Theorem elnn0rabdioph 40622

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 ∈ (ℕ0 ↑m (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.

Step	Hyp	Ref	Expression
1		risset 3196	. . . . 5 ⊢ (𝐴 ∈ ℕ0 ↔ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴)
2	1	rabbii 3406	. . . 4 ⊢ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ∈ ℕ0} = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴}
3	2	a1i 11	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ∈ ℕ0} = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴})
4		nfcv 2909	. . . 4 ⊢ Ⅎ𝑡(ℕ0 ↑m (1...𝑁))
5		nfcv 2909	. . . 4 ⊢ Ⅎ𝑎(ℕ0 ↑m (1...𝑁))
6		nfv 1921	. . . 4 ⊢ Ⅎ𝑎∃𝑏 ∈ ℕ0 𝑏 = 𝐴
7		nfcv 2909	. . . . 5 ⊢ Ⅎ𝑡ℕ0
8		nfcsb1v 3862	. . . . . 6 ⊢ Ⅎ𝑡⦋𝑎 / 𝑡⦌𝐴
9	8	nfeq2 2926	. . . . 5 ⊢ Ⅎ𝑡 𝑏 = ⦋𝑎 / 𝑡⦌𝐴
10	7, 9	nfrex 3240	. . . 4 ⊢ Ⅎ𝑡∃𝑏 ∈ ℕ0 𝑏 = ⦋𝑎 / 𝑡⦌𝐴
11		csbeq1a 3851	. . . . . 6 ⊢ (𝑡 = 𝑎 → 𝐴 = ⦋𝑎 / 𝑡⦌𝐴)
12	11	eqeq2d 2751	. . . . 5 ⊢ (𝑡 = 𝑎 → (𝑏 = 𝐴 ↔ 𝑏 = ⦋𝑎 / 𝑡⦌𝐴))
13	12	rexbidv 3228	. . . 4 ⊢ (𝑡 = 𝑎 → (∃𝑏 ∈ ℕ0 𝑏 = 𝐴 ↔ ∃𝑏 ∈ ℕ0 𝑏 = ⦋𝑎 / 𝑡⦌𝐴))
14	4, 5, 6, 10, 13	cbvrabw 3423	. . 3 ⊢ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴} = {𝑎 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = ⦋𝑎 / 𝑡⦌𝐴}
15	3, 14	eqtrdi 2796	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ∈ ℕ0} = {𝑎 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = ⦋𝑎 / 𝑡⦌𝐴})
16		peano2nn0 12273	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
17	16	adantr 481	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ ℕ0)
18		ovex 7304	. . . . 5 ⊢ (1...(𝑁 + 1)) ∈ V
19		nn0p1nn 12272	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
20		elfz1end 13285	. . . . . . 7 ⊢ ((𝑁 + 1) ∈ ℕ ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
21	19, 20	sylib 217	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
22	21	adantr 481	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
23		mzpproj 40556	. . . . 5 ⊢ (((1...(𝑁 + 1)) ∈ V ∧ (𝑁 + 1) ∈ (1...(𝑁 + 1))) → (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))))
24	18, 22, 23	sylancr 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 2740	. . . . 5 ⊢ (𝑁 + 1) = (𝑁 + 1)
26	25	rabdiophlem2 40621	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴) ∈ (mzPoly‘(1...(𝑁 + 1))))
27		eqrabdioph 40596	. . . 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)) / 𝑡⦌𝐴) ∈ (mzPoly‘(1...(𝑁 + 1)))) → {𝑐 ∈ (ℕ0 ↑m (1...(𝑁 + 1))) ∣ (𝑐‘(𝑁 + 1)) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴} ∈ (Dioph‘(𝑁 + 1)))
28	17, 24, 26, 27	syl3anc 1370	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑐 ∈ (ℕ0 ↑m (1...(𝑁 + 1))) ∣ (𝑐‘(𝑁 + 1)) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴} ∈ (Dioph‘(𝑁 + 1)))
29		eqeq1 2744	. . . 4 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → (𝑏 = ⦋𝑎 / 𝑡⦌𝐴 ↔ (𝑐‘(𝑁 + 1)) = ⦋𝑎 / 𝑡⦌𝐴))
30		csbeq1 3840	. . . . 5 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ⦋𝑎 / 𝑡⦌𝐴 = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴)
31	30	eqeq2d 2751	. . . 4 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ((𝑐‘(𝑁 + 1)) = ⦋𝑎 / 𝑡⦌𝐴 ↔ (𝑐‘(𝑁 + 1)) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴))
32	25, 29, 31	rexrabdioph 40613	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑐 ∈ (ℕ0 ↑m (1...(𝑁 + 1))) ∣ (𝑐‘(𝑁 + 1)) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴} ∈ (Dioph‘(𝑁 + 1))) → {𝑎 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = ⦋𝑎 / 𝑡⦌𝐴} ∈ (Dioph‘𝑁))
33	28, 32	syldan 591	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑎 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 𝑏 = ⦋𝑎 / 𝑡⦌𝐴} ∈ (Dioph‘𝑁))
34	15, 33	eqeltrd 2841	1 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ∈ ℕ0} ∈ (Dioph‘𝑁))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1542   ∈ wcel 2110  ∃wrex 3067  {crab 3070  Vcvv 3431  ⦋csb 3837   ↦ cmpt 5162   ↾ cres 5592  ‘cfv 6432  (class class class)co 7271   ↑_m cmap 8598  1c1 10873   + caddc 10875  ℕcn 11973  ℕ₀cn0 12233  ℤcz 12319  ...cfz 13238  mzPolycmzp 40541  Diophcdioph 40574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-inf2 9377  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-of 7527  df-om 7707  df-1st 7824  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-1o 8288  df-oadd 8292  df-er 8481  df-map 8600  df-en 8717  df-dom 8718  df-sdom 8719  df-fin 8720  df-dju 9660  df-card 9698  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12582  df-fz 13239  df-hash 14043  df-mzpcl 40542  df-mzp 40543  df-dioph 40575

This theorem is referenced by:  lerabdioph  40624