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Theorem eldiophss 42785
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 42776 . 2 (𝐴 ∈ (Dioph‘𝐵) ↔ (𝐵 ∈ ℕ0 ∧ ∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)}))
2 simpr 484 . . . 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 3484 . . . . . . . 8 𝑑 ∈ V
4 eqeq1 2741 . . . . . . . . . 10 (𝑏 = 𝑑 → (𝑏 = (𝑐 ↾ (1...𝐵)) ↔ 𝑑 = (𝑐 ↾ (1...𝐵))))
54anbi1d 631 . . . . . . . . 9 (𝑏 = 𝑑 → ((𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) ↔ (𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)))
65rexbidv 3179 . . . . . . . 8 (𝑏 = 𝑑 → (∃𝑐 ∈ (ℕ0m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) ↔ ∃𝑐 ∈ (ℕ0m ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)))
73, 6elab 3679 . . . . . . 7 (𝑑 ∈ {𝑏 ∣ ∃𝑐 ∈ (ℕ0m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)} ↔ ∃𝑐 ∈ (ℕ0m ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0))
8 simpr 484 . . . . . . . . . . 11 ((((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0m ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → 𝑑 = (𝑐 ↾ (1...𝐵)))
9 elfznn 13593 . . . . . . . . . . . . . 14 (𝑎 ∈ (1...𝐵) → 𝑎 ∈ ℕ)
109ssriv 3987 . . . . . . . . . . . . 13 (1...𝐵) ⊆ ℕ
11 elmapssres 8907 . . . . . . . . . . . . 13 ((𝑐 ∈ (ℕ0m ℕ) ∧ (1...𝐵) ⊆ ℕ) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0m (1...𝐵)))
1210, 11mpan2 691 . . . . . . . . . . . 12 (𝑐 ∈ (ℕ0m ℕ) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0m (1...𝐵)))
1312ad2antlr 727 . . . . . . . . . . 11 ((((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0m ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0m (1...𝐵)))
148, 13eqeltrd 2841 . . . . . . . . . 10 ((((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0m ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → 𝑑 ∈ (ℕ0m (1...𝐵)))
1514ex 412 . . . . . . . . 9 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0m ℕ)) → (𝑑 = (𝑐 ↾ (1...𝐵)) → 𝑑 ∈ (ℕ0m (1...𝐵))))
1615adantrd 491 . . . . . . . 8 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0m ℕ)) → ((𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) → 𝑑 ∈ (ℕ0m (1...𝐵))))
1716rexlimdva 3155 . . . . . . 7 ((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) → (∃𝑐 ∈ (ℕ0m ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) → 𝑑 ∈ (ℕ0m (1...𝐵))))
187, 17biimtrid 242 . . . . . 6 ((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) → (𝑑 ∈ {𝑏 ∣ ∃𝑐 ∈ (ℕ0m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)} → 𝑑 ∈ (ℕ0m (1...𝐵))))
1918ssrdv 3989 . . . . 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 4018 . . 3 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)}) → 𝐴 ⊆ (ℕ0m (1...𝐵)))
2221r19.29an 3158 . 2 ((𝐵 ∈ ℕ0 ∧ ∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)}) → 𝐴 ⊆ (ℕ0m (1...𝐵)))
231, 22sylbi 217 1 (𝐴 ∈ (Dioph‘𝐵) → 𝐴 ⊆ (ℕ0m (1...𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  wss 3951  cres 5687  cfv 6561  (class class class)co 7431  m cmap 8866  0cc0 11155  1c1 11156  cn 12266  0cn0 12526  ...cfz 13547  mzPolycmzp 42733  Diophcdioph 42766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370  df-mzpcl 42734  df-mzp 42735  df-dioph 42767
This theorem is referenced by: (None)
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