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Theorem pm2mpval 22144
Description: Value of the transformation of a polynomial matrix into a polynomial over matrices. (Contributed by AV, 5-Dec-2019.)
Hypotheses
Ref Expression
pm2mpval.p 𝑃 = (Poly1𝑅)
pm2mpval.c 𝐶 = (𝑁 Mat 𝑃)
pm2mpval.b 𝐵 = (Base‘𝐶)
pm2mpval.m = ( ·𝑠𝑄)
pm2mpval.e = (.g‘(mulGrp‘𝑄))
pm2mpval.x 𝑋 = (var1𝐴)
pm2mpval.a 𝐴 = (𝑁 Mat 𝑅)
pm2mpval.q 𝑄 = (Poly1𝐴)
pm2mpval.t 𝑇 = (𝑁 pMatToMatPoly 𝑅)
Assertion
Ref Expression
pm2mpval ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑇 = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))
Distinct variable groups:   𝐵,𝑚   𝑘,𝑁,𝑚   𝑅,𝑘,𝑚   𝑚,𝑉
Allowed substitution hints:   𝐴(𝑘,𝑚)   𝐵(𝑘)   𝐶(𝑘,𝑚)   𝑃(𝑘,𝑚)   𝑄(𝑘,𝑚)   𝑇(𝑘,𝑚)   (𝑘,𝑚)   (𝑘,𝑚)   𝑉(𝑘)   𝑋(𝑘,𝑚)

Proof of Theorem pm2mpval
Dummy variables 𝑛 𝑟 𝑎 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pm2mpval.t . 2 𝑇 = (𝑁 pMatToMatPoly 𝑅)
2 df-pm2mp 22142 . . . 4 pMatToMatPoly = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat (Poly1𝑟))) ↦ (𝑛 Mat 𝑟) / 𝑎(Poly1𝑎) / 𝑞(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎)))))))
32a1i 11 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → pMatToMatPoly = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat (Poly1𝑟))) ↦ (𝑛 Mat 𝑟) / 𝑎(Poly1𝑎) / 𝑞(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎))))))))
4 simpl 483 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
5 fveq2 6842 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
65adantl 482 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = (Poly1𝑅))
74, 6oveq12d 7375 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat (Poly1𝑟)) = (𝑁 Mat (Poly1𝑅)))
87fveq2d 6846 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat (Poly1𝑟))) = (Base‘(𝑁 Mat (Poly1𝑅))))
9 pm2mpval.b . . . . . . 7 𝐵 = (Base‘𝐶)
10 pm2mpval.c . . . . . . . . 9 𝐶 = (𝑁 Mat 𝑃)
11 pm2mpval.p . . . . . . . . . 10 𝑃 = (Poly1𝑅)
1211oveq2i 7368 . . . . . . . . 9 (𝑁 Mat 𝑃) = (𝑁 Mat (Poly1𝑅))
1310, 12eqtri 2764 . . . . . . . 8 𝐶 = (𝑁 Mat (Poly1𝑅))
1413fveq2i 6845 . . . . . . 7 (Base‘𝐶) = (Base‘(𝑁 Mat (Poly1𝑅)))
159, 14eqtri 2764 . . . . . 6 𝐵 = (Base‘(𝑁 Mat (Poly1𝑅)))
168, 15eqtr4di 2794 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat (Poly1𝑟))) = 𝐵)
1716adantl 482 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → (Base‘(𝑛 Mat (Poly1𝑟))) = 𝐵)
18 ovex 7390 . . . . . 6 (𝑛 Mat 𝑟) ∈ V
19 fvexd 6857 . . . . . . 7 (𝑎 = (𝑛 Mat 𝑟) → (Poly1𝑎) ∈ V)
20 simpr 485 . . . . . . . . 9 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → 𝑞 = (Poly1𝑎))
21 fveq2 6842 . . . . . . . . . 10 (𝑎 = (𝑛 Mat 𝑟) → (Poly1𝑎) = (Poly1‘(𝑛 Mat 𝑟)))
2221adantr 481 . . . . . . . . 9 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (Poly1𝑎) = (Poly1‘(𝑛 Mat 𝑟)))
2320, 22eqtrd 2776 . . . . . . . 8 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → 𝑞 = (Poly1‘(𝑛 Mat 𝑟)))
2423fveq2d 6846 . . . . . . . . . 10 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → ( ·𝑠𝑞) = ( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟))))
25 eqidd 2737 . . . . . . . . . 10 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (𝑚 decompPMat 𝑘) = (𝑚 decompPMat 𝑘))
2623fveq2d 6846 . . . . . . . . . . . 12 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (mulGrp‘𝑞) = (mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))
2726fveq2d 6846 . . . . . . . . . . 11 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (.g‘(mulGrp‘𝑞)) = (.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟)))))
28 eqidd 2737 . . . . . . . . . . 11 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → 𝑘 = 𝑘)
29 fveq2 6842 . . . . . . . . . . . 12 (𝑎 = (𝑛 Mat 𝑟) → (var1𝑎) = (var1‘(𝑛 Mat 𝑟)))
3029adantr 481 . . . . . . . . . . 11 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (var1𝑎) = (var1‘(𝑛 Mat 𝑟)))
3127, 28, 30oveq123d 7378 . . . . . . . . . 10 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (𝑘(.g‘(mulGrp‘𝑞))(var1𝑎)) = (𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟))))
3224, 25, 31oveq123d 7378 . . . . . . . . 9 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎))) = ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟)))))
3332mpteq2dv 5207 . . . . . . . 8 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎)))) = (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟))))))
3423, 33oveq12d 7375 . . . . . . 7 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎))))) = ((Poly1‘(𝑛 Mat 𝑟)) Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟)))))))
3519, 34csbied 3893 . . . . . 6 (𝑎 = (𝑛 Mat 𝑟) → (Poly1𝑎) / 𝑞(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎))))) = ((Poly1‘(𝑛 Mat 𝑟)) Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟)))))))
3618, 35csbie 3891 . . . . 5 (𝑛 Mat 𝑟) / 𝑎(Poly1𝑎) / 𝑞(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎))))) = ((Poly1‘(𝑛 Mat 𝑟)) Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟))))))
37 oveq12 7366 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
3837fveq2d 6846 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1‘(𝑛 Mat 𝑟)) = (Poly1‘(𝑁 Mat 𝑅)))
39 pm2mpval.q . . . . . . . . 9 𝑄 = (Poly1𝐴)
40 pm2mpval.a . . . . . . . . . 10 𝐴 = (𝑁 Mat 𝑅)
4140fveq2i 6845 . . . . . . . . 9 (Poly1𝐴) = (Poly1‘(𝑁 Mat 𝑅))
4239, 41eqtri 2764 . . . . . . . 8 𝑄 = (Poly1‘(𝑁 Mat 𝑅))
4338, 42eqtr4di 2794 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1‘(𝑛 Mat 𝑟)) = 𝑄)
4438fveq2d 6846 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟))) = ( ·𝑠 ‘(Poly1‘(𝑁 Mat 𝑅))))
45 pm2mpval.m . . . . . . . . . . 11 = ( ·𝑠𝑄)
4642fveq2i 6845 . . . . . . . . . . 11 ( ·𝑠𝑄) = ( ·𝑠 ‘(Poly1‘(𝑁 Mat 𝑅)))
4745, 46eqtri 2764 . . . . . . . . . 10 = ( ·𝑠 ‘(Poly1‘(𝑁 Mat 𝑅)))
4844, 47eqtr4di 2794 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟))) = )
49 eqidd 2737 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 decompPMat 𝑘) = (𝑚 decompPMat 𝑘))
5038fveq2d 6846 . . . . . . . . . . . 12 ((𝑛 = 𝑁𝑟 = 𝑅) → (mulGrp‘(Poly1‘(𝑛 Mat 𝑟))) = (mulGrp‘(Poly1‘(𝑁 Mat 𝑅))))
5150fveq2d 6846 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟)))) = (.g‘(mulGrp‘(Poly1‘(𝑁 Mat 𝑅)))))
52 pm2mpval.e . . . . . . . . . . . 12 = (.g‘(mulGrp‘𝑄))
5342fveq2i 6845 . . . . . . . . . . . . 13 (mulGrp‘𝑄) = (mulGrp‘(Poly1‘(𝑁 Mat 𝑅)))
5453fveq2i 6845 . . . . . . . . . . . 12 (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘(Poly1‘(𝑁 Mat 𝑅))))
5552, 54eqtri 2764 . . . . . . . . . . 11 = (.g‘(mulGrp‘(Poly1‘(𝑁 Mat 𝑅))))
5651, 55eqtr4di 2794 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟)))) = )
57 eqidd 2737 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑘 = 𝑘)
5837fveq2d 6846 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (var1‘(𝑛 Mat 𝑟)) = (var1‘(𝑁 Mat 𝑅)))
59 pm2mpval.x . . . . . . . . . . . 12 𝑋 = (var1𝐴)
6040fveq2i 6845 . . . . . . . . . . . 12 (var1𝐴) = (var1‘(𝑁 Mat 𝑅))
6159, 60eqtri 2764 . . . . . . . . . . 11 𝑋 = (var1‘(𝑁 Mat 𝑅))
6258, 61eqtr4di 2794 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (var1‘(𝑛 Mat 𝑟)) = 𝑋)
6356, 57, 62oveq123d 7378 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟))) = (𝑘 𝑋))
6448, 49, 63oveq123d 7378 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟)))) = ((𝑚 decompPMat 𝑘) (𝑘 𝑋)))
6564mpteq2dv 5207 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟))))) = (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))
6643, 65oveq12d 7375 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → ((Poly1‘(𝑛 Mat 𝑟)) Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟)))))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋)))))
6766adantl 482 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → ((Poly1‘(𝑛 Mat 𝑟)) Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟)))))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋)))))
6836, 67eqtrid 2788 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → (𝑛 Mat 𝑟) / 𝑎(Poly1𝑎) / 𝑞(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎))))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋)))))
6917, 68mpteq12dv 5196 . . 3 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → (𝑚 ∈ (Base‘(𝑛 Mat (Poly1𝑟))) ↦ (𝑛 Mat 𝑟) / 𝑎(Poly1𝑎) / 𝑞(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎)))))) = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))
70 simpl 483 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
71 elex 3463 . . . 4 (𝑅𝑉𝑅 ∈ V)
7271adantl 482 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
739fvexi 6856 . . . . 5 𝐵 ∈ V
7473mptex 7173 . . . 4 (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))) ∈ V
7574a1i 11 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))) ∈ V)
763, 69, 70, 72, 75ovmpod 7507 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 pMatToMatPoly 𝑅) = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))
771, 76eqtrid 2788 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑇 = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  csb 3855  cmpt 5188  cfv 6496  (class class class)co 7357  cmpo 7359  Fincfn 8883  0cn0 12413  Basecbs 17083   ·𝑠 cvsca 17137   Σg cgsu 17322  .gcmg 18872  mulGrpcmgp 19896  var1cv1 21547  Poly1cpl1 21548   Mat cmat 21754   decompPMat cdecpmat 22111   pMatToMatPoly cpm2mp 22141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-pm2mp 22142
This theorem is referenced by:  pm2mpfval  22145  pm2mpf  22147
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