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Theorem lincop 44608
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 44606 . 2 linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥)))))
2 2fveq3 6651 . . . 4 (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘𝑀)))
32oveq1d 7148 . . 3 (𝑚 = 𝑀 → ((Base‘(Scalar‘𝑚)) ↑m 𝑣) = ((Base‘(Scalar‘𝑀)) ↑m 𝑣))
4 fveq2 6646 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
54pweqd 4534 . . 3 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
6 id 22 . . . 4 (𝑚 = 𝑀𝑚 = 𝑀)
7 fveq2 6646 . . . . . 6 (𝑚 = 𝑀 → ( ·𝑠𝑚) = ( ·𝑠𝑀))
87oveqd 7150 . . . . 5 (𝑚 = 𝑀 → ((𝑠𝑥)( ·𝑠𝑚)𝑥) = ((𝑠𝑥)( ·𝑠𝑀)𝑥))
98mpteq2dv 5138 . . . 4 (𝑚 = 𝑀 → (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥)) = (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))
106, 9oveq12d 7151 . . 3 (𝑚 = 𝑀 → (𝑚 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥))) = (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥))))
113, 5, 10mpoeq123dv 7206 . 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 3491 . 2 (𝑀𝑋𝑀 ∈ V)
13 fvex 6659 . . . 4 (Base‘𝑀) ∈ V
1413pwex 5257 . . 3 𝒫 (Base‘𝑀) ∈ V
15 ovexd 7168 . . . 4 (𝑀𝑋 → ((Base‘(Scalar‘𝑀)) ↑m 𝑣) ∈ V)
1615ralrimivw 3170 . . 3 (𝑀𝑋 → ∀𝑣 ∈ 𝒫 (Base‘𝑀)((Base‘(Scalar‘𝑀)) ↑m 𝑣) ∈ V)
17 eqid 2820 . . . 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 44531 . . 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 589 . 2 (𝑀𝑋 → (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))) ∈ V)
201, 11, 12, 19fvmptd3 6767 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 1537  wcel 2114  wral 3125  Vcvv 3473  𝒫 cpw 4515  cmpt 5122  cfv 6331  (class class class)co 7133  cmpo 7135  m cmap 8384  Basecbs 16462  Scalarcsca 16547   ·𝑠 cvsca 16548   Σg cgsu 16693   linC clinc 44604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-1st 7667  df-2nd 7668  df-linc 44606
This theorem is referenced by:  lincval  44609
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