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Theorem lincop 48330
Description: A linear combination as operation. (Contributed by AV, 30-Mar-2019.)
Assertion
Ref Expression
lincop (𝑀𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))))
Distinct variable groups:   𝑀,𝑠,𝑣,𝑥   𝑣,𝑋
Allowed substitution hints:   𝑋(𝑥,𝑠)

Proof of Theorem lincop
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 df-linc 48328 . 2 linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥)))))
2 2fveq3 6910 . . . 4 (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘𝑀)))
32oveq1d 7447 . . 3 (𝑚 = 𝑀 → ((Base‘(Scalar‘𝑚)) ↑m 𝑣) = ((Base‘(Scalar‘𝑀)) ↑m 𝑣))
4 fveq2 6905 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
54pweqd 4616 . . 3 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
6 id 22 . . . 4 (𝑚 = 𝑀𝑚 = 𝑀)
7 fveq2 6905 . . . . . 6 (𝑚 = 𝑀 → ( ·𝑠𝑚) = ( ·𝑠𝑀))
87oveqd 7449 . . . . 5 (𝑚 = 𝑀 → ((𝑠𝑥)( ·𝑠𝑚)𝑥) = ((𝑠𝑥)( ·𝑠𝑀)𝑥))
98mpteq2dv 5243 . . . 4 (𝑚 = 𝑀 → (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥)) = (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))
106, 9oveq12d 7450 . . 3 (𝑚 = 𝑀 → (𝑚 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥))) = (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))))
113, 5, 10mpoeq123dv 7509 . 2 (𝑚 = 𝑀 → (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥)))) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))))
12 elex 3500 . 2 (𝑀𝑋𝑀 ∈ V)
13 fvex 6918 . . . 4 (Base‘𝑀) ∈ V
1413pwex 5379 . . 3 𝒫 (Base‘𝑀) ∈ V
15 ovexd 7467 . . . 4 (𝑀𝑋 → ((Base‘(Scalar‘𝑀)) ↑m 𝑣) ∈ V)
1615ralrimivw 3149 . . 3 (𝑀𝑋 → ∀𝑣 ∈ 𝒫 (Base‘𝑀)((Base‘(Scalar‘𝑀)) ↑m 𝑣) ∈ V)
17 eqid 2736 . . . 4 (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))))
1817mpoexxg2 48259 . . 3 ((𝒫 (Base‘𝑀) ∈ V ∧ ∀𝑣 ∈ 𝒫 (Base‘𝑀)((Base‘(Scalar‘𝑀)) ↑m 𝑣) ∈ V) → (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))) ∈ V)
1914, 16, 18sylancr 587 . 2 (𝑀𝑋 → (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))) ∈ V)
201, 11, 12, 19fvmptd3 7038 1 (𝑀𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479  𝒫 cpw 4599  cmpt 5224  cfv 6560  (class class class)co 7432  cmpo 7434  m cmap 8867  Basecbs 17248  Scalarcsca 17301   ·𝑠 cvsca 17302   Σg cgsu 17486   linC clinc 48326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-linc 48328
This theorem is referenced by:  lincval  48331
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