dvdsrabdioph - Mathbox for Stefan O'Rear

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Theorem dvdsrabdioph 42142

Description: Divisibility is a Diophantine relation. (Contributed by Stefan O'Rear, 11-Oct-2014.)

**Assertion**
Ref	Expression
dvdsrabdioph	⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝐴 ∥ 𝐵} ∈ (Dioph‘𝑁))

Distinct variable group: 𝑡,𝑁 Allowed substitution hints: 𝐴(𝑡) 𝐵(𝑡)

Proof of Theorem dvdsrabdioph Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabdiophlem1 42133	. . . 4 ⊢ ((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))𝐴 ∈ ℤ)
2		rabdiophlem1 42133	. . . 4 ⊢ ((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))𝐵 ∈ ℤ)
3		divides 16218	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∥ 𝐵 ↔ ∃𝑎 ∈ ℤ (𝑎 · 𝐴) = 𝐵))
4		oveq1 7421	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑎 · 𝐴) = (𝑏 · 𝐴))
5	4	eqeq1d 2729	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ((𝑎 · 𝐴) = 𝐵 ↔ (𝑏 · 𝐴) = 𝐵))
6		oveq1 7421	. . . . . . . . 9 ⊢ (𝑎 = -𝑏 → (𝑎 · 𝐴) = (-𝑏 · 𝐴))
7	6	eqeq1d 2729	. . . . . . . 8 ⊢ (𝑎 = -𝑏 → ((𝑎 · 𝐴) = 𝐵 ↔ (-𝑏 · 𝐴) = 𝐵))
8	5, 7	rexzrexnn0 42136	. . . . . . 7 ⊢ (∃𝑎 ∈ ℤ (𝑎 · 𝐴) = 𝐵 ↔ ∃𝑏 ∈ ℕ₀ ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵))
9	3, 8	bitrdi 287	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∥ 𝐵 ↔ ∃𝑏 ∈ ℕ₀ ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)))
10	9	ralimi 3078	. . . . 5 ⊢ (∀𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))(𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ∀𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))(𝐴 ∥ 𝐵 ↔ ∃𝑏 ∈ ℕ₀ ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)))
11		r19.26 3106	. . . . 5 ⊢ (∀𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))(𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ↔ (∀𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))𝐴 ∈ ℤ ∧ ∀𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))𝐵 ∈ ℤ))
12		rabbi 3457	. . . . 5 ⊢ (∀𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))(𝐴 ∥ 𝐵 ↔ ∃𝑏 ∈ ℕ₀ ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)) ↔ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝐴 ∥ 𝐵} = {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ₀ ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)})
13	10, 11, 12	3imtr3i 291	. . . 4 ⊢ ((∀𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))𝐴 ∈ ℤ ∧ ∀𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))𝐵 ∈ ℤ) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝐴 ∥ 𝐵} = {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ₀ ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)})
14	1, 2, 13	syl2an 595	. . 3 ⊢ (((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝐴 ∥ 𝐵} = {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ₀ ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)})
15	14	3adant1 1128	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝐴 ∥ 𝐵} = {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ₀ ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)})
16		nfcv 2898	. . . 4 ⊢ Ⅎ𝑡(ℕ₀ ↑_m (1...𝑁))
17		nfcv 2898	. . . 4 ⊢ Ⅎ𝑎(ℕ₀ ↑_m (1...𝑁))
18		nfv 1910	. . . 4 ⊢ Ⅎ𝑎∃𝑏 ∈ ℕ₀ ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)
19		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑡ℕ₀
20		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑡𝑏
21		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑡 ·
22		nfcsb1v 3914	. . . . . . . 8 ⊢ Ⅎ𝑡⦋𝑎 / 𝑡⦌𝐴
23	20, 21, 22	nfov 7444	. . . . . . 7 ⊢ Ⅎ𝑡(𝑏 · ⦋𝑎 / 𝑡⦌𝐴)
24		nfcsb1v 3914	. . . . . . 7 ⊢ Ⅎ𝑡⦋𝑎 / 𝑡⦌𝐵
25	23, 24	nfeq 2911	. . . . . 6 ⊢ Ⅎ𝑡(𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵
26		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑡-𝑏
27	26, 21, 22	nfov 7444	. . . . . . 7 ⊢ Ⅎ𝑡(-𝑏 · ⦋𝑎 / 𝑡⦌𝐴)
28	27, 24	nfeq 2911	. . . . . 6 ⊢ Ⅎ𝑡(-𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵
29	25, 28	nfor 1900	. . . . 5 ⊢ Ⅎ𝑡((𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ∨ (-𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵)
30	19, 29	nfrexw 3305	. . . 4 ⊢ Ⅎ𝑡∃𝑏 ∈ ℕ₀ ((𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ∨ (-𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵)
31		csbeq1a 3903	. . . . . . . 8 ⊢ (𝑡 = 𝑎 → 𝐴 = ⦋𝑎 / 𝑡⦌𝐴)
32	31	oveq2d 7430	. . . . . . 7 ⊢ (𝑡 = 𝑎 → (𝑏 · 𝐴) = (𝑏 · ⦋𝑎 / 𝑡⦌𝐴))
33		csbeq1a 3903	. . . . . . 7 ⊢ (𝑡 = 𝑎 → 𝐵 = ⦋𝑎 / 𝑡⦌𝐵)
34	32, 33	eqeq12d 2743	. . . . . 6 ⊢ (𝑡 = 𝑎 → ((𝑏 · 𝐴) = 𝐵 ↔ (𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵))
35	31	oveq2d 7430	. . . . . . 7 ⊢ (𝑡 = 𝑎 → (-𝑏 · 𝐴) = (-𝑏 · ⦋𝑎 / 𝑡⦌𝐴))
36	35, 33	eqeq12d 2743	. . . . . 6 ⊢ (𝑡 = 𝑎 → ((-𝑏 · 𝐴) = 𝐵 ↔ (-𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵))
37	34, 36	orbi12d 917	. . . . 5 ⊢ (𝑡 = 𝑎 → (((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵) ↔ ((𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ∨ (-𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵)))
38	37	rexbidv 3173	. . . 4 ⊢ (𝑡 = 𝑎 → (∃𝑏 ∈ ℕ₀ ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵) ↔ ∃𝑏 ∈ ℕ₀ ((𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ∨ (-𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵)))
39	16, 17, 18, 30, 38	cbvrabw 3462	. . 3 ⊢ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ₀ ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)} = {𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ₀ ((𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ∨ (-𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵)}
40		simp1 1134	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ ℕ₀)
41		peano2nn0 12528	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
42	41	3ad2ant1 1131	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → (𝑁 + 1) ∈ ℕ₀)
43		ovex 7447	. . . . . . . . . 10 ⊢ (1...(𝑁 + 1)) ∈ V
44		nn0p1nn 12527	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ)
45		elfz1end 13549	. . . . . . . . . . 11 ⊢ ((𝑁 + 1) ∈ ℕ ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
46	44, 45	sylib 217	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
47		mzpproj 42069	. . . . . . . . . 10 ⊢ (((1...(𝑁 + 1)) ∈ V ∧ (𝑁 + 1) ∈ (1...(𝑁 + 1))) → (𝑐 ∈ (ℤ ↑_m (1...(𝑁 + 1))) ↦ (𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))))
48	43, 46, 47	sylancr 586	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (𝑐 ∈ (ℤ ↑_m (1...(𝑁 + 1))) ↦ (𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))))
49	48	adantr 480	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑐 ∈ (ℤ ↑_m (1...(𝑁 + 1))) ↦ (𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))))
50		eqid 2727	. . . . . . . . 9 ⊢ (𝑁 + 1) = (𝑁 + 1)
51	50	rabdiophlem2 42134	. . . . . . . 8 ⊢ ((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))))
52		mzpmulmpt 42074	. . . . . . . 8 ⊢ (((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 ∈ (ℤ ↑_m (1...(𝑁 + 1))) ↦ ((𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴)) ∈ (mzPoly‘(1...(𝑁 + 1))))
53	49, 51, 52	syl2anc 583	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑐 ∈ (ℤ ↑_m (1...(𝑁 + 1))) ↦ ((𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴)) ∈ (mzPoly‘(1...(𝑁 + 1))))
54	53	3adant3 1130	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → (𝑐 ∈ (ℤ ↑_m (1...(𝑁 + 1))) ↦ ((𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴)) ∈ (mzPoly‘(1...(𝑁 + 1))))
55	50	rabdiophlem2 42134	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → (𝑐 ∈ (ℤ ↑_m (1...(𝑁 + 1))) ↦ ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵) ∈ (mzPoly‘(1...(𝑁 + 1))))
56	55	3adant2 1129	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → (𝑐 ∈ (ℤ ↑_m (1...(𝑁 + 1))) ↦ ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵) ∈ (mzPoly‘(1...(𝑁 + 1))))
57		eqrabdioph 42109	. . . . . 6 ⊢ (((𝑁 + 1) ∈ ℕ₀ ∧ (𝑐 ∈ (ℤ ↑_m (1...(𝑁 + 1))) ↦ ((𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴)) ∈ (mzPoly‘(1...(𝑁 + 1))) ∧ (𝑐 ∈ (ℤ ↑_m (1...(𝑁 + 1))) ↦ ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵) ∈ (mzPoly‘(1...(𝑁 + 1)))) → {𝑐 ∈ (ℕ₀ ↑_m (1...(𝑁 + 1))) ∣ ((𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵} ∈ (Dioph‘(𝑁 + 1)))
58	42, 54, 56, 57	syl3anc 1369	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑐 ∈ (ℕ₀ ↑_m (1...(𝑁 + 1))) ∣ ((𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵} ∈ (Dioph‘(𝑁 + 1)))
59		mzpnegmpt 42076	. . . . . . . . 9 ⊢ ((𝑐 ∈ (ℤ ↑_m (1...(𝑁 + 1))) ↦ (𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))) → (𝑐 ∈ (ℤ ↑_m (1...(𝑁 + 1))) ↦ -(𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))))
60	49, 59	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑐 ∈ (ℤ ↑_m (1...(𝑁 + 1))) ↦ -(𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))))
61		mzpmulmpt 42074	. . . . . . . 8 ⊢ (((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 ∈ (ℤ ↑_m (1...(𝑁 + 1))) ↦ (-(𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴)) ∈ (mzPoly‘(1...(𝑁 + 1))))
62	60, 51, 61	syl2anc 583	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑐 ∈ (ℤ ↑_m (1...(𝑁 + 1))) ↦ (-(𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴)) ∈ (mzPoly‘(1...(𝑁 + 1))))
63	62	3adant3 1130	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → (𝑐 ∈ (ℤ ↑_m (1...(𝑁 + 1))) ↦ (-(𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴)) ∈ (mzPoly‘(1...(𝑁 + 1))))
64		eqrabdioph 42109	. . . . . 6 ⊢ (((𝑁 + 1) ∈ ℕ₀ ∧ (𝑐 ∈ (ℤ ↑_m (1...(𝑁 + 1))) ↦ (-(𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴)) ∈ (mzPoly‘(1...(𝑁 + 1))) ∧ (𝑐 ∈ (ℤ ↑_m (1...(𝑁 + 1))) ↦ ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵) ∈ (mzPoly‘(1...(𝑁 + 1)))) → {𝑐 ∈ (ℕ₀ ↑_m (1...(𝑁 + 1))) ∣ (-(𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵} ∈ (Dioph‘(𝑁 + 1)))
65	42, 63, 56, 64	syl3anc 1369	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑐 ∈ (ℕ₀ ↑_m (1...(𝑁 + 1))) ∣ (-(𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵} ∈ (Dioph‘(𝑁 + 1)))
66		orrabdioph 42113	. . . . 5 ⊢ (({𝑐 ∈ (ℕ₀ ↑_m (1...(𝑁 + 1))) ∣ ((𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵} ∈ (Dioph‘(𝑁 + 1)) ∧ {𝑐 ∈ (ℕ₀ ↑_m (1...(𝑁 + 1))) ∣ (-(𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵} ∈ (Dioph‘(𝑁 + 1))) → {𝑐 ∈ (ℕ₀ ↑_m (1...(𝑁 + 1))) ∣ (((𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵 ∨ (-(𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵)} ∈ (Dioph‘(𝑁 + 1)))
67	58, 65, 66	syl2anc 583	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑐 ∈ (ℕ₀ ↑_m (1...(𝑁 + 1))) ∣ (((𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵 ∨ (-(𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵)} ∈ (Dioph‘(𝑁 + 1)))
68		oveq1 7421	. . . . . . 7 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → (𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ((𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴))
69	68	eqeq1d 2729	. . . . . 6 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → ((𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ↔ ((𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵))
70		negeq 11468	. . . . . . . 8 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → -𝑏 = -(𝑐‘(𝑁 + 1)))
71	70	oveq1d 7429	. . . . . . 7 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → (-𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = (-(𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴))
72	71	eqeq1d 2729	. . . . . 6 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → ((-𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ↔ (-(𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵))
73	69, 72	orbi12d 917	. . . . 5 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → (((𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ∨ (-𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵) ↔ (((𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ∨ (-(𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵)))
74		csbeq1 3892	. . . . . . . 8 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ⦋𝑎 / 𝑡⦌𝐴 = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴)
75	74	oveq2d 7430	. . . . . . 7 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ((𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴) = ((𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴))
76		csbeq1 3892	. . . . . . 7 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ⦋𝑎 / 𝑡⦌𝐵 = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵)
77	75, 76	eqeq12d 2743	. . . . . 6 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → (((𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ↔ ((𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵))
78	74	oveq2d 7430	. . . . . . 7 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → (-(𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴) = (-(𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴))
79	78, 76	eqeq12d 2743	. . . . . 6 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ((-(𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ↔ (-(𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵))
80	77, 79	orbi12d 917	. . . . 5 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ((((𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ∨ (-(𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵) ↔ (((𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵 ∨ (-(𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵)))
81	50, 73, 80	rexrabdioph 42126	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑐 ∈ (ℕ₀ ↑_m (1...(𝑁 + 1))) ∣ (((𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵 ∨ (-(𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴) = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵)} ∈ (Dioph‘(𝑁 + 1))) → {𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ₀ ((𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ∨ (-𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵)} ∈ (Dioph‘𝑁))
82	40, 67, 81	syl2anc 583	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ₀ ((𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ∨ (-𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵)} ∈ (Dioph‘𝑁))
83	39, 82	eqeltrid 2832	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ₀ ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)} ∈ (Dioph‘𝑁))
84	15, 83	eqeltrd 2828	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝐴 ∥ 𝐵} ∈ (Dioph‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 {crab 3427 Vcvv 3469 ⦋csb 3889 class class class wbr 5142 ↦ cmpt 5225 ↾ cres 5674 ‘cfv 6542 (class class class)co 7414 ↑_m cmap 8834 1c1 11125 + caddc 11127 · cmul 11129 -cneg 11461 ℕcn 12228 ℕ₀cn0 12488 ℤcz 12574 ...cfz 13502 ∥ cdvds 16216 mzPolycmzp 42054 Diophcdioph 42087

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-oadd 8482 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-dju 9910 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-n0 12489 df-z 12575 df-uz 12839 df-fz 13503 df-hash 14308 df-dvds 16217 df-mzpcl 42055 df-mzp 42056 df-dioph 42088

This theorem is referenced by: rmydioph 42347 expdiophlem2 42355

W3C validator