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Theorem eldiophss 36853
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 36843 . 2 (𝐴 ∈ (Dioph‘𝐵) ↔ (𝐵 ∈ ℕ0 ∧ ∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)}))
2 simpr 477 . . . 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 3192 . . . . . . . 8 𝑑 ∈ V
4 eqeq1 2625 . . . . . . . . . 10 (𝑏 = 𝑑 → (𝑏 = (𝑐 ↾ (1...𝐵)) ↔ 𝑑 = (𝑐 ↾ (1...𝐵))))
54anbi1d 740 . . . . . . . . 9 (𝑏 = 𝑑 → ((𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) ↔ (𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)))
65rexbidv 3046 . . . . . . . 8 (𝑏 = 𝑑 → (∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) ↔ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)))
73, 6elab 3337 . . . . . . 7 (𝑑 ∈ {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)} ↔ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0))
8 simpr 477 . . . . . . . . . . 11 ((((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0𝑚 ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → 𝑑 = (𝑐 ↾ (1...𝐵)))
9 elfznn 12320 . . . . . . . . . . . . . 14 (𝑎 ∈ (1...𝐵) → 𝑎 ∈ ℕ)
109ssriv 3591 . . . . . . . . . . . . 13 (1...𝐵) ⊆ ℕ
11 elmapssres 7834 . . . . . . . . . . . . 13 ((𝑐 ∈ (ℕ0𝑚 ℕ) ∧ (1...𝐵) ⊆ ℕ) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0𝑚 (1...𝐵)))
1210, 11mpan2 706 . . . . . . . . . . . 12 (𝑐 ∈ (ℕ0𝑚 ℕ) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0𝑚 (1...𝐵)))
1312ad2antlr 762 . . . . . . . . . . 11 ((((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0𝑚 ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0𝑚 (1...𝐵)))
148, 13eqeltrd 2698 . . . . . . . . . 10 ((((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0𝑚 ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → 𝑑 ∈ (ℕ0𝑚 (1...𝐵)))
1514ex 450 . . . . . . . . 9 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0𝑚 ℕ)) → (𝑑 = (𝑐 ↾ (1...𝐵)) → 𝑑 ∈ (ℕ0𝑚 (1...𝐵))))
1615adantrd 484 . . . . . . . 8 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0𝑚 ℕ)) → ((𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) → 𝑑 ∈ (ℕ0𝑚 (1...𝐵))))
1716rexlimdva 3025 . . . . . . 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 3593 . . . . 5 ((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) → {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)} ⊆ (ℕ0𝑚 (1...𝐵)))
2019adantr 481 . . . 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 3623 . . 3 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)}) → 𝐴 ⊆ (ℕ0𝑚 (1...𝐵)))
2221r19.29an 3071 . 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 384   = wceq 1480  wcel 1987  {cab 2607  wrex 2908  wss 3559  cres 5081  cfv 5852  (class class class)co 6610  𝑚 cmap 7809  0cc0 9888  1c1 9889  cn 10972  0cn0 11244  ...cfz 12276  mzPolycmzp 36800  Diophcdioph 36833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-card 8717  df-cda 8942  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-hash 13066  df-mzpcl 36801  df-mzp 36802  df-dioph 36834
This theorem is referenced by: (None)
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