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Theorem eq0rabdioph 42787
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 1914 . . . . . . . 8 𝑡 𝑁 ∈ ℕ0
2 nfmpt1 5250 . . . . . . . . 9 𝑡(𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)
32nfel1 2922 . . . . . . . 8 𝑡(𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))
41, 3nfan 1899 . . . . . . 7 𝑡(𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
5 zex 12622 . . . . . . . . . . . . . 14 ℤ ∈ V
6 nn0ssz 12636 . . . . . . . . . . . . . 14 0 ⊆ ℤ
7 mapss 8929 . . . . . . . . . . . . . 14 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0m (1...𝑁)) ⊆ (ℤ ↑m (1...𝑁)))
85, 6, 7mp2an 692 . . . . . . . . . . . . 13 (ℕ0m (1...𝑁)) ⊆ (ℤ ↑m (1...𝑁))
98sseli 3979 . . . . . . . . . . . 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 42747 . . . . . . . . . . . . 13 ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ)
12 mptfcl 42731 . . . . . . . . . . . . . 14 ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ → (𝑡 ∈ (ℤ ↑m (1...𝑁)) → 𝐴 ∈ ℤ))
1312imp 406 . . . . . . . . . . . . 13 (((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ ∧ 𝑡 ∈ (ℤ ↑m (1...𝑁))) → 𝐴 ∈ ℤ)
1411, 9, 13syl2an 596 . . . . . . . . . . . 12 (((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ 𝑡 ∈ (ℕ0m (1...𝑁))) → 𝐴 ∈ ℤ)
1514adantll 714 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0m (1...𝑁))) → 𝐴 ∈ ℤ)
16 eqid 2737 . . . . . . . . . . . 12 (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) = (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)
1716fvmpt2 7027 . . . . . . . . . . 11 ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ∧ 𝐴 ∈ ℤ) → ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡) = 𝐴)
1810, 15, 17syl2anc 584 . . . . . . . . . 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 2743 . . . . . . . . 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 2739 . . . . . . . 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 412 . . . . . . 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 3257 . . . . . 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 3467 . . . . . 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 218 . . . . 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 2905 . . . . . 6 𝑡(ℕ0m (1...𝑁))
26 nfcv 2905 . . . . . 6 𝑎(ℕ0m (1...𝑁))
27 nfv 1914 . . . . . 6 𝑎((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡) = 0
28 nffvmpt1 6917 . . . . . . 7 𝑡((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎)
2928nfeq1 2921 . . . . . 6 𝑡((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0
30 fveqeq2 6915 . . . . . 6 (𝑡 = 𝑎 → (((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡) = 0 ↔ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0))
3125, 26, 27, 29, 30cbvrabw 3473 . . . . 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, 31eqtrdi 2793 . . . 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 3437 . . . 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, 33eqtrdi 2793 . . 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 8889 . . . . . . . . . 10 (𝑏 ∈ (ℕ0m (1...𝑁)) → 𝑏:(1...𝑁)⟶ℕ0)
36 ffn 6736 . . . . . . . . . 10 (𝑏:(1...𝑁)⟶ℕ0𝑏 Fn (1...𝑁))
37 fnresdm 6687 . . . . . . . . . 10 (𝑏 Fn (1...𝑁) → (𝑏 ↾ (1...𝑁)) = 𝑏)
3835, 36, 373syl 18 . . . . . . . . 9 (𝑏 ∈ (ℕ0m (1...𝑁)) → (𝑏 ↾ (1...𝑁)) = 𝑏)
3938eqeq2d 2748 . . . . . . . 8 (𝑏 ∈ (ℕ0m (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = 𝑏))
40 equcom 2017 . . . . . . . 8 (𝑎 = 𝑏𝑏 = 𝑎)
4139, 40bitrdi 287 . . . . . . 7 (𝑏 ∈ (ℕ0m (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑏 = 𝑎))
4241anbi1d 631 . . . . . 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 3092 . . . . 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 6915 . . . . . 6 (𝑏 = 𝑎 → (((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑏) = 0 ↔ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0))
4544ceqsrexbv 3656 . . . . 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 276 . . . 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 2809 . . 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, 47eqtrdi 2793 . 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 482 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ ℕ0)
50 nn0z 12638 . . . . 5 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
51 uzid 12893 . . . . 5 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
5250, 51syl 17 . . . 4 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ𝑁))
5352adantr 480 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ (ℤ𝑁))
54 simpr 484 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
55 eldioph 42769 . . 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 1373 . 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 2841 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 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  wss 3951  cmpt 5225  cres 5687   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  0cc0 11155  1c1 11156  0cn0 12526  cz 12613  cuz 12878  ...cfz 13547  mzPolycmzp 42733  Diophcdioph 42766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-mzpcl 42734  df-mzp 42735  df-dioph 42767
This theorem is referenced by:  eqrabdioph  42788  0dioph  42789  vdioph  42790  rmydioph  43026  expdioph  43035
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