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Theorem eq0rabdioph 37657
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 1883 . . . . . . . 8 𝑡 𝑁 ∈ ℕ0
2 nfmpt1 4780 . . . . . . . . 9 𝑡(𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)
32nfel1 2808 . . . . . . . 8 𝑡(𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))
41, 3nfan 1868 . . . . . . 7 𝑡(𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
5 zex 11424 . . . . . . . . . . . . . 14 ℤ ∈ V
6 nn0ssz 11436 . . . . . . . . . . . . . 14 0 ⊆ ℤ
7 mapss 7942 . . . . . . . . . . . . . 14 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0𝑚 (1...𝑁)) ⊆ (ℤ ↑𝑚 (1...𝑁)))
85, 6, 7mp2an 708 . . . . . . . . . . . . 13 (ℕ0𝑚 (1...𝑁)) ⊆ (ℤ ↑𝑚 (1...𝑁))
98sseli 3632 . . . . . . . . . . . 12 (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) → 𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)))
109adantl 481 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)))
11 mzpf 37616 . . . . . . . . . . . . 13 ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ)
12 mptfcl 37600 . . . . . . . . . . . . . 14 ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) → 𝐴 ∈ ℤ))
1312imp 444 . . . . . . . . . . . . 13 (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ ∧ 𝑡 ∈ (ℤ ↑𝑚 (1...𝑁))) → 𝐴 ∈ ℤ)
1411, 9, 13syl2an 493 . . . . . . . . . . . 12 (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → 𝐴 ∈ ℤ)
1514adantll 750 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → 𝐴 ∈ ℤ)
16 eqid 2651 . . . . . . . . . . . 12 (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) = (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)
1716fvmpt2 6330 . . . . . . . . . . 11 ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ∧ 𝐴 ∈ ℤ) → ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 𝐴)
1810, 15, 17syl2anc 694 . . . . . . . . . 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 2657 . . . . . . . . 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 2653 . . . . . . . 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 449 . . . . . . 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 2986 . . . . . 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 3150 . . . . . 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 2793 . . . . . 6 𝑡(ℕ0𝑚 (1...𝑁))
26 nfcv 2793 . . . . . 6 𝑎(ℕ0𝑚 (1...𝑁))
27 nfv 1883 . . . . . 6 𝑎((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 0
28 nffvmpt1 6237 . . . . . . 7 𝑡((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎)
2928nfeq1 2807 . . . . . 6 𝑡((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0
30 fveq2 6229 . . . . . . 7 (𝑡 = 𝑎 → ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎))
3130eqeq1d 2653 . . . . . 6 (𝑡 = 𝑎 → (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 0 ↔ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0))
3225, 26, 27, 29, 31cbvrab 3229 . . . . 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 2701 . . . 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 2950 . . . 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 2701 . . 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 7921 . . . . . . . . . 10 (𝑏 ∈ (ℕ0𝑚 (1...𝑁)) → 𝑏:(1...𝑁)⟶ℕ0)
37 ffn 6083 . . . . . . . . . 10 (𝑏:(1...𝑁)⟶ℕ0𝑏 Fn (1...𝑁))
38 fnresdm 6038 . . . . . . . . . 10 (𝑏 Fn (1...𝑁) → (𝑏 ↾ (1...𝑁)) = 𝑏)
3936, 37, 383syl 18 . . . . . . . . 9 (𝑏 ∈ (ℕ0𝑚 (1...𝑁)) → (𝑏 ↾ (1...𝑁)) = 𝑏)
4039eqeq2d 2661 . . . . . . . 8 (𝑏 ∈ (ℕ0𝑚 (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = 𝑏))
41 equcom 1991 . . . . . . . 8 (𝑎 = 𝑏𝑏 = 𝑎)
4240, 41syl6bb 276 . . . . . . 7 (𝑏 ∈ (ℕ0𝑚 (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑏 = 𝑎))
4342anbi1d 741 . . . . . 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 3069 . . . . 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 6229 . . . . . . 7 (𝑏 = 𝑎 → ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎))
4645eqeq1d 2653 . . . . . 6 (𝑏 = 𝑎 → (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = 0 ↔ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0))
4746ceqsrexbv 3368 . . . . 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 2768 . . 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 2701 . 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 472 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ ℕ0)
52 nn0z 11438 . . . . 5 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
53 uzid 11740 . . . . 5 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
5452, 53syl 17 . . . 4 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ𝑁))
5554adantr 480 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ (ℤ𝑁))
56 simpr 476 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
57 eldioph 37638 . . 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 1366 . 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 2730 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 383   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  cmpt 4762  cres 5145   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  0cc0 9974  1c1 9975  0cn0 11330  cz 11415  cuz 11725  ...cfz 12364  mzPolycmzp 37602  Diophcdioph 37635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-mzpcl 37603  df-mzp 37604  df-dioph 37636
This theorem is referenced by:  eqrabdioph  37658  0dioph  37659  vdioph  37660  rmydioph  37898  expdioph  37907
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