Theorem elnn0rabdioph 41541

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 3231	. . . . 5 ⊢ (𝐴 ∈ ℕ0 ↔ ∃𝑏 ∈ ℕ0 𝑏 = 𝐴)
2	1	rabbii 3439	. . . 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 2904	. . . 4 ⊢ Ⅎ𝑡(ℕ0 ↑m (1...𝑁))
5		nfcv 2904	. . . 4 ⊢ Ⅎ𝑎(ℕ0 ↑m (1...𝑁))
6		nfv 1918	. . . 4 ⊢ Ⅎ𝑎∃𝑏 ∈ ℕ0 𝑏 = 𝐴
7		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑡ℕ0
8		nfcsb1v 3919	. . . . . 6 ⊢ Ⅎ𝑡⦋𝑎 / 𝑡⦌𝐴
9	8	nfeq2 2921	. . . . 5 ⊢ Ⅎ𝑡 𝑏 = ⦋𝑎 / 𝑡⦌𝐴
10	7, 9	nfrexw 3311	. . . 4 ⊢ Ⅎ𝑡∃𝑏 ∈ ℕ0 𝑏 = ⦋𝑎 / 𝑡⦌𝐴
11		csbeq1a 3908	. . . . . 6 ⊢ (𝑡 = 𝑎 → 𝐴 = ⦋𝑎 / 𝑡⦌𝐴)
12	11	eqeq2d 2744	. . . . 5 ⊢ (𝑡 = 𝑎 → (𝑏 = 𝐴 ↔ 𝑏 = ⦋𝑎 / 𝑡⦌𝐴))
13	12	rexbidv 3179	. . . 4 ⊢ (𝑡 = 𝑎 → (∃𝑏 ∈ ℕ0 𝑏 = 𝐴 ↔ ∃𝑏 ∈ ℕ0 𝑏 = ⦋𝑎 / 𝑡⦌𝐴))
14	4, 5, 6, 10, 13	cbvrabw 3468	. . 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 2789	. 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 12512	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
17	16	adantr 482	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ ℕ0)
18		ovex 7442	. . . . 5 ⊢ (1...(𝑁 + 1)) ∈ V
19		nn0p1nn 12511	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
20		elfz1end 13531	. . . . . . 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 482	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
23		mzpproj 41475	. . . . 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 588	. . . 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 2733	. . . . 5 ⊢ (𝑁 + 1) = (𝑁 + 1)
26	25	rabdiophlem2 41540	. . . 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 41515	. . . 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 1372	. . 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 2737	. . . 4 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → (𝑏 = ⦋𝑎 / 𝑡⦌𝐴 ↔ (𝑐‘(𝑁 + 1)) = ⦋𝑎 / 𝑡⦌𝐴))
30		csbeq1 3897	. . . . 5 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ⦋𝑎 / 𝑡⦌𝐴 = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴)
31	30	eqeq2d 2744	. . . 4 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ((𝑐‘(𝑁 + 1)) = ⦋𝑎 / 𝑡⦌𝐴 ↔ (𝑐‘(𝑁 + 1)) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴))
32	25, 29, 31	rexrabdioph 41532	. . 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 592	. 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 397   = wceq 1542   ∈ wcel 2107  ∃wrex 3071  {crab 3433  Vcvv 3475  ⦋csb 3894   ↦ cmpt 5232   ↾ cres 5679  ‘cfv 6544  (class class class)co 7409   ↑_m cmap 8820  1c1 11111   + caddc 11113  ℕcn 12212  ℕ₀cn0 12472  ℤcz 12558  ...cfz 13484  mzPolycmzp 41460  Diophcdioph 41493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9896  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-hash 14291  df-mzpcl 41461  df-mzp 41462  df-dioph 41494

This theorem is referenced by:  lerabdioph  41543