Theorem elnn0rabdioph 42987

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 3209	. . . . 5 ⊢ (𝐴 ∈ ℕ0 ↔ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴)
2	1	rabbii 3402	. . . 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 2896	. . . 4 ⊢ Ⅎ𝑡(ℕ0 ↑m (1...𝑁))
5		nfcv 2896	. . . 4 ⊢ Ⅎ𝑎(ℕ0 ↑m (1...𝑁))
6		nfv 1915	. . . 4 ⊢ Ⅎ𝑎∃𝑏 ∈ ℕ0 𝑏 = 𝐴
7		nfcv 2896	. . . . 5 ⊢ Ⅎ𝑡ℕ0
8		nfcsb1v 3871	. . . . . 6 ⊢ Ⅎ𝑡⦋𝑎 / 𝑡⦌𝐴
9	8	nfeq2 2914	. . . . 5 ⊢ Ⅎ𝑡 𝑏 = ⦋𝑎 / 𝑡⦌𝐴
10	7, 9	nfrexw 3282	. . . 4 ⊢ Ⅎ𝑡∃𝑏 ∈ ℕ0 𝑏 = ⦋𝑎 / 𝑡⦌𝐴
11		csbeq1a 3861	. . . . . 6 ⊢ (𝑡 = 𝑎 → 𝐴 = ⦋𝑎 / 𝑡⦌𝐴)
12	11	eqeq2d 2745	. . . . 5 ⊢ (𝑡 = 𝑎 → (𝑏 = 𝐴 ↔ 𝑏 = ⦋𝑎 / 𝑡⦌𝐴))
13	12	rexbidv 3158	. . . 4 ⊢ (𝑡 = 𝑎 → (∃𝑏 ∈ ℕ0 𝑏 = 𝐴 ↔ ∃𝑏 ∈ ℕ0 𝑏 = ⦋𝑎 / 𝑡⦌𝐴))
14	4, 5, 6, 10, 13	cbvrabw 3432	. . 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 2785	. 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 12439	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
17	16	adantr 480	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ ℕ0)
18		ovex 7389	. . . . 5 ⊢ (1...(𝑁 + 1)) ∈ V
19		nn0p1nn 12438	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
20		elfz1end 13468	. . . . . . 7 ⊢ ((𝑁 + 1) ∈ ℕ ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
21	19, 20	sylib 218	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
22	21	adantr 480	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
23		mzpproj 42921	. . . . 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 2734	. . . . 5 ⊢ (𝑁 + 1) = (𝑁 + 1)
26	25	rabdiophlem2 42986	. . . 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 42961	. . . 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 1373	. . 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 2738	. . . 4 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → (𝑏 = ⦋𝑎 / 𝑡⦌𝐴 ↔ (𝑐‘(𝑁 + 1)) = ⦋𝑎 / 𝑡⦌𝐴))
30		csbeq1 3850	. . . . 5 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ⦋𝑎 / 𝑡⦌𝐴 = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴)
31	30	eqeq2d 2745	. . . 4 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ((𝑐‘(𝑁 + 1)) = ⦋𝑎 / 𝑡⦌𝐴 ↔ (𝑐‘(𝑁 + 1)) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴))
32	25, 29, 31	rexrabdioph 42978	. . 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 2834	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 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3058  {crab 3397  Vcvv 3438  ⦋csb 3847   ↦ cmpt 5177   ↾ cres 5624  ‘cfv 6490  (class class class)co 7356   ↑_m cmap 8761  1c1 11025   + caddc 11027  ℕcn 12143  ℕ₀cn0 12399  ℤcz 12486  ...cfz 13421  mzPolycmzp 42906  Diophcdioph 42939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-inf2 9548  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-er 8633  df-map 8763  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-dju 9811  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-n0 12400  df-z 12487  df-uz 12750  df-fz 13422  df-hash 14252  df-mzpcl 42907  df-mzp 42908  df-dioph 42940

This theorem is referenced by:  lerabdioph  42989