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Theorem eldioph 41798
Description: Condition for a set to be Diophantine (unpacking existential quantifier). (Contributed by Stefan O'Rear, 5-Oct-2014.)
Assertion
Ref Expression
eldioph ((𝑁 ∈ β„•0 ∧ 𝐾 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑃 ∈ (mzPolyβ€˜(1...𝐾))) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))
Distinct variable groups:   𝑑,𝑁,𝑒   𝑑,𝐾,𝑒   𝑑,𝑃,𝑒
Proof of Theorem eldioph
Dummy variables π‘˜ 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1136 . 2 ((𝑁 ∈ β„•0 ∧ 𝐾 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑃 ∈ (mzPolyβ€˜(1...𝐾))) β†’ 𝑁 ∈ β„•0)
2 simp2 1137 . . 3 ((𝑁 ∈ β„•0 ∧ 𝐾 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑃 ∈ (mzPolyβ€˜(1...𝐾))) β†’ 𝐾 ∈ (β„€β‰₯β€˜π‘))
3 simp3 1138 . . . 4 ((𝑁 ∈ β„•0 ∧ 𝐾 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑃 ∈ (mzPolyβ€˜(1...𝐾))) β†’ 𝑃 ∈ (mzPolyβ€˜(1...𝐾)))
4 eqidd 2733 . . . 4 ((𝑁 ∈ β„•0 ∧ 𝐾 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑃 ∈ (mzPolyβ€˜(1...𝐾))) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)})
5 fveq1 6890 . . . . . . . . 9 (𝑝 = 𝑃 β†’ (π‘β€˜π‘’) = (π‘ƒβ€˜π‘’))
65eqeq1d 2734 . . . . . . . 8 (𝑝 = 𝑃 β†’ ((π‘β€˜π‘’) = 0 ↔ (π‘ƒβ€˜π‘’) = 0))
76anbi2d 629 . . . . . . 7 (𝑝 = 𝑃 β†’ ((𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0) ↔ (𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)))
87rexbidv 3178 . . . . . 6 (𝑝 = 𝑃 β†’ (βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0) ↔ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)))
98abbidv 2801 . . . . 5 (𝑝 = 𝑃 β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)})
109rspceeqv 3633 . . . 4 ((𝑃 ∈ (mzPolyβ€˜(1...𝐾)) ∧ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)}) β†’ βˆƒπ‘ ∈ (mzPolyβ€˜(1...𝐾)){𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)})
113, 4, 10syl2anc 584 . . 3 ((𝑁 ∈ β„•0 ∧ 𝐾 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑃 ∈ (mzPolyβ€˜(1...𝐾))) β†’ βˆƒπ‘ ∈ (mzPolyβ€˜(1...𝐾)){𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)})
12 oveq2 7419 . . . . . 6 (π‘˜ = 𝐾 β†’ (1...π‘˜) = (1...𝐾))
1312fveq2d 6895 . . . . 5 (π‘˜ = 𝐾 β†’ (mzPolyβ€˜(1...π‘˜)) = (mzPolyβ€˜(1...𝐾)))
1412oveq2d 7427 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (β„•0 ↑m (1...π‘˜)) = (β„•0 ↑m (1...𝐾)))
1514rexeqdv 3326 . . . . . . 7 (π‘˜ = 𝐾 β†’ (βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0) ↔ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)))
1615abbidv 2801 . . . . . 6 (π‘˜ = 𝐾 β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)})
1716eqeq2d 2743 . . . . 5 (π‘˜ = 𝐾 β†’ ({𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} ↔ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
1813, 17rexeqbidv 3343 . . . 4 (π‘˜ = 𝐾 β†’ (βˆƒπ‘ ∈ (mzPolyβ€˜(1...π‘˜)){𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} ↔ βˆƒπ‘ ∈ (mzPolyβ€˜(1...𝐾)){𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
1918rspcev 3612 . . 3 ((𝐾 ∈ (β„€β‰₯β€˜π‘) ∧ βˆƒπ‘ ∈ (mzPolyβ€˜(1...𝐾)){𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}) β†’ βˆƒπ‘˜ ∈ (β„€β‰₯β€˜π‘)βˆƒπ‘ ∈ (mzPolyβ€˜(1...π‘˜)){𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)})
202, 11, 19syl2anc 584 . 2 ((𝑁 ∈ β„•0 ∧ 𝐾 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑃 ∈ (mzPolyβ€˜(1...𝐾))) β†’ βˆƒπ‘˜ ∈ (β„€β‰₯β€˜π‘)βˆƒπ‘ ∈ (mzPolyβ€˜(1...π‘˜)){𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)})
21 eldiophb 41797 . 2 ({𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} ∈ (Diophβ€˜π‘) ↔ (𝑁 ∈ β„•0 ∧ βˆƒπ‘˜ ∈ (β„€β‰₯β€˜π‘)βˆƒπ‘ ∈ (mzPolyβ€˜(1...π‘˜)){𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
221, 20, 21sylanbrc 583 1 ((𝑁 ∈ β„•0 ∧ 𝐾 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑃 ∈ (mzPolyβ€˜(1...𝐾))) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2709  βˆƒwrex 3070   β†Ύ cres 5678  β€˜cfv 6543  (class class class)co 7411   ↑m cmap 8822  0cc0 11112  1c1 11113  β„•0cn0 12476  β„€β‰₯cuz 12826  ...cfz 13488  mzPolycmzp 41762  Diophcdioph 41795
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-addcl 11172  ax-pre-lttri 11186  ax-pre-lttrn 11187
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-dioph 41796
This theorem is referenced by:  eldioph2  41802  eq0rabdioph  41816
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