Theorem elnn0rabdioph 43590

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 3242	. . . . 5 ⊢ (𝐴 ∈ ℕ0 ↔ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴)
2	1	rabbii 3423	. . . 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 2927	. . . 4 ⊢ Ⅎ𝑡(ℕ0 ↑m (1...𝑁))
5		nfcv 2927	. . . 4 ⊢ Ⅎ𝑎(ℕ0 ↑m (1...𝑁))
6		nfv 1947	. . . 4 ⊢ Ⅎ𝑎∃𝑏 ∈ ℕ0 𝑏 = 𝐴
7		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑡ℕ0
8		nfcsb1v 3878	. . . . . 6 ⊢ Ⅎ𝑡⦋𝑎 / 𝑡⦌𝐴
9	8	nfeq2 2944	. . . . 5 ⊢ Ⅎ𝑡 𝑏 = ⦋𝑎 / 𝑡⦌𝐴
10	7, 9	nfrexw 3315	. . . 4 ⊢ Ⅎ𝑡∃𝑏 ∈ ℕ0 𝑏 = ⦋𝑎 / 𝑡⦌𝐴
11		csbeq1a 3868	. . . . . 6 ⊢ (𝑡 = 𝑎 → 𝐴 = ⦋𝑎 / 𝑡⦌𝐴)
12	11	eqeq2d 2776	. . . . 5 ⊢ (𝑡 = 𝑎 → (𝑏 = 𝐴 ↔ 𝑏 = ⦋𝑎 / 𝑡⦌𝐴))
13	12	rexbidv 3191	. . . 4 ⊢ (𝑡 = 𝑎 → (∃𝑏 ∈ ℕ0 𝑏 = 𝐴 ↔ ∃𝑏 ∈ ℕ0 𝑏 = ⦋𝑎 / 𝑡⦌𝐴))
14	4, 5, 6, 10, 13	cbvrabw 3453	. . 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 2816	. 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 12561	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
17	16	adantr 486	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ ℕ0)
18		ovex 7452	. . . . 5 ⊢ (1...(𝑁 + 1)) ∈ V
19		nn0p1nn 12560	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
20		elfz1end 13601	. . . . . . 7 ⊢ ((𝑁 + 1) ∈ ℕ ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
21	19, 20	sylib 221	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
22	21	adantr 486	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
23		mzpproj 43528	. . . . 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 599	. . . 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 2765	. . . . 5 ⊢ (𝑁 + 1) = (𝑁 + 1)
26	25	rabdiophlem2 43589	. . . 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 43568	. . . 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 1398	. . 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 2769	. . . 4 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → (𝑏 = ⦋𝑎 / 𝑡⦌𝐴 ↔ (𝑐‘(𝑁 + 1)) = ⦋𝑎 / 𝑡⦌𝐴))
30		csbeq1 3857	. . . . 5 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ⦋𝑎 / 𝑡⦌𝐴 = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴)
31	30	eqeq2d 2776	. . . 4 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ((𝑐‘(𝑁 + 1)) = ⦋𝑎 / 𝑡⦌𝐴 ↔ (𝑐‘(𝑁 + 1)) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴))
32	25, 29, 31	rexrabdioph 43581	. . 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 603	. 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 2865	1 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ∈ ℕ0} ∈ (Dioph‘𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3091  {crab 3418  Vcvv 3457  ⦋csb 3854   ↦ cmpt 5194   ↾ cres 5665  ‘cfv 6540  (class class class)co 7419   ↑_m cmap 8830  1c1 11118   + caddc 11120  ℕcn 12250  ℕ₀cn0 12521  ℤcz 12608  ...cfz 13553  mzPolycmzp 43513  Diophcdioph 43546

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-inf2 9617  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-oadd 8463  df-er 8700  df-map 8832  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-dju 9903  df-card 9941  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-nn 12251  df-n0 12522  df-z 12609  df-uz 12881  df-fz 13554  df-hash 14387  df-mzpcl 43514  df-mzp 43515  df-dioph 43547

This theorem is used by:  lerabdioph  43592