Theorem nerabdioph 43387

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 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ≠ 𝐵} ∈ (Dioph‘𝑁))

Distinct variable group:   𝑡,𝑁

Allowed substitution hints:   𝐴(𝑡)   𝐵(𝑡)

Proof of Theorem nerabdioph

Step	Hyp	Ref	Expression
1		rabdiophlem1 43379	. . . 4 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))𝐴 ∈ ℤ)
2		rabdiophlem1 43379	. . . 4 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))𝐵 ∈ ℤ)
3		zre 12573	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
4		zre 12573	. . . . . . 7 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
5		lttri2 11266	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
6	3, 4, 5	syl2an 605	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
7	6	ralimi 3100	. . . . 5 ⊢ (∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))(𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))(𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
8		r19.26 3123	. . . . 5 ⊢ (∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))(𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ↔ (∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))𝐴 ∈ ℤ ∧ ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))𝐵 ∈ ℤ))
9		rabbi 3445	. . . . 5 ⊢ (∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))(𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) ↔ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ≠ 𝐵} = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)})
10	7, 8, 9	3imtr3i 293	. . . 4 ⊢ ((∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))𝐴 ∈ ℤ ∧ ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))𝐵 ∈ ℤ) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ≠ 𝐵} = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)})
11	1, 2, 10	syl2an 605	. . 3 ⊢ (((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ≠ 𝐵} = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)})
12	11	3adant1 1144	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ≠ 𝐵} = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)})
13		ltrabdioph 43386	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 < 𝐵} ∈ (Dioph‘𝑁))
14		ltrabdioph 43386	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐵 < 𝐴} ∈ (Dioph‘𝑁))
15	14	3com23 1140	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐵 < 𝐴} ∈ (Dioph‘𝑁))
16		orrabdioph 43363	. . 3 ⊢ (({𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 < 𝐵} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐵 < 𝐴} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)} ∈ (Dioph‘𝑁))
17	13, 15, 16	syl2anc 593	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)} ∈ (Dioph‘𝑁))
18	12, 17	eqeltrd 2863	1 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ≠ 𝐵} ∈ (Dioph‘𝑁))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  ∀wral 3077  {crab 3415   class class class wbr 5101   ↦ cmpt 5182  ‘cfv 6522  (class class class)co 7397   ↑_m cmap 8809  ℝcr 11073  1c1 11075   < clt 11217  ℕ₀cn0 12482  ℤcz 12569  ...cfz 13513  mzPolycmzp 43304  Diophcdioph 43337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-inf2 9597  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-of 7661  df-om 7848  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-1o 8438  df-oadd 8442  df-er 8679  df-map 8811  df-en 8929  df-dom 8930  df-sdom 8931  df-fin 8932  df-dju 9860  df-card 9898  df-pnf 11219  df-mnf 11220  df-xr 11221  df-ltxr 11222  df-le 11223  df-sub 11417  df-neg 11418  df-nn 12212  df-n0 12483  df-z 12570  df-uz 12841  df-fz 13514  df-hash 14345  df-mzpcl 43305  df-mzp 43306  df-dioph 43338

This theorem is referenced by: (None)