Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elnn0rabdioph Structured version   Visualization version   GIF version

Theorem elnn0rabdioph 42791
Description: Diophantine set builder for nonnegativity constraints. The first builder which uses a witness variable internally; an expression is nonnegative if there is a nonnegative integer equal to it. (Contributed by Stefan O'Rear, 11-Oct-2014.)
Assertion
Ref Expression
elnn0rabdioph ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴 ∈ ℕ0} ∈ (Dioph‘𝑁))
Distinct variable group:   𝑡,𝑁
Allowed substitution hint:   𝐴(𝑡)

Proof of Theorem elnn0rabdioph
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 risset 3231 . . . . 5 (𝐴 ∈ ℕ0 ↔ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴)
21rabbii 3439 . . . 4 {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴 ∈ ℕ0} = {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴}
32a1i 11 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴 ∈ ℕ0} = {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴})
4 nfcv 2903 . . . 4 𝑡(ℕ0m (1...𝑁))
5 nfcv 2903 . . . 4 𝑎(ℕ0m (1...𝑁))
6 nfv 1912 . . . 4 𝑎𝑏 ∈ ℕ0 𝑏 = 𝐴
7 nfcv 2903 . . . . 5 𝑡0
8 nfcsb1v 3933 . . . . . 6 𝑡𝑎 / 𝑡𝐴
98nfeq2 2921 . . . . 5 𝑡 𝑏 = 𝑎 / 𝑡𝐴
107, 9nfrexw 3311 . . . 4 𝑡𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴
11 csbeq1a 3922 . . . . . 6 (𝑡 = 𝑎𝐴 = 𝑎 / 𝑡𝐴)
1211eqeq2d 2746 . . . . 5 (𝑡 = 𝑎 → (𝑏 = 𝐴𝑏 = 𝑎 / 𝑡𝐴))
1312rexbidv 3177 . . . 4 (𝑡 = 𝑎 → (∃𝑏 ∈ ℕ0 𝑏 = 𝐴 ↔ ∃𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴))
144, 5, 6, 10, 13cbvrabw 3471 . . 3 {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴} = {𝑎 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴}
153, 14eqtrdi 2791 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴 ∈ ℕ0} = {𝑎 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴})
16 peano2nn0 12564 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
1716adantr 480 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ ℕ0)
18 ovex 7464 . . . . 5 (1...(𝑁 + 1)) ∈ V
19 nn0p1nn 12563 . . . . . . 7 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
20 elfz1end 13591 . . . . . . 7 ((𝑁 + 1) ∈ ℕ ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
2119, 20sylib 218 . . . . . 6 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
2221adantr 480 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
23 mzpproj 42725 . . . . 5 (((1...(𝑁 + 1)) ∈ V ∧ (𝑁 + 1) ∈ (1...(𝑁 + 1))) → (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))))
2418, 22, 23sylancr 587 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))))
25 eqid 2735 . . . . 5 (𝑁 + 1) = (𝑁 + 1)
2625rabdiophlem2 42790 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) ∈ (mzPoly‘(1...(𝑁 + 1))))
27 eqrabdioph 42765 . . . 4 (((𝑁 + 1) ∈ ℕ0 ∧ (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))) ∧ (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) ∈ (mzPoly‘(1...(𝑁 + 1)))) → {𝑐 ∈ (ℕ0m (1...(𝑁 + 1))) ∣ (𝑐‘(𝑁 + 1)) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐴} ∈ (Dioph‘(𝑁 + 1)))
2817, 24, 26, 27syl3anc 1370 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑐 ∈ (ℕ0m (1...(𝑁 + 1))) ∣ (𝑐‘(𝑁 + 1)) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐴} ∈ (Dioph‘(𝑁 + 1)))
29 eqeq1 2739 . . . 4 (𝑏 = (𝑐‘(𝑁 + 1)) → (𝑏 = 𝑎 / 𝑡𝐴 ↔ (𝑐‘(𝑁 + 1)) = 𝑎 / 𝑡𝐴))
30 csbeq1 3911 . . . . 5 (𝑎 = (𝑐 ↾ (1...𝑁)) → 𝑎 / 𝑡𝐴 = (𝑐 ↾ (1...𝑁)) / 𝑡𝐴)
3130eqeq2d 2746 . . . 4 (𝑎 = (𝑐 ↾ (1...𝑁)) → ((𝑐‘(𝑁 + 1)) = 𝑎 / 𝑡𝐴 ↔ (𝑐‘(𝑁 + 1)) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐴))
3225, 29, 31rexrabdioph 42782 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑐 ∈ (ℕ0m (1...(𝑁 + 1))) ∣ (𝑐‘(𝑁 + 1)) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐴} ∈ (Dioph‘(𝑁 + 1))) → {𝑎 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴} ∈ (Dioph‘𝑁))
3328, 32syldan 591 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑎 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴} ∈ (Dioph‘𝑁))
3415, 33eqeltrd 2839 1 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴 ∈ ℕ0} ∈ (Dioph‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  {crab 3433  Vcvv 3478  csb 3908  cmpt 5231  cres 5691  cfv 6563  (class class class)co 7431  m cmap 8865  1c1 11154   + caddc 11156  cn 12264  0cn0 12524  cz 12611  ...cfz 13544  mzPolycmzp 42710  Diophcdioph 42743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-hash 14367  df-mzpcl 42711  df-mzp 42712  df-dioph 42744
This theorem is referenced by:  lerabdioph  42793
  Copyright terms: Public domain W3C validator