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Theorem elnn0rabdioph 39731
 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 3229 . . . . 5 (𝐴 ∈ ℕ0 ↔ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴)
21rabbii 3423 . . . 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 2958 . . . 4 𝑡(ℕ0m (1...𝑁))
5 nfcv 2958 . . . 4 𝑎(ℕ0m (1...𝑁))
6 nfv 1915 . . . 4 𝑎𝑏 ∈ ℕ0 𝑏 = 𝐴
7 nfcv 2958 . . . . 5 𝑡0
8 nfcsb1v 3855 . . . . . 6 𝑡𝑎 / 𝑡𝐴
98nfeq2 2975 . . . . 5 𝑡 𝑏 = 𝑎 / 𝑡𝐴
107, 9nfrex 3271 . . . 4 𝑡𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴
11 csbeq1a 3845 . . . . . 6 (𝑡 = 𝑎𝐴 = 𝑎 / 𝑡𝐴)
1211eqeq2d 2812 . . . . 5 (𝑡 = 𝑎 → (𝑏 = 𝐴𝑏 = 𝑎 / 𝑡𝐴))
1312rexbidv 3259 . . . 4 (𝑡 = 𝑎 → (∃𝑏 ∈ ℕ0 𝑏 = 𝐴 ↔ ∃𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴))
144, 5, 6, 10, 13cbvrabw 3440 . . 3 {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴} = {𝑎 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴}
153, 14eqtrdi 2852 . 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 11929 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
1716adantr 484 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ ℕ0)
18 ovex 7172 . . . . 5 (1...(𝑁 + 1)) ∈ V
19 nn0p1nn 11928 . . . . . . 7 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
20 elfz1end 12936 . . . . . . 7 ((𝑁 + 1) ∈ ℕ ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
2119, 20sylib 221 . . . . . 6 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
2221adantr 484 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
23 mzpproj 39665 . . . . 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 590 . . . 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 2801 . . . . 5 (𝑁 + 1) = (𝑁 + 1)
2625rabdiophlem2 39730 . . . 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 39705 . . . 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 1368 . . 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 2805 . . . 4 (𝑏 = (𝑐‘(𝑁 + 1)) → (𝑏 = 𝑎 / 𝑡𝐴 ↔ (𝑐‘(𝑁 + 1)) = 𝑎 / 𝑡𝐴))
30 csbeq1 3834 . . . . 5 (𝑎 = (𝑐 ↾ (1...𝑁)) → 𝑎 / 𝑡𝐴 = (𝑐 ↾ (1...𝑁)) / 𝑡𝐴)
3130eqeq2d 2812 . . . 4 (𝑎 = (𝑐 ↾ (1...𝑁)) → ((𝑐‘(𝑁 + 1)) = 𝑎 / 𝑡𝐴 ↔ (𝑐‘(𝑁 + 1)) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐴))
3225, 29, 31rexrabdioph 39722 . . 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 594 . 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 2893 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 399   = wceq 1538   ∈ wcel 2112  ∃wrex 3110  {crab 3113  Vcvv 3444  ⦋csb 3831   ↦ cmpt 5113   ↾ cres 5525  ‘cfv 6328  (class class class)co 7139   ↑m cmap 8393  1c1 10531   + caddc 10533  ℕcn 11629  ℕ0cn0 11889  ℤcz 11973  ...cfz 12889  mzPolycmzp 39650  Diophcdioph 39683 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-dju 9318  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-hash 13691  df-mzpcl 39651  df-mzp 39652  df-dioph 39684 This theorem is referenced by:  lerabdioph  39733
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