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Theorem nerabdioph 42820
Description: Diophantine set builder for inequality. This not quite trivial theorem touches on something important; Diophantine sets are not closed under negation, but they contain an important subclass that is, namely the recursive sets. With this theorem and De Morgan's laws, all quantifier-free formulas can be negated. (Contributed by Stefan O'Rear, 11-Oct-2014.)
Assertion
Ref Expression
nerabdioph ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴𝐵} ∈ (Dioph‘𝑁))
Distinct variable group:   𝑡,𝑁
Allowed substitution hints:   𝐴(𝑡)   𝐵(𝑡)

Proof of Theorem nerabdioph
StepHypRef Expression
1 rabdiophlem1 42812 . . . 4 ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0m (1...𝑁))𝐴 ∈ ℤ)
2 rabdiophlem1 42812 . . . 4 ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0m (1...𝑁))𝐵 ∈ ℤ)
3 zre 12617 . . . . . . 7 (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
4 zre 12617 . . . . . . 7 (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
5 lttri2 11343 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
63, 4, 5syl2an 596 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
76ralimi 3083 . . . . 5 (∀𝑡 ∈ (ℕ0m (1...𝑁))(𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ∀𝑡 ∈ (ℕ0m (1...𝑁))(𝐴𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
8 r19.26 3111 . . . . 5 (∀𝑡 ∈ (ℕ0m (1...𝑁))(𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ↔ (∀𝑡 ∈ (ℕ0m (1...𝑁))𝐴 ∈ ℤ ∧ ∀𝑡 ∈ (ℕ0m (1...𝑁))𝐵 ∈ ℤ))
9 rabbi 3467 . . . . 5 (∀𝑡 ∈ (ℕ0m (1...𝑁))(𝐴𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)) ↔ {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴𝐵} = {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ (𝐴 < 𝐵𝐵 < 𝐴)})
107, 8, 93imtr3i 291 . . . 4 ((∀𝑡 ∈ (ℕ0m (1...𝑁))𝐴 ∈ ℤ ∧ ∀𝑡 ∈ (ℕ0m (1...𝑁))𝐵 ∈ ℤ) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴𝐵} = {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ (𝐴 < 𝐵𝐵 < 𝐴)})
111, 2, 10syl2an 596 . . 3 (((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴𝐵} = {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ (𝐴 < 𝐵𝐵 < 𝐴)})
12113adant1 1131 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴𝐵} = {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ (𝐴 < 𝐵𝐵 < 𝐴)})
13 ltrabdioph 42819 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴 < 𝐵} ∈ (Dioph‘𝑁))
14 ltrabdioph 42819 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐵 < 𝐴} ∈ (Dioph‘𝑁))
15143com23 1127 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐵 < 𝐴} ∈ (Dioph‘𝑁))
16 orrabdioph 42792 . . 3 (({𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴 < 𝐵} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐵 < 𝐴} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ (𝐴 < 𝐵𝐵 < 𝐴)} ∈ (Dioph‘𝑁))
1713, 15, 16syl2anc 584 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ (𝐴 < 𝐵𝐵 < 𝐴)} ∈ (Dioph‘𝑁))
1812, 17eqeltrd 2841 1 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴𝐵} ∈ (Dioph‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  {crab 3436   class class class wbr 5143  cmpt 5225  cfv 6561  (class class class)co 7431  m cmap 8866  cr 11154  1c1 11156   < clt 11295  0cn0 12526  cz 12613  ...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-inf2 9681  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-1o 8506  df-oadd 8510  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  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-hash 14370  df-mzpcl 42734  df-mzp 42735  df-dioph 42767
This theorem is referenced by: (None)
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