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Theorem elnn0rabdioph 36882
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 3056 . . . . . 6 (𝐴 ∈ ℕ0 ↔ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴)
21a1i 11 . . . . 5 (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) → (𝐴 ∈ ℕ0 ↔ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴))
32rabbiia 3176 . . . 4 {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 ∈ ℕ0} = {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴}
43a1i 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})
5 nfcv 2761 . . . 4 𝑡(ℕ0𝑚 (1...𝑁))
6 nfcv 2761 . . . 4 𝑎(ℕ0𝑚 (1...𝑁))
7 nfv 1840 . . . 4 𝑎𝑏 ∈ ℕ0 𝑏 = 𝐴
8 nfcv 2761 . . . . 5 𝑡0
9 nfcsb1v 3534 . . . . . 6 𝑡𝑎 / 𝑡𝐴
109nfeq2 2776 . . . . 5 𝑡 𝑏 = 𝑎 / 𝑡𝐴
118, 10nfrex 3002 . . . 4 𝑡𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴
12 csbeq1a 3527 . . . . . 6 (𝑡 = 𝑎𝐴 = 𝑎 / 𝑡𝐴)
1312eqeq2d 2631 . . . . 5 (𝑡 = 𝑎 → (𝑏 = 𝐴𝑏 = 𝑎 / 𝑡𝐴))
1413rexbidv 3046 . . . 4 (𝑡 = 𝑎 → (∃𝑏 ∈ ℕ0 𝑏 = 𝐴 ↔ ∃𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴))
155, 6, 7, 11, 14cbvrab 3187 . . 3 {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴} = {𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴}
164, 15syl6eq 2671 . 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})
17 peano2nn0 11285 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
1817adantr 481 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ ℕ0)
19 ovex 6638 . . . . 5 (1...(𝑁 + 1)) ∈ V
20 nn0p1nn 11284 . . . . . . 7 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
21 elfz1end 12321 . . . . . . 7 ((𝑁 + 1) ∈ ℕ ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
2220, 21sylib 208 . . . . . 6 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
2322adantr 481 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
24 mzpproj 36815 . . . . 5 (((1...(𝑁 + 1)) ∈ V ∧ (𝑁 + 1) ∈ (1...(𝑁 + 1))) → (𝑐 ∈ (ℤ ↑𝑚 (1...(𝑁 + 1))) ↦ (𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))))
2519, 23, 24sylancr 694 . . . 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))))
26 eqid 2621 . . . . 5 (𝑁 + 1) = (𝑁 + 1)
2726rabdiophlem2 36881 . . . 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))))
28 eqrabdioph 36856 . . . 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)))
2918, 25, 27, 28syl3anc 1323 . . 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)))
30 eqeq1 2625 . . . 4 (𝑏 = (𝑐‘(𝑁 + 1)) → (𝑏 = 𝑎 / 𝑡𝐴 ↔ (𝑐‘(𝑁 + 1)) = 𝑎 / 𝑡𝐴))
31 csbeq1 3521 . . . . 5 (𝑎 = (𝑐 ↾ (1...𝑁)) → 𝑎 / 𝑡𝐴 = (𝑐 ↾ (1...𝑁)) / 𝑡𝐴)
3231eqeq2d 2631 . . . 4 (𝑎 = (𝑐 ↾ (1...𝑁)) → ((𝑐‘(𝑁 + 1)) = 𝑎 / 𝑡𝐴 ↔ (𝑐‘(𝑁 + 1)) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐴))
3326, 30, 32rexrabdioph 36873 . . 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))
3429, 33syldan 487 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝑎 / 𝑡𝐴} ∈ (Dioph‘𝑁))
3516, 34eqeltrd 2698 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  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  {crab 2911  Vcvv 3189  csb 3518  cmpt 4678  cres 5081  cfv 5852  (class class class)co 6610  𝑚 cmap 7809  1c1 9889   + caddc 9891  cn 10972  0cn0 11244  cz 11329  ...cfz 12276  mzPolycmzp 36800  Diophcdioph 36833
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-card 8717  df-cda 8942  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-hash 13066  df-mzpcl 36801  df-mzp 36802  df-dioph 36834
This theorem is referenced by:  lerabdioph  36884
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