1312eqcomd 2831 . . . . 5 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖)) = (𝑠𝑓 ( ·𝑠𝑚)( I ↾ 𝑣)))
1413oveq2d 6921 . . . 4 ((𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) ∧ 𝑣 ∈ 𝒫 (Base‘𝑚)) → (𝑚 Σg (𝑖𝑣 ↦ ((𝑠𝑖)( ·𝑠𝑚)𝑖))) = (𝑚 Σg (𝑠𝑓 ( ·𝑠𝑚)( I ↾ 𝑣))))
1514mpt2eq3ia 6980 . . 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 4964 . 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 2849 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 386   = wceq 1656  wcel 2164  Vcvv 3414  𝒫 cpw 4378  cmpt 4952   I cid 5249  cres 5344   Fn wfn 6118  cfv 6123  (class class class)co 6905  cmpt2 6907  𝑓 cof 7155  𝑚 cmap 8122  Basecbs 16222  Scalarcsca 16308   ·𝑠 cvsca 16309   Σg cgsu 16454   linC clinc 43033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-of 7157  df-1st 7428  df-2nd 7429  df-map 8124  df-linc 43035
This theorem is referenced by: (None)
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