Theorem nerabdioph 42294

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 42286	. . . 4 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))𝐴 ∈ ℤ)
2		rabdiophlem1 42286	. . . 4 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))𝐵 ∈ ℤ)
3		zre 12592	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
4		zre 12592	. . . . . . 7 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
5		lttri2 11326	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
6	3, 4, 5	syl2an 594	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
7	6	ralimi 3073	. . . . 5 ⊢ (∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))(𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))(𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
8		r19.26 3101	. . . . 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 3450	. . . . 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 290	. . . 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 594	. . 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 1127	. 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 42293	. . 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 42293	. . . 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 1123	. . 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 42266	. . 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 582	. 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 2825	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 205   ∧ wa 394   ∨ wo 845   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2930  ∀wral 3051  {crab 3419   class class class wbr 5143   ↦ cmpt 5226  ‘cfv 6543  (class class class)co 7416   ↑_m cmap 8843  ℝcr 11137  1c1 11139   < clt 11278  ℕ₀cn0 12502  ℤcz 12588  ...cfz 13516  mzPolycmzp 42207  Diophcdioph 42240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-inf2 9664  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-oadd 8489  df-er 8723  df-map 8845  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-dju 9924  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-n0 12503  df-z 12589  df-uz 12853  df-fz 13517  df-hash 14322  df-mzpcl 42208  df-mzp 42209  df-dioph 42241

This theorem is referenced by: (None)