Theorem dvdsrabdioph 44077

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 44068	. . . 4 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))𝐴 ∈ ℤ)
2		rabdiophlem1 44068	. . . 4 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))𝐵 ∈ ℤ)
3		divides 16465	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∥ 𝐵 ↔ ∃𝑎 ∈ ℤ (𝑎 · 𝐴) = 𝐵))
4		oveq1 7456	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑎 · 𝐴) = (𝑏 · 𝐴))
5	4	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ((𝑎 · 𝐴) = 𝐵 ↔ (𝑏 · 𝐴) = 𝐵))
6		oveq1 7456	. . . . . . . . 9 ⊢ (𝑎 = -𝑏 → (𝑎 · 𝐴) = ( -𝑏 · 𝐴))
7	6	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑎 = -𝑏 → ((𝑎 · 𝐴) = 𝐵 ↔ ( -𝑏 · 𝐴) = 𝐵))
8	5, 7	rexzrexnn0 44071	. . . . . . 7 ⊢ (∃𝑎 ∈ ℤ (𝑎 · 𝐴) = 𝐵 ↔ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ ( -𝑏 · 𝐴) = 𝐵))
9	3, 8	bitrdi 290	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∥ 𝐵 ↔ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ ( -𝑏 · 𝐴) = 𝐵)))
10	9	ralimi 3103	. . . . 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 3126	. . . . 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 3445	. . . . 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 294	. . . 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 608	. . 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 1148	. 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 2926	. . . 4 ⊢ Ⅎ𝑡(ℕ0 ↑m (1...𝑁))
17		nfcv 2926	. . . 4 ⊢ Ⅎ𝑎(ℕ0 ↑m (1...𝑁))
18		nfv 1949	. . . 4 ⊢ Ⅎ𝑎∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ ( -𝑏 · 𝐴) = 𝐵)
19		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑡ℕ0
20		nfcv 2926	. . . . . . . 8 ⊢ Ⅎ𝑡𝑏
21		nfcv 2926	. . . . . . . 8 ⊢ Ⅎ𝑡 ·
22		nfcsb1v 3875	. . . . . . . 8 ⊢ Ⅎ𝑡⦋𝑎 / 𝑡⦌𝐴
23	20, 21, 22	nfov 7479	. . . . . . 7 ⊢ Ⅎ𝑡(𝑏 · ⦋𝑎 / 𝑡⦌𝐴)
24		nfcsb1v 3875	. . . . . . 7 ⊢ Ⅎ𝑡⦋𝑎 / 𝑡⦌𝐵
25	23, 24	nfeq 2939	. . . . . 6 ⊢ Ⅎ𝑡(𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵
26		nfcv 2926	. . . . . . . 8 ⊢ Ⅎ𝑡 -𝑏
27	26, 21, 22	nfov 7479	. . . . . . 7 ⊢ Ⅎ𝑡( -𝑏 · ⦋𝑎 / 𝑡⦌𝐴)
28	27, 24	nfeq 2939	. . . . . 6 ⊢ Ⅎ𝑡( -𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵
29	25, 28	nfor 1939	. . . . 5 ⊢ Ⅎ𝑡((𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ∨ ( -𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵)
30	19, 29	nfrexw 3314	. . . 4 ⊢ Ⅎ𝑡∃𝑏 ∈ ℕ0 ((𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ∨ ( -𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵)
31		csbeq1a 3865	. . . . . . . 8 ⊢ (𝑡 = 𝑎 → 𝐴 = ⦋𝑎 / 𝑡⦌𝐴)
32	31	oveq2d 7465	. . . . . . 7 ⊢ (𝑡 = 𝑎 → (𝑏 · 𝐴) = (𝑏 · ⦋𝑎 / 𝑡⦌𝐴))
33		csbeq1a 3865	. . . . . . 7 ⊢ (𝑡 = 𝑎 → 𝐵 = ⦋𝑎 / 𝑡⦌𝐵)
34	32, 33	eqeq12d 2779	. . . . . 6 ⊢ (𝑡 = 𝑎 → ((𝑏 · 𝐴) = 𝐵 ↔ (𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵))
35	31	oveq2d 7465	. . . . . . 7 ⊢ (𝑡 = 𝑎 → ( -𝑏 · 𝐴) = ( -𝑏 · ⦋𝑎 / 𝑡⦌𝐴))
36	35, 33	eqeq12d 2779	. . . . . 6 ⊢ (𝑡 = 𝑎 → (( -𝑏 · 𝐴) = 𝐵 ↔ ( -𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵))
37	34, 36	orbi12d 932	. . . . 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 3190	. . . 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 3450	. . 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 1154	. . . 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 12684	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
42	41	3ad2ant1 1151	. . . . . 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 7482	. . . . . . . . . 10 ⊢ (1...(𝑁 + 1)) ∈ V
44		nn0p1nn 12683	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
45		elfz1end 13726	. . . . . . . . . . 11 ⊢ ((𝑁 + 1) ∈ ℕ ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
46	44, 45	sylib 221	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
47		mzpproj 44008	. . . . . . . . . 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 599	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))))
49	48	adantr 486	. . . . . . . 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 2763	. . . . . . . . 9 ⊢ (𝑁 + 1) = (𝑁 + 1)
51	50	rabdiophlem2 44069	. . . . . . . 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 44013	. . . . . . . 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 596	. . . . . . 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 1150	. . . . . 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 44069	. . . . . . 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 1149	. . . . . 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 44048	. . . . . 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 1399	. . . . 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 44015	. . . . . . . . 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 18	. . . . . . . 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 44013	. . . . . . . 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 596	. . . . . . 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 1150	. . . . . 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 44048	. . . . . 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 1399	. . . . 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 44052	. . . . 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 596	. . . 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 7456	. . . . . . 7 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → (𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ((𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴))
69	68	eqeq1d 2765	. . . . . 6 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → ((𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ↔ ((𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵))
70		negeq 11586	. . . . . . . 8 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → -𝑏 = -(𝑐‘(𝑁 + 1)))
71	70	oveq1d 7464	. . . . . . 7 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → ( -𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ( -(𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴))
72	71	eqeq1d 2765	. . . . . 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 932	. . . . 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 3854	. . . . . . . 8 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ⦋𝑎 / 𝑡⦌𝐴 = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴)
75	74	oveq2d 7465	. . . . . . 7 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ((𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴) = ((𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴))
76		csbeq1 3854	. . . . . . 7 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ⦋𝑎 / 𝑡⦌𝐵 = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵)
77	75, 76	eqeq12d 2779	. . . . . 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 7465	. . . . . . 7 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ( -(𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴) = ( -(𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴))
79	78, 76	eqeq12d 2779	. . . . . 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 932	. . . . 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 44061	. . . 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 596	. . 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 2868	. 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 2864	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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454  ⦋csb 3851   class class class wbr 5106   ↦ cmpt 5190   ↾ cres 5658  ‘cfv 6564  (class class class)co 7449   ↑_m cmap 8883  1c1 11238   + caddc 11240   · cmul 11242   -cneg 11579  ℕcn 12373  ℕ₀cn0 12644  ℤcz 12731  ...cfz 13677   ∥ cdvds 16463  mzPolycmzp 43993  Diophcdioph 44026

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781  ax-inf2 9677  ax-cnex 11293  ax-resscn 11294  ax-1cn 11295  ax-icn 11296  ax-addcl 11297  ax-addrcl 11298  ax-mulcl 11299  ax-mulrcl 11300  ax-mulcom 11301  ax-addass 11302  ax-mulass 11303  ax-distr 11304  ax-i2m1 11305  ax-1ne0 11306  ax-1rid 11307  ax-rnegex 11308  ax-rrecex 11309  ax-cnre 11310  ax-pre-lttri 11311  ax-pre-lttrn 11312  ax-pre-ltadd 11313  ax-pre-mulgt0 11314

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-of 7723  df-om 7910  df-1st 8033  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-rdg 8454  df-1o 8512  df-oadd 8516  df-er 8753  df-map 8885  df-en 9010  df-dom 9011  df-sdom 9012  df-fin 9013  df-dju 10017  df-card 10055  df-pnf 11382  df-mnf 11383  df-xr 11384  df-ltxr 11385  df-le 11386  df-sub 11580  df-neg 11581  df-nn 12374  df-n0 12645  df-z 12732  df-uz 13004  df-fz 13678  df-hash 14514  df-dvds 16464  df-mzpcl 43994  df-mzp 43995  df-dioph 44027

This theorem is used by:  rmydioph  44281  expdiophlem2  44289