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Theorem eq0rabdioph 36841
Description: This is the first of a number of theorems which allow sets to be proven Diophantine by syntactic induction, and models the correspondence between Diophantine sets and monotone existential first-order logic. This first theorem shows that the zero set of an implicit polynomial is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Assertion
Ref Expression
eq0rabdioph ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 = 0} ∈ (Dioph‘𝑁))
Distinct variable group:   𝑡,𝑁
Allowed substitution hint:   𝐴(𝑡)

Proof of Theorem eq0rabdioph
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1840 . . . . . . . 8 𝑡 𝑁 ∈ ℕ0
2 nfmpt1 4709 . . . . . . . . 9 𝑡(𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)
32nfel1 2775 . . . . . . . 8 𝑡(𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))
41, 3nfan 1825 . . . . . . 7 𝑡(𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
5 zex 11333 . . . . . . . . . . . . . 14 ℤ ∈ V
6 nn0ssz 11345 . . . . . . . . . . . . . 14 0 ⊆ ℤ
7 mapss 7847 . . . . . . . . . . . . . 14 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0𝑚 (1...𝑁)) ⊆ (ℤ ↑𝑚 (1...𝑁)))
85, 6, 7mp2an 707 . . . . . . . . . . . . 13 (ℕ0𝑚 (1...𝑁)) ⊆ (ℤ ↑𝑚 (1...𝑁))
98sseli 3580 . . . . . . . . . . . 12 (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) → 𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)))
109adantl 482 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)))
11 mzpf 36800 . . . . . . . . . . . . 13 ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ)
12 mptfcl 36784 . . . . . . . . . . . . . 14 ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) → 𝐴 ∈ ℤ))
1312imp 445 . . . . . . . . . . . . 13 (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ ∧ 𝑡 ∈ (ℤ ↑𝑚 (1...𝑁))) → 𝐴 ∈ ℤ)
1411, 9, 13syl2an 494 . . . . . . . . . . . 12 (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → 𝐴 ∈ ℤ)
1514adantll 749 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → 𝐴 ∈ ℤ)
16 eqid 2621 . . . . . . . . . . . 12 (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) = (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)
1716fvmpt2 6250 . . . . . . . . . . 11 ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ∧ 𝐴 ∈ ℤ) → ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 𝐴)
1810, 15, 17syl2anc 692 . . . . . . . . . 10 (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 𝐴)
1918eqcomd 2627 . . . . . . . . 9 (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → 𝐴 = ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡))
2019eqeq1d 2623 . . . . . . . 8 (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → (𝐴 = 0 ↔ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 0))
2120ex 450 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) → (𝐴 = 0 ↔ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 0)))
224, 21ralrimi 2951 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → ∀𝑡 ∈ (ℕ0𝑚 (1...𝑁))(𝐴 = 0 ↔ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 0))
23 rabbi 3109 . . . . . 6 (∀𝑡 ∈ (ℕ0𝑚 (1...𝑁))(𝐴 = 0 ↔ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 0) ↔ {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 = 0} = {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 0})
2422, 23sylib 208 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 = 0} = {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 0})
25 nfcv 2761 . . . . . 6 𝑡(ℕ0𝑚 (1...𝑁))
26 nfcv 2761 . . . . . 6 𝑎(ℕ0𝑚 (1...𝑁))
27 nfv 1840 . . . . . 6 𝑎((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 0
28 nffvmpt1 6158 . . . . . . 7 𝑡((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎)
2928nfeq1 2774 . . . . . 6 𝑡((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0
30 fveq2 6150 . . . . . . 7 (𝑡 = 𝑎 → ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎))
3130eqeq1d 2623 . . . . . 6 (𝑡 = 𝑎 → (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 0 ↔ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0))
3225, 26, 27, 29, 31cbvrab 3184 . . . . 5 {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 0} = {𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0}
3324, 32syl6eq 2671 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 = 0} = {𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0})
34 df-rab 2916 . . . 4 {𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0} = {𝑎 ∣ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0)}
3533, 34syl6eq 2671 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 = 0} = {𝑎 ∣ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0)})
36 elmapi 7826 . . . . . . . . . 10 (𝑏 ∈ (ℕ0𝑚 (1...𝑁)) → 𝑏:(1...𝑁)⟶ℕ0)
37 ffn 6004 . . . . . . . . . 10 (𝑏:(1...𝑁)⟶ℕ0𝑏 Fn (1...𝑁))
38 fnresdm 5960 . . . . . . . . . 10 (𝑏 Fn (1...𝑁) → (𝑏 ↾ (1...𝑁)) = 𝑏)
3936, 37, 383syl 18 . . . . . . . . 9 (𝑏 ∈ (ℕ0𝑚 (1...𝑁)) → (𝑏 ↾ (1...𝑁)) = 𝑏)
4039eqeq2d 2631 . . . . . . . 8 (𝑏 ∈ (ℕ0𝑚 (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = 𝑏))
41 equcom 1942 . . . . . . . 8 (𝑎 = 𝑏𝑏 = 𝑎)
4240, 41syl6bb 276 . . . . . . 7 (𝑏 ∈ (ℕ0𝑚 (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑏 = 𝑎))
4342anbi1d 740 . . . . . 6 (𝑏 ∈ (ℕ0𝑚 (1...𝑁)) → ((𝑎 = (𝑏 ↾ (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = 0) ↔ (𝑏 = 𝑎 ∧ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = 0)))
4443rexbiia 3033 . . . . 5 (∃𝑏 ∈ (ℕ0𝑚 (1...𝑁))(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = 0) ↔ ∃𝑏 ∈ (ℕ0𝑚 (1...𝑁))(𝑏 = 𝑎 ∧ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = 0))
45 fveq2 6150 . . . . . . 7 (𝑏 = 𝑎 → ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎))
4645eqeq1d 2623 . . . . . 6 (𝑏 = 𝑎 → (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = 0 ↔ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0))
4746ceqsrexbv 3321 . . . . 5 (∃𝑏 ∈ (ℕ0𝑚 (1...𝑁))(𝑏 = 𝑎 ∧ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = 0) ↔ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0))
4844, 47bitr2i 265 . . . 4 ((𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0) ↔ ∃𝑏 ∈ (ℕ0𝑚 (1...𝑁))(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = 0))
4948abbii 2736 . . 3 {𝑎 ∣ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 (1...𝑁))(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = 0)}
5035, 49syl6eq 2671 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 = 0} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 (1...𝑁))(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = 0)})
51 simpl 473 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ ℕ0)
52 nn0z 11347 . . . . 5 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
53 uzid 11649 . . . . 5 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
5452, 53syl 17 . . . 4 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ𝑁))
5554adantr 481 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ (ℤ𝑁))
56 simpr 477 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
57 eldioph 36822 . . 3 ((𝑁 ∈ ℕ0𝑁 ∈ (ℤ𝑁) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 (1...𝑁))(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = 0)} ∈ (Dioph‘𝑁))
5851, 55, 56, 57syl3anc 1323 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 (1...𝑁))(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = 0)} ∈ (Dioph‘𝑁))
5950, 58eqeltrd 2698 1 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 = 0} ∈ (Dioph‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  wss 3556  cmpt 4675  cres 5078   Fn wfn 5844  wf 5845  cfv 5849  (class class class)co 6607  𝑚 cmap 7805  0cc0 9883  1c1 9884  0cn0 11239  cz 11324  cuz 11634  ...cfz 12271  mzPolycmzp 36786  Diophcdioph 36819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-of 6853  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-er 7690  df-map 7807  df-en 7903  df-dom 7904  df-sdom 7905  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-n0 11240  df-z 11325  df-uz 11635  df-fz 12272  df-mzpcl 36787  df-mzp 36788  df-dioph 36820
This theorem is referenced by:  eqrabdioph  36842  0dioph  36843  vdioph  36844  rmydioph  37082  expdioph  37091
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