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Theorem eldioph3 36806
 Description: Inference version of eldioph3b 36805 with quantifier expanded. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Assertion
Ref Expression
eldioph3 ((𝑁 ∈ ℕ0𝑃 ∈ (mzPoly‘ℕ)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} ∈ (Dioph‘𝑁))
Distinct variable groups:   𝑢,𝑡,𝑁   𝑡,𝑃,𝑢

Proof of Theorem eldioph3
Dummy variables 𝑎 𝑏 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 473 . 2 ((𝑁 ∈ ℕ0𝑃 ∈ (mzPoly‘ℕ)) → 𝑁 ∈ ℕ0)
2 simpr 477 . . 3 ((𝑁 ∈ ℕ0𝑃 ∈ (mzPoly‘ℕ)) → 𝑃 ∈ (mzPoly‘ℕ))
3 eqidd 2622 . . 3 ((𝑁 ∈ ℕ0𝑃 ∈ (mzPoly‘ℕ)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)})
4 fveq1 6147 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝑝𝑏) = (𝑃𝑏))
54eqeq1d 2623 . . . . . . . . 9 (𝑝 = 𝑃 → ((𝑝𝑏) = 0 ↔ (𝑃𝑏) = 0))
65anbi2d 739 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝𝑏) = 0) ↔ (𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0)))
76rexbidv 3045 . . . . . . 7 (𝑝 = 𝑃 → (∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝𝑏) = 0) ↔ ∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0)))
87abbidv 2738 . . . . . 6 (𝑝 = 𝑃 → {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝𝑏) = 0)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0)})
9 eqeq1 2625 . . . . . . . . . 10 (𝑎 = 𝑡 → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑏 ↾ (1...𝑁))))
109anbi1d 740 . . . . . . . . 9 (𝑎 = 𝑡 → ((𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0) ↔ (𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0)))
1110rexbidv 3045 . . . . . . . 8 (𝑎 = 𝑡 → (∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0) ↔ ∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0)))
12 reseq1 5350 . . . . . . . . . . 11 (𝑏 = 𝑢 → (𝑏 ↾ (1...𝑁)) = (𝑢 ↾ (1...𝑁)))
1312eqeq2d 2631 . . . . . . . . . 10 (𝑏 = 𝑢 → (𝑡 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
14 fveq2 6148 . . . . . . . . . . 11 (𝑏 = 𝑢 → (𝑃𝑏) = (𝑃𝑢))
1514eqeq1d 2623 . . . . . . . . . 10 (𝑏 = 𝑢 → ((𝑃𝑏) = 0 ↔ (𝑃𝑢) = 0))
1613, 15anbi12d 746 . . . . . . . . 9 (𝑏 = 𝑢 → ((𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)))
1716cbvrexv 3160 . . . . . . . 8 (∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0) ↔ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0))
1811, 17syl6bb 276 . . . . . . 7 (𝑎 = 𝑡 → (∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0) ↔ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)))
1918cbvabv 2744 . . . . . 6 {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)}
208, 19syl6eq 2671 . . . . 5 (𝑝 = 𝑃 → {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝𝑏) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)})
2120eqeq2d 2631 . . . 4 (𝑝 = 𝑃 → ({𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝𝑏) = 0)} ↔ {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)}))
2221rspcev 3295 . . 3 ((𝑃 ∈ (mzPoly‘ℕ) ∧ {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)}) → ∃𝑝 ∈ (mzPoly‘ℕ){𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝𝑏) = 0)})
232, 3, 22syl2anc 692 . 2 ((𝑁 ∈ ℕ0𝑃 ∈ (mzPoly‘ℕ)) → ∃𝑝 ∈ (mzPoly‘ℕ){𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝𝑏) = 0)})
24 eldioph3b 36805 . 2 ({𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑝 ∈ (mzPoly‘ℕ){𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝𝑏) = 0)}))
251, 23, 24sylanbrc 697 1 ((𝑁 ∈ ℕ0𝑃 ∈ (mzPoly‘ℕ)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} ∈ (Dioph‘𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  {cab 2607  ∃wrex 2908   ↾ cres 5076  ‘cfv 5847  (class class class)co 6604   ↑𝑚 cmap 7802  0cc0 9880  1c1 9881  ℕcn 10964  ℕ0cn0 11236  ...cfz 12268  mzPolycmzp 36762  Diophcdioph 36795 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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957 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 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-hash 13058  df-mzpcl 36763  df-mzp 36764  df-dioph 36796 This theorem is referenced by:  diophrex  36816
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