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Theorem lincop 43216
 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 43214 . 2 linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥)))))
2 2fveq3 6451 . . . 4 (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘𝑀)))
32oveq1d 6937 . . 3 (𝑚 = 𝑀 → ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) = ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣))
4 fveq2 6446 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
54pweqd 4384 . . 3 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
6 id 22 . . . 4 (𝑚 = 𝑀𝑚 = 𝑀)
7 fveq2 6446 . . . . . 6 (𝑚 = 𝑀 → ( ·𝑠𝑚) = ( ·𝑠𝑀))
87oveqd 6939 . . . . 5 (𝑚 = 𝑀 → ((𝑠𝑥)( ·𝑠𝑚)𝑥) = ((𝑠𝑥)( ·𝑠𝑀)𝑥))
98mpteq2dv 4980 . . . 4 (𝑚 = 𝑀 → (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥)) = (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))
106, 9oveq12d 6940 . . 3 (𝑚 = 𝑀 → (𝑚 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥))) = (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))))
113, 5, 10mpt2eq123dv 6994 . 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 3414 . 2 (𝑀𝑋𝑀 ∈ V)
13 fvex 6459 . . . 4 (Base‘𝑀) ∈ V
1413pwex 5092 . . 3 𝒫 (Base‘𝑀) ∈ V
15 ovexd 6956 . . . 4 (𝑀𝑋 → ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣) ∈ V)
1615ralrimivw 3149 . . 3 (𝑀𝑋 → ∀𝑣 ∈ 𝒫 (Base‘𝑀)((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣) ∈ V)
17 eqid 2778 . . . 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))))
1817mpt2exxg2 43135 . . 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 581 . 2 (𝑀𝑋 → (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))) ∈ V)
201, 11, 12, 19fvmptd3 6564 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 1601   ∈ wcel 2107  ∀wral 3090  Vcvv 3398  𝒫 cpw 4379   ↦ cmpt 4965  ‘cfv 6135  (class class class)co 6922   ↦ cmpt2 6924   ↑𝑚 cmap 8140  Basecbs 16255  Scalarcsca 16341   ·𝑠 cvsca 16342   Σg cgsu 16487   linC clinc 43212 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-linc 43214 This theorem is referenced by:  lincval  43217
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