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Theorem eq0rabdioph 38183
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 2013 . . . . . . . 8 𝑡 𝑁 ∈ ℕ0
2 nfmpt1 4972 . . . . . . . . 9 𝑡(𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)
32nfel1 2984 . . . . . . . 8 𝑡(𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))
41, 3nfan 2002 . . . . . . 7 𝑡(𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
5 zex 11720 . . . . . . . . . . . . . 14 ℤ ∈ V
6 nn0ssz 11733 . . . . . . . . . . . . . 14 0 ⊆ ℤ
7 mapss 8173 . . . . . . . . . . . . . 14 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0𝑚 (1...𝑁)) ⊆ (ℤ ↑𝑚 (1...𝑁)))
85, 6, 7mp2an 683 . . . . . . . . . . . . 13 (ℕ0𝑚 (1...𝑁)) ⊆ (ℤ ↑𝑚 (1...𝑁))
98sseli 3823 . . . . . . . . . . . 12 (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) → 𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)))
109adantl 475 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)))
11 mzpf 38142 . . . . . . . . . . . . 13 ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ)
12 mptfcl 38126 . . . . . . . . . . . . . 14 ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) → 𝐴 ∈ ℤ))
1312imp 397 . . . . . . . . . . . . 13 (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ ∧ 𝑡 ∈ (ℤ ↑𝑚 (1...𝑁))) → 𝐴 ∈ ℤ)
1411, 9, 13syl2an 589 . . . . . . . . . . . 12 (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → 𝐴 ∈ ℤ)
1514adantll 705 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → 𝐴 ∈ ℤ)
16 eqid 2825 . . . . . . . . . . . 12 (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) = (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)
1716fvmpt2 6543 . . . . . . . . . . 11 ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ∧ 𝐴 ∈ ℤ) → ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 𝐴)
1810, 15, 17syl2anc 579 . . . . . . . . . 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 2831 . . . . . . . . 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 2827 . . . . . . . 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 403 . . . . . . 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 3166 . . . . . 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 3331 . . . . . 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 210 . . . . 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 2969 . . . . . 6 𝑡(ℕ0𝑚 (1...𝑁))
26 nfcv 2969 . . . . . 6 𝑎(ℕ0𝑚 (1...𝑁))
27 nfv 2013 . . . . . 6 𝑎((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 0
28 nffvmpt1 6448 . . . . . . 7 𝑡((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎)
2928nfeq1 2983 . . . . . 6 𝑡((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0
30 fveqeq2 6446 . . . . . 6 (𝑡 = 𝑎 → (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 0 ↔ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0))
3125, 26, 27, 29, 30cbvrab 3411 . . . . 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}
3224, 31syl6eq 2877 . . . 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})
33 df-rab 3126 . . . 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)}
3432, 33syl6eq 2877 . . 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)})
35 elmapi 8149 . . . . . . . . . 10 (𝑏 ∈ (ℕ0𝑚 (1...𝑁)) → 𝑏:(1...𝑁)⟶ℕ0)
36 ffn 6282 . . . . . . . . . 10 (𝑏:(1...𝑁)⟶ℕ0𝑏 Fn (1...𝑁))
37 fnresdm 6237 . . . . . . . . . 10 (𝑏 Fn (1...𝑁) → (𝑏 ↾ (1...𝑁)) = 𝑏)
3835, 36, 373syl 18 . . . . . . . . 9 (𝑏 ∈ (ℕ0𝑚 (1...𝑁)) → (𝑏 ↾ (1...𝑁)) = 𝑏)
3938eqeq2d 2835 . . . . . . . 8 (𝑏 ∈ (ℕ0𝑚 (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = 𝑏))
40 equcom 2122 . . . . . . . 8 (𝑎 = 𝑏𝑏 = 𝑎)
4139, 40syl6bb 279 . . . . . . 7 (𝑏 ∈ (ℕ0𝑚 (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑏 = 𝑎))
4241anbi1d 623 . . . . . 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)))
4342rexbiia 3250 . . . . 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))
44 fveqeq2 6446 . . . . . 6 (𝑏 = 𝑎 → (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = 0 ↔ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0))
4544ceqsrexbv 3555 . . . . 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))
4643, 45bitr2i 268 . . . 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))
4746abbii 2944 . . 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)}
4834, 47syl6eq 2877 . 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)})
49 simpl 476 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ ℕ0)
50 nn0z 11735 . . . . 5 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
51 uzid 11990 . . . . 5 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
5250, 51syl 17 . . . 4 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ𝑁))
5352adantr 474 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ (ℤ𝑁))
54 simpr 479 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
55 eldioph 38164 . . 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))
5649, 53, 54, 55syl3anc 1494 . 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))
5748, 56eqeltrd 2906 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 198  wa 386   = wceq 1656  wcel 2164  {cab 2811  wral 3117  wrex 3118  {crab 3121  Vcvv 3414  wss 3798  cmpt 4954  cres 5348   Fn wfn 6122  wf 6123  cfv 6127  (class class class)co 6910  𝑚 cmap 8127  0cc0 10259  1c1 10260  0cn0 11625  cz 11711  cuz 11975  ...cfz 12626  mzPolycmzp 38128  Diophcdioph 38161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-of 7162  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-er 8014  df-map 8129  df-en 8229  df-dom 8230  df-sdom 8231  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-n0 11626  df-z 11712  df-uz 11976  df-fz 12627  df-mzpcl 38129  df-mzp 38130  df-dioph 38162
This theorem is referenced by:  eqrabdioph  38184  0dioph  38185  vdioph  38186  rmydioph  38423  expdioph  38432
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