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Theorem eq0rabdioph 37881
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 1998 . . . . . . . 8 𝑡 𝑁 ∈ ℕ0
2 nfmpt1 4894 . . . . . . . . 9 𝑡(𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)
32nfel1 2931 . . . . . . . 8 𝑡(𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))
41, 3nfan 1983 . . . . . . 7 𝑡(𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
5 zex 11610 . . . . . . . . . . . . . 14 ℤ ∈ V
6 nn0ssz 11622 . . . . . . . . . . . . . 14 0 ⊆ ℤ
7 mapss 8075 . . . . . . . . . . . . . 14 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0𝑚 (1...𝑁)) ⊆ (ℤ ↑𝑚 (1...𝑁)))
85, 6, 7mp2an 673 . . . . . . . . . . . . 13 (ℕ0𝑚 (1...𝑁)) ⊆ (ℤ ↑𝑚 (1...𝑁))
98sseli 3754 . . . . . . . . . . . 12 (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) → 𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)))
109adantl 468 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)))
11 mzpf 37840 . . . . . . . . . . . . 13 ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ)
12 mptfcl 37824 . . . . . . . . . . . . . 14 ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) → 𝐴 ∈ ℤ))
1312imp 394 . . . . . . . . . . . . 13 (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ ∧ 𝑡 ∈ (ℤ ↑𝑚 (1...𝑁))) → 𝐴 ∈ ℤ)
1411, 9, 13syl2an 584 . . . . . . . . . . . 12 (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → 𝐴 ∈ ℤ)
1514adantll 694 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → 𝐴 ∈ ℤ)
16 eqid 2774 . . . . . . . . . . . 12 (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) = (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)
1716fvmpt2 6450 . . . . . . . . . . 11 ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ∧ 𝐴 ∈ ℤ) → ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 𝐴)
1810, 15, 17syl2anc 574 . . . . . . . . . 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 2780 . . . . . . . . 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 2776 . . . . . . . 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 398 . . . . . . 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 3109 . . . . . 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 3273 . . . . . 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 209 . . . . 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 2916 . . . . . 6 𝑡(ℕ0𝑚 (1...𝑁))
26 nfcv 2916 . . . . . 6 𝑎(ℕ0𝑚 (1...𝑁))
27 nfv 1998 . . . . . 6 𝑎((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 0
28 nffvmpt1 6357 . . . . . . 7 𝑡((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎)
2928nfeq1 2930 . . . . . 6 𝑡((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0
30 fveq2 6348 . . . . . . 7 (𝑡 = 𝑎 → ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎))
3130eqeq1d 2776 . . . . . 6 (𝑡 = 𝑎 → (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 0 ↔ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0))
3225, 26, 27, 29, 31cbvrab 3352 . . . . 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 2824 . . . 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 3073 . . . 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 2824 . . 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 8052 . . . . . . . . . 10 (𝑏 ∈ (ℕ0𝑚 (1...𝑁)) → 𝑏:(1...𝑁)⟶ℕ0)
37 ffn 6196 . . . . . . . . . 10 (𝑏:(1...𝑁)⟶ℕ0𝑏 Fn (1...𝑁))
38 fnresdm 6151 . . . . . . . . . 10 (𝑏 Fn (1...𝑁) → (𝑏 ↾ (1...𝑁)) = 𝑏)
3936, 37, 383syl 18 . . . . . . . . 9 (𝑏 ∈ (ℕ0𝑚 (1...𝑁)) → (𝑏 ↾ (1...𝑁)) = 𝑏)
4039eqeq2d 2784 . . . . . . . 8 (𝑏 ∈ (ℕ0𝑚 (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = 𝑏))
41 equcom 2106 . . . . . . . 8 (𝑎 = 𝑏𝑏 = 𝑎)
4240, 41syl6bb 277 . . . . . . 7 (𝑏 ∈ (ℕ0𝑚 (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑏 = 𝑎))
4342anbi1d 616 . . . . . 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 3192 . . . . 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 6348 . . . . . . 7 (𝑏 = 𝑎 → ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎))
4645eqeq1d 2776 . . . . . 6 (𝑏 = 𝑎 → (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = 0 ↔ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0))
4746ceqsrexbv 3493 . . . . 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 266 . . . 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 2891 . . 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 2824 . 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 469 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ ℕ0)
52 nn0z 11624 . . . . 5 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
53 uzid 11925 . . . . 5 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
5452, 53syl 17 . . . 4 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ𝑁))
5554adantr 467 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ (ℤ𝑁))
56 simpr 472 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
57 eldioph 37862 . . 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 1480 . 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 2853 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 197  wa 383   = wceq 1634  wcel 2148  {cab 2760  wral 3064  wrex 3065  {crab 3068  Vcvv 3355  wss 3729  cmpt 4876  cres 5265   Fn wfn 6037  wf 6038  cfv 6042  (class class class)co 6812  𝑚 cmap 8030  0cc0 10159  1c1 10160  0cn0 11516  cz 11601  cuz 11910  ...cfz 12555  mzPolycmzp 37826  Diophcdioph 37859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-rep 4917  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117  ax-cnex 10215  ax-resscn 10216  ax-1cn 10217  ax-icn 10218  ax-addcl 10219  ax-addrcl 10220  ax-mulcl 10221  ax-mulrcl 10222  ax-mulcom 10223  ax-addass 10224  ax-mulass 10225  ax-distr 10226  ax-i2m1 10227  ax-1ne0 10228  ax-1rid 10229  ax-rnegex 10230  ax-rrecex 10231  ax-cnre 10232  ax-pre-lttri 10233  ax-pre-lttrn 10234  ax-pre-ltadd 10235  ax-pre-mulgt0 10236
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-int 4623  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-pred 5834  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-riota 6773  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-of 7065  df-om 7234  df-1st 7336  df-2nd 7337  df-wrecs 7580  df-recs 7642  df-rdg 7680  df-er 7917  df-map 8032  df-en 8131  df-dom 8132  df-sdom 8133  df-pnf 10299  df-mnf 10300  df-xr 10301  df-ltxr 10302  df-le 10303  df-sub 10491  df-neg 10492  df-nn 11244  df-n0 11517  df-z 11602  df-uz 11911  df-fz 12556  df-mzpcl 37827  df-mzp 37828  df-dioph 37860
This theorem is referenced by:  eqrabdioph  37882  0dioph  37883  vdioph  37884  rmydioph  38122  expdioph  38131
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