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Theorem elnn0rabdioph 39278
Description: Diophantine set builder for nonnegativity constraints. The first builder which uses a witness variable internally; an expression is nonnegative if there is a nonnegative integer equal to it. (Contributed by Stefan O'Rear, 11-Oct-2014.)
Assertion
Ref Expression
elnn0rabdioph ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴 ∈ ℕ0} ∈ (Dioph‘𝑁))
Distinct variable group:   𝑡,𝑁
Allowed substitution hint:   𝐴(𝑡)

Proof of Theorem elnn0rabdioph
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 risset 3264 . . . . 5 (𝐴 ∈ ℕ0 ↔ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴)
21rabbii 3471 . . . 4 {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴 ∈ ℕ0} = {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴}
32a1i 11 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴 ∈ ℕ0} = {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴})
4 nfcv 2974 . . . 4 𝑡(ℕ0m (1...𝑁))
5 nfcv 2974 . . . 4 𝑎(ℕ0m (1...𝑁))
6 nfv 1906 . . . 4 𝑎𝑏 ∈ ℕ0 𝑏 = 𝐴
7 nfcv 2974 . . . . 5 𝑡0
8 nfcsb1v 3904 . . . . . 6 𝑡𝑎 / 𝑡𝐴
98nfeq2 2992 . . . . 5 𝑡 𝑏 = 𝑎 / 𝑡𝐴
107, 9nfrex 3306 . . . 4 𝑡𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴
11 csbeq1a 3894 . . . . . 6 (𝑡 = 𝑎𝐴 = 𝑎 / 𝑡𝐴)
1211eqeq2d 2829 . . . . 5 (𝑡 = 𝑎 → (𝑏 = 𝐴𝑏 = 𝑎 / 𝑡𝐴))
1312rexbidv 3294 . . . 4 (𝑡 = 𝑎 → (∃𝑏 ∈ ℕ0 𝑏 = 𝐴 ↔ ∃𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴))
144, 5, 6, 10, 13cbvrabw 3487 . . 3 {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴} = {𝑎 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴}
153, 14syl6eq 2869 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴 ∈ ℕ0} = {𝑎 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴})
16 peano2nn0 11925 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
1716adantr 481 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ ℕ0)
18 ovex 7178 . . . . 5 (1...(𝑁 + 1)) ∈ V
19 nn0p1nn 11924 . . . . . . 7 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
20 elfz1end 12925 . . . . . . 7 ((𝑁 + 1) ∈ ℕ ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
2119, 20sylib 219 . . . . . 6 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
2221adantr 481 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
23 mzpproj 39212 . . . . 5 (((1...(𝑁 + 1)) ∈ V ∧ (𝑁 + 1) ∈ (1...(𝑁 + 1))) → (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))))
2418, 22, 23sylancr 587 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))))
25 eqid 2818 . . . . 5 (𝑁 + 1) = (𝑁 + 1)
2625rabdiophlem2 39277 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) ∈ (mzPoly‘(1...(𝑁 + 1))))
27 eqrabdioph 39252 . . . 4 (((𝑁 + 1) ∈ ℕ0 ∧ (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))) ∧ (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) ∈ (mzPoly‘(1...(𝑁 + 1)))) → {𝑐 ∈ (ℕ0m (1...(𝑁 + 1))) ∣ (𝑐‘(𝑁 + 1)) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐴} ∈ (Dioph‘(𝑁 + 1)))
2817, 24, 26, 27syl3anc 1363 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑐 ∈ (ℕ0m (1...(𝑁 + 1))) ∣ (𝑐‘(𝑁 + 1)) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐴} ∈ (Dioph‘(𝑁 + 1)))
29 eqeq1 2822 . . . 4 (𝑏 = (𝑐‘(𝑁 + 1)) → (𝑏 = 𝑎 / 𝑡𝐴 ↔ (𝑐‘(𝑁 + 1)) = 𝑎 / 𝑡𝐴))
30 csbeq1 3883 . . . . 5 (𝑎 = (𝑐 ↾ (1...𝑁)) → 𝑎 / 𝑡𝐴 = (𝑐 ↾ (1...𝑁)) / 𝑡𝐴)
3130eqeq2d 2829 . . . 4 (𝑎 = (𝑐 ↾ (1...𝑁)) → ((𝑐‘(𝑁 + 1)) = 𝑎 / 𝑡𝐴 ↔ (𝑐‘(𝑁 + 1)) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐴))
3225, 29, 31rexrabdioph 39269 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑐 ∈ (ℕ0m (1...(𝑁 + 1))) ∣ (𝑐‘(𝑁 + 1)) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐴} ∈ (Dioph‘(𝑁 + 1))) → {𝑎 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴} ∈ (Dioph‘𝑁))
3328, 32syldan 591 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑎 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴} ∈ (Dioph‘𝑁))
3415, 33eqeltrd 2910 1 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴 ∈ ℕ0} ∈ (Dioph‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wrex 3136  {crab 3139  Vcvv 3492  csb 3880  cmpt 5137  cres 5550  cfv 6348  (class class class)co 7145  m cmap 8395  1c1 10526   + caddc 10528  cn 11626  0cn0 11885  cz 11969  ...cfz 12880  mzPolycmzp 39197  Diophcdioph 39230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-hash 13679  df-mzpcl 39198  df-mzp 39199  df-dioph 39231
This theorem is referenced by:  lerabdioph  39280
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