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Theorem eldioph3 39704
Description: Inference version of eldioph3b 39703 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 486 . 2 ((𝑁 ∈ ℕ0𝑃 ∈ (mzPoly‘ℕ)) → 𝑁 ∈ ℕ0)
2 simpr 488 . . 3 ((𝑁 ∈ ℕ0𝑃 ∈ (mzPoly‘ℕ)) → 𝑃 ∈ (mzPoly‘ℕ))
3 eqidd 2802 . . 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 6648 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝𝑏) = (𝑃𝑏))
54eqeq1d 2803 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑝𝑏) = 0 ↔ (𝑃𝑏) = 0))
65anbi2d 631 . . . . . . 7 (𝑝 = 𝑃 → ((𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝𝑏) = 0) ↔ (𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0)))
76rexbidv 3259 . . . . . 6 (𝑝 = 𝑃 → (∃𝑏 ∈ (ℕ0m ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝𝑏) = 0) ↔ ∃𝑏 ∈ (ℕ0m ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0)))
87abbidv 2865 . . . . 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 2805 . . . . . . . . 9 (𝑎 = 𝑡 → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑏 ↾ (1...𝑁))))
109anbi1d 632 . . . . . . . 8 (𝑎 = 𝑡 → ((𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0) ↔ (𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0)))
1110rexbidv 3259 . . . . . . 7 (𝑎 = 𝑡 → (∃𝑏 ∈ (ℕ0m ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0) ↔ ∃𝑏 ∈ (ℕ0m ℕ)(𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0)))
12 reseq1 5816 . . . . . . . . . 10 (𝑏 = 𝑢 → (𝑏 ↾ (1...𝑁)) = (𝑢 ↾ (1...𝑁)))
1312eqeq2d 2812 . . . . . . . . 9 (𝑏 = 𝑢 → (𝑡 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
14 fveqeq2 6658 . . . . . . . . 9 (𝑏 = 𝑢 → ((𝑃𝑏) = 0 ↔ (𝑃𝑢) = 0))
1513, 14anbi12d 633 . . . . . . . 8 (𝑏 = 𝑢 → ((𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)))
1615cbvrexvw 3400 . . . . . . 7 (∃𝑏 ∈ (ℕ0m ℕ)(𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0) ↔ ∃𝑢 ∈ (ℕ0m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0))
1711, 16syl6bb 290 . . . . . 6 (𝑎 = 𝑡 → (∃𝑏 ∈ (ℕ0m ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0) ↔ ∃𝑢 ∈ (ℕ0m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)))
1817cbvabv 2869 . . . . 5 {𝑎 ∣ ∃𝑏 ∈ (ℕ0m ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃𝑏) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)}
198, 18eqtrdi 2852 . . . 4 (𝑝 = 𝑃 → {𝑎 ∣ ∃𝑏 ∈ (ℕ0m ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝𝑏) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)})
2019rspceeqv 3589 . . 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 587 . 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 39703 . 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 586 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 399   = wceq 1538  wcel 2112  {cab 2779  wrex 3110  cres 5525  cfv 6328  (class class class)co 7139  m cmap 8393  0cc0 10530  1c1 10531  cn 11629  0cn0 11889  ...cfz 12889  mzPolycmzp 39660  Diophcdioph 39693
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-dju 9318  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-hash 13691  df-mzpcl 39661  df-mzp 39662  df-dioph 39694
This theorem is referenced by:  diophrex  39713
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