Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eqrabdioph Structured version   Visualization version   GIF version

Theorem eqrabdioph 38767
Description: Diophantine set builder for equality of polynomial expressions. Note that the two expressions need not be nonnegative; only variables are so constrained. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Assertion
Ref Expression
eqrabdioph ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 = 𝐵} ∈ (Dioph‘𝑁))
Distinct variable group:   𝑡,𝑁
Allowed substitution hints:   𝐴(𝑡)   𝐵(𝑡)

Proof of Theorem eqrabdioph
StepHypRef Expression
1 nfmpt1 5025 . . . . . . 7 𝑡(𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)
21nfel1 2947 . . . . . 6 𝑡(𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))
3 nfmpt1 5025 . . . . . . 7 𝑡(𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵)
43nfel1 2947 . . . . . 6 𝑡(𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))
52, 4nfan 1862 . . . . 5 𝑡((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)))
6 mzpf 38725 . . . . . . . . . . 11 ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ)
76ad2antrr 713 . . . . . . . . . 10 ((((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ)
8 zex 11802 . . . . . . . . . . . . 13 ℤ ∈ V
9 nn0ssz 11816 . . . . . . . . . . . . 13 0 ⊆ ℤ
10 mapss 8251 . . . . . . . . . . . . 13 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0𝑚 (1...𝑁)) ⊆ (ℤ ↑𝑚 (1...𝑁)))
118, 9, 10mp2an 679 . . . . . . . . . . . 12 (ℕ0𝑚 (1...𝑁)) ⊆ (ℤ ↑𝑚 (1...𝑁))
1211sseli 3855 . . . . . . . . . . 11 (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) → 𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)))
1312adantl 474 . . . . . . . . . 10 ((((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)))
14 mptfcl 38709 . . . . . . . . . 10 ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) → 𝐴 ∈ ℤ))
157, 13, 14sylc 65 . . . . . . . . 9 ((((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → 𝐴 ∈ ℤ)
1615zcnd 11901 . . . . . . . 8 ((((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → 𝐴 ∈ ℂ)
17 mzpf 38725 . . . . . . . . . . 11 ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵):(ℤ ↑𝑚 (1...𝑁))⟶ℤ)
1817ad2antlr 714 . . . . . . . . . 10 ((((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵):(ℤ ↑𝑚 (1...𝑁))⟶ℤ)
19 mptfcl 38709 . . . . . . . . . 10 ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵):(ℤ ↑𝑚 (1...𝑁))⟶ℤ → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) → 𝐵 ∈ ℤ))
2018, 13, 19sylc 65 . . . . . . . . 9 ((((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → 𝐵 ∈ ℤ)
2120zcnd 11901 . . . . . . . 8 ((((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → 𝐵 ∈ ℂ)
2216, 21subeq0ad 10808 . . . . . . 7 ((((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → ((𝐴𝐵) = 0 ↔ 𝐴 = 𝐵))
2322bicomd 215 . . . . . 6 ((((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → (𝐴 = 𝐵 ↔ (𝐴𝐵) = 0))
2423ex 405 . . . . 5 (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) → (𝐴 = 𝐵 ↔ (𝐴𝐵) = 0)))
255, 24ralrimi 3167 . . . 4 (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → ∀𝑡 ∈ (ℕ0𝑚 (1...𝑁))(𝐴 = 𝐵 ↔ (𝐴𝐵) = 0))
26 rabbi 3323 . . . 4 (∀𝑡 ∈ (ℕ0𝑚 (1...𝑁))(𝐴 = 𝐵 ↔ (𝐴𝐵) = 0) ↔ {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 = 𝐵} = {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ (𝐴𝐵) = 0})
2725, 26sylib 210 . . 3 (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 = 𝐵} = {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ (𝐴𝐵) = 0})
28273adant1 1110 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 = 𝐵} = {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ (𝐴𝐵) = 0})
29 simp1 1116 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ ℕ0)
30 mzpsubmpt 38732 . . . 4 (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ (𝐴𝐵)) ∈ (mzPoly‘(1...𝑁)))
31303adant1 1110 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ (𝐴𝐵)) ∈ (mzPoly‘(1...𝑁)))
32 eq0rabdioph 38766 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ (𝐴𝐵)) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ (𝐴𝐵) = 0} ∈ (Dioph‘𝑁))
3329, 31, 32syl2anc 576 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ (𝐴𝐵) = 0} ∈ (Dioph‘𝑁))
3428, 33eqeltrd 2867 1 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 = 𝐵} ∈ (Dioph‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wral 3089  {crab 3093  Vcvv 3416  wss 3830  cmpt 5008  wf 6184  cfv 6188  (class class class)co 6976  𝑚 cmap 8206  0cc0 10335  1c1 10336  cmin 10670  0cn0 11707  cz 11793  ...cfz 12708  mzPolycmzp 38711  Diophcdioph 38744
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 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412
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 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-pss 3846  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-of 7227  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-er 8089  df-map 8208  df-en 8307  df-dom 8308  df-sdom 8309  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-nn 11440  df-n0 11708  df-z 11794  df-uz 12059  df-fz 12709  df-mzpcl 38712  df-mzp 38713  df-dioph 38745
This theorem is referenced by:  elnn0rabdioph  38793  dvdsrabdioph  38800  rmydioph  39004  rmxdioph  39006  expdiophlem2  39012  expdioph  39013
  Copyright terms: Public domain W3C validator