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Theorem eldiophss 37655
Description: Diophantine sets are sets of tuples of nonnegative integers. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Assertion
Ref Expression
eldiophss (𝐴 ∈ (Dioph‘𝐵) → 𝐴 ⊆ (ℕ0𝑚 (1...𝐵)))

Proof of Theorem eldiophss
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldioph3b 37645 . 2 (𝐴 ∈ (Dioph‘𝐵) ↔ (𝐵 ∈ ℕ0 ∧ ∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)}))
2 simpr 476 . . . 4 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)}) → 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)})
3 vex 3234 . . . . . . . 8 𝑑 ∈ V
4 eqeq1 2655 . . . . . . . . . 10 (𝑏 = 𝑑 → (𝑏 = (𝑐 ↾ (1...𝐵)) ↔ 𝑑 = (𝑐 ↾ (1...𝐵))))
54anbi1d 741 . . . . . . . . 9 (𝑏 = 𝑑 → ((𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) ↔ (𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)))
65rexbidv 3081 . . . . . . . 8 (𝑏 = 𝑑 → (∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) ↔ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)))
73, 6elab 3382 . . . . . . 7 (𝑑 ∈ {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)} ↔ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0))
8 simpr 476 . . . . . . . . . . 11 ((((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0𝑚 ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → 𝑑 = (𝑐 ↾ (1...𝐵)))
9 elfznn 12408 . . . . . . . . . . . . . 14 (𝑎 ∈ (1...𝐵) → 𝑎 ∈ ℕ)
109ssriv 3640 . . . . . . . . . . . . 13 (1...𝐵) ⊆ ℕ
11 elmapssres 7924 . . . . . . . . . . . . 13 ((𝑐 ∈ (ℕ0𝑚 ℕ) ∧ (1...𝐵) ⊆ ℕ) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0𝑚 (1...𝐵)))
1210, 11mpan2 707 . . . . . . . . . . . 12 (𝑐 ∈ (ℕ0𝑚 ℕ) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0𝑚 (1...𝐵)))
1312ad2antlr 763 . . . . . . . . . . 11 ((((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0𝑚 ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0𝑚 (1...𝐵)))
148, 13eqeltrd 2730 . . . . . . . . . 10 ((((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0𝑚 ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → 𝑑 ∈ (ℕ0𝑚 (1...𝐵)))
1514ex 449 . . . . . . . . 9 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0𝑚 ℕ)) → (𝑑 = (𝑐 ↾ (1...𝐵)) → 𝑑 ∈ (ℕ0𝑚 (1...𝐵))))
1615adantrd 483 . . . . . . . 8 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0𝑚 ℕ)) → ((𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) → 𝑑 ∈ (ℕ0𝑚 (1...𝐵))))
1716rexlimdva 3060 . . . . . . 7 ((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) → (∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) → 𝑑 ∈ (ℕ0𝑚 (1...𝐵))))
187, 17syl5bi 232 . . . . . 6 ((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) → (𝑑 ∈ {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)} → 𝑑 ∈ (ℕ0𝑚 (1...𝐵))))
1918ssrdv 3642 . . . . 5 ((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) → {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)} ⊆ (ℕ0𝑚 (1...𝐵)))
2019adantr 480 . . . 4 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)}) → {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)} ⊆ (ℕ0𝑚 (1...𝐵)))
212, 20eqsstrd 3672 . . 3 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)}) → 𝐴 ⊆ (ℕ0𝑚 (1...𝐵)))
2221r19.29an 3106 . 2 ((𝐵 ∈ ℕ0 ∧ ∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)}) → 𝐴 ⊆ (ℕ0𝑚 (1...𝐵)))
231, 22sylbi 207 1 (𝐴 ∈ (Dioph‘𝐵) → 𝐴 ⊆ (ℕ0𝑚 (1...𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {cab 2637  wrex 2942  wss 3607  cres 5145  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  0cc0 9974  1c1 9975  cn 11058  0cn0 11330  ...cfz 12364  mzPolycmzp 37602  Diophcdioph 37635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158  df-mzpcl 37603  df-mzp 37604  df-dioph 37636
This theorem is referenced by: (None)
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