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Theorem pm2mpval 22717
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 22715 . . . 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 481 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
5 fveq2 6902 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
65adantl 480 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = (Poly1𝑅))
74, 6oveq12d 7444 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat (Poly1𝑟)) = (𝑁 Mat (Poly1𝑅)))
87fveq2d 6906 . . . . . 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 7437 . . . . . . . . 9 (𝑁 Mat 𝑃) = (𝑁 Mat (Poly1𝑅))
1310, 12eqtri 2756 . . . . . . . 8 𝐶 = (𝑁 Mat (Poly1𝑅))
1413fveq2i 6905 . . . . . . 7 (Base‘𝐶) = (Base‘(𝑁 Mat (Poly1𝑅)))
159, 14eqtri 2756 . . . . . 6 𝐵 = (Base‘(𝑁 Mat (Poly1𝑅)))
168, 15eqtr4di 2786 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat (Poly1𝑟))) = 𝐵)
1716adantl 480 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → (Base‘(𝑛 Mat (Poly1𝑟))) = 𝐵)
18 ovex 7459 . . . . . 6 (𝑛 Mat 𝑟) ∈ V
19 fvexd 6917 . . . . . . 7 (𝑎 = (𝑛 Mat 𝑟) → (Poly1𝑎) ∈ V)
20 simpr 483 . . . . . . . . 9 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → 𝑞 = (Poly1𝑎))
21 fveq2 6902 . . . . . . . . . 10 (𝑎 = (𝑛 Mat 𝑟) → (Poly1𝑎) = (Poly1‘(𝑛 Mat 𝑟)))
2221adantr 479 . . . . . . . . 9 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (Poly1𝑎) = (Poly1‘(𝑛 Mat 𝑟)))
2320, 22eqtrd 2768 . . . . . . . 8 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → 𝑞 = (Poly1‘(𝑛 Mat 𝑟)))
2423fveq2d 6906 . . . . . . . . . 10 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → ( ·𝑠𝑞) = ( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟))))
25 eqidd 2729 . . . . . . . . . 10 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (𝑚 decompPMat 𝑘) = (𝑚 decompPMat 𝑘))
2623fveq2d 6906 . . . . . . . . . . . 12 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (mulGrp‘𝑞) = (mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))
2726fveq2d 6906 . . . . . . . . . . 11 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (.g‘(mulGrp‘𝑞)) = (.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟)))))
28 eqidd 2729 . . . . . . . . . . 11 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → 𝑘 = 𝑘)
29 fveq2 6902 . . . . . . . . . . . 12 (𝑎 = (𝑛 Mat 𝑟) → (var1𝑎) = (var1‘(𝑛 Mat 𝑟)))
3029adantr 479 . . . . . . . . . . 11 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (var1𝑎) = (var1‘(𝑛 Mat 𝑟)))
3127, 28, 30oveq123d 7447 . . . . . . . . . 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 7447 . . . . . . . . 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 5254 . . . . . . . 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 7444 . . . . . . 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 3932 . . . . . 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 3930 . . . . 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 7435 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
3837fveq2d 6906 . . . . . . . 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 6905 . . . . . . . . 9 (Poly1𝐴) = (Poly1‘(𝑁 Mat 𝑅))
4239, 41eqtri 2756 . . . . . . . 8 𝑄 = (Poly1‘(𝑁 Mat 𝑅))
4338, 42eqtr4di 2786 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1‘(𝑛 Mat 𝑟)) = 𝑄)
4438fveq2d 6906 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟))) = ( ·𝑠 ‘(Poly1‘(𝑁 Mat 𝑅))))
45 pm2mpval.m . . . . . . . . . . 11 = ( ·𝑠𝑄)
4642fveq2i 6905 . . . . . . . . . . 11 ( ·𝑠𝑄) = ( ·𝑠 ‘(Poly1‘(𝑁 Mat 𝑅)))
4745, 46eqtri 2756 . . . . . . . . . 10 = ( ·𝑠 ‘(Poly1‘(𝑁 Mat 𝑅)))
4844, 47eqtr4di 2786 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟))) = )
49 eqidd 2729 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 decompPMat 𝑘) = (𝑚 decompPMat 𝑘))
5038fveq2d 6906 . . . . . . . . . . . 12 ((𝑛 = 𝑁𝑟 = 𝑅) → (mulGrp‘(Poly1‘(𝑛 Mat 𝑟))) = (mulGrp‘(Poly1‘(𝑁 Mat 𝑅))))
5150fveq2d 6906 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟)))) = (.g‘(mulGrp‘(Poly1‘(𝑁 Mat 𝑅)))))
52 pm2mpval.e . . . . . . . . . . . 12 = (.g‘(mulGrp‘𝑄))
5342fveq2i 6905 . . . . . . . . . . . . 13 (mulGrp‘𝑄) = (mulGrp‘(Poly1‘(𝑁 Mat 𝑅)))
5453fveq2i 6905 . . . . . . . . . . . 12 (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘(Poly1‘(𝑁 Mat 𝑅))))
5552, 54eqtri 2756 . . . . . . . . . . 11 = (.g‘(mulGrp‘(Poly1‘(𝑁 Mat 𝑅))))
5651, 55eqtr4di 2786 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟)))) = )
57 eqidd 2729 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑘 = 𝑘)
5837fveq2d 6906 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (var1‘(𝑛 Mat 𝑟)) = (var1‘(𝑁 Mat 𝑅)))
59 pm2mpval.x . . . . . . . . . . . 12 𝑋 = (var1𝐴)
6040fveq2i 6905 . . . . . . . . . . . 12 (var1𝐴) = (var1‘(𝑁 Mat 𝑅))
6159, 60eqtri 2756 . . . . . . . . . . 11 𝑋 = (var1‘(𝑁 Mat 𝑅))
6258, 61eqtr4di 2786 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (var1‘(𝑛 Mat 𝑟)) = 𝑋)
6356, 57, 62oveq123d 7447 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟))) = (𝑘 𝑋))
6448, 49, 63oveq123d 7447 . . . . . . . 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 5254 . . . . . . 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 7444 . . . . . 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 480 . . . . 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 2780 . . . 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 5243 . . 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 481 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
71 elex 3492 . . . 4 (𝑅𝑉𝑅 ∈ V)
7271adantl 480 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
739fvexi 6916 . . . . 5 𝐵 ∈ V
7473mptex 7241 . . . 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 7579 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 pMatToMatPoly 𝑅) = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))
771, 76eqtrid 2780 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 394   = wceq 1533  wcel 2098  Vcvv 3473  csb 3894  cmpt 5235  cfv 6553  (class class class)co 7426  cmpo 7428  Fincfn 8970  0cn0 12510  Basecbs 17187   ·𝑠 cvsca 17244   Σg cgsu 17429  .gcmg 19030  mulGrpcmgp 20081  var1cv1 22102  Poly1cpl1 22103   Mat cmat 22327   decompPMat cdecpmat 22684   pMatToMatPoly cpm2mp 22714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-pm2mp 22715
This theorem is referenced by:  pm2mpfval  22718  pm2mpf  22720
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