Theorem elnn0rabdioph 43530

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 3240	. . . . 5 ⊢ (𝐴 ∈ ℕ0 ↔ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴)
2	1	rabbii 3421	. . . 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 2925	. . . 4 ⊢ Ⅎ𝑡(ℕ0 ↑m (1...𝑁))
5		nfcv 2925	. . . 4 ⊢ Ⅎ𝑎(ℕ0 ↑m (1...𝑁))
6		nfv 1944	. . . 4 ⊢ Ⅎ𝑎∃𝑏 ∈ ℕ0 𝑏 = 𝐴
7		nfcv 2925	. . . . 5 ⊢ Ⅎ𝑡ℕ0
8		nfcsb1v 3877	. . . . . 6 ⊢ Ⅎ𝑡⦋𝑎 / 𝑡⦌𝐴
9	8	nfeq2 2942	. . . . 5 ⊢ Ⅎ𝑡 𝑏 = ⦋𝑎 / 𝑡⦌𝐴
10	7, 9	nfrexw 3313	. . . 4 ⊢ Ⅎ𝑡∃𝑏 ∈ ℕ0 𝑏 = ⦋𝑎 / 𝑡⦌𝐴
11		csbeq1a 3867	. . . . . 6 ⊢ (𝑡 = 𝑎 → 𝐴 = ⦋𝑎 / 𝑡⦌𝐴)
12	11	eqeq2d 2774	. . . . 5 ⊢ (𝑡 = 𝑎 → (𝑏 = 𝐴 ↔ 𝑏 = ⦋𝑎 / 𝑡⦌𝐴))
13	12	rexbidv 3189	. . . 4 ⊢ (𝑡 = 𝑎 → (∃𝑏 ∈ ℕ0 𝑏 = 𝐴 ↔ ∃𝑏 ∈ ℕ0 𝑏 = ⦋𝑎 / 𝑡⦌𝐴))
14	4, 5, 6, 10, 13	cbvrabw 3451	. . 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 2814	. 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 12539	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
17	16	adantr 485	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ ℕ0)
18		ovex 7443	. . . . 5 ⊢ (1...(𝑁 + 1)) ∈ V
19		nn0p1nn 12538	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
20		elfz1end 13578	. . . . . . 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 485	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
23		mzpproj 43468	. . . . 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 598	. . . 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 2763	. . . . 5 ⊢ (𝑁 + 1) = (𝑁 + 1)
26	25	rabdiophlem2 43529	. . . 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 43508	. . . 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 2767	. . . 4 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → (𝑏 = ⦋𝑎 / 𝑡⦌𝐴 ↔ (𝑐‘(𝑁 + 1)) = ⦋𝑎 / 𝑡⦌𝐴))
30		csbeq1 3856	. . . . 5 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ⦋𝑎 / 𝑡⦌𝐴 = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴)
31	30	eqeq2d 2774	. . . 4 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ((𝑐‘(𝑁 + 1)) = ⦋𝑎 / 𝑡⦌𝐴 ↔ (𝑐‘(𝑁 + 1)) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴))
32	25, 29, 31	rexrabdioph 43521	. . 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 602	. 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 2863	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 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  {crab 3416  Vcvv 3455  ⦋csb 3853   ↦ cmpt 5192   ↾ cres 5663  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8820  1c1 11096   + caddc 11098  ℕcn 12228  ℕ₀cn0 12499  ℤcz 12586  ...cfz 13530  mzPolycmzp 43453  Diophcdioph 43486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-oadd 8453  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9883  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-n0 12500  df-z 12587  df-uz 12858  df-fz 13531  df-hash 14363  df-mzpcl 43454  df-mzp 43455  df-dioph 43487

This theorem is referenced by:  lerabdioph  43532