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Theorem eldioph3 38703
Description: Inference version of eldioph3b 38702 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:   𝑡,𝑁,𝑢   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 475 . 2 ((𝑁 ∈ ℕ0𝑃 ∈ (mzPoly‘ℕ)) → 𝑁 ∈ ℕ0)
2 simpr 477 . . 3 ((𝑁 ∈ ℕ0𝑃 ∈ (mzPoly‘ℕ)) → 𝑃 ∈ (mzPoly‘ℕ))
3 eqidd 2773 . . 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 6492 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝𝑏) = (𝑃𝑏))
54eqeq1d 2774 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑝𝑏) = 0 ↔ (𝑃𝑏) = 0))
65anbi2d 619 . . . . . . 7 (𝑝 = 𝑃 → ((𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝𝑏) = 0) ↔ (𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0)))
76rexbidv 3236 . . . . . 6 (𝑝 = 𝑃 → (∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝𝑏) = 0) ↔ ∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0)))
87abbidv 2837 . . . . 5 (𝑝 = 𝑃 → {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝𝑏) = 0)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0)})
9 eqeq1 2776 . . . . . . . . 9 (𝑎 = 𝑡 → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑏 ↾ (1...𝑁))))
109anbi1d 620 . . . . . . . 8 (𝑎 = 𝑡 → ((𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0) ↔ (𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0)))
1110rexbidv 3236 . . . . . . 7 (𝑎 = 𝑡 → (∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0) ↔ ∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0)))
12 reseq1 5682 . . . . . . . . . 10 (𝑏 = 𝑢 → (𝑏 ↾ (1...𝑁)) = (𝑢 ↾ (1...𝑁)))
1312eqeq2d 2782 . . . . . . . . 9 (𝑏 = 𝑢 → (𝑡 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
14 fveqeq2 6502 . . . . . . . . 9 (𝑏 = 𝑢 → ((𝑃𝑏) = 0 ↔ (𝑃𝑢) = 0))
1513, 14anbi12d 621 . . . . . . . 8 (𝑏 = 𝑢 → ((𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)))
1615cbvrexv 3378 . . . . . . 7 (∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0) ↔ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0))
1711, 16syl6bb 279 . . . . . 6 (𝑎 = 𝑡 → (∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0) ↔ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)))
1817cbvabv 2904 . . . . 5 {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)}
198, 18syl6eq 2824 . . . 4 (𝑝 = 𝑃 → {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝𝑏) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)})
2019rspceeqv 3547 . . 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)})
212, 3, 20syl2anc 576 . 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)})
22 eldioph3b 38702 . 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)}))
231, 21, 22sylanbrc 575 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 387   = wceq 1507  wcel 2048  {cab 2753  wrex 3083  cres 5402  cfv 6182  (class class class)co 6970  𝑚 cmap 8198  0cc0 10327  1c1 10328  cn 11431  0cn0 11700  ...cfz 12701  mzPolycmzp 38659  Diophcdioph 38692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-inf2 8890  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-of 7221  df-om 7391  df-1st 7494  df-2nd 7495  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-oadd 7901  df-er 8081  df-map 8200  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-dju 9116  df-card 9154  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-nn 11432  df-n0 11701  df-z 11787  df-uz 12052  df-fz 12702  df-hash 13499  df-mzpcl 38660  df-mzp 38661  df-dioph 38693
This theorem is referenced by:  diophrex  38713
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