Theorem dvdsrabdioph 42798

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 42789	. . . 4 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))𝐴 ∈ ℤ)
2		rabdiophlem1 42789	. . . 4 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))𝐵 ∈ ℤ)
3		divides 16289	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∥ 𝐵 ↔ ∃𝑎 ∈ ℤ (𝑎 · 𝐴) = 𝐵))
4		oveq1 7438	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑎 · 𝐴) = (𝑏 · 𝐴))
5	4	eqeq1d 2737	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ((𝑎 · 𝐴) = 𝐵 ↔ (𝑏 · 𝐴) = 𝐵))
6		oveq1 7438	. . . . . . . . 9 ⊢ (𝑎 = -𝑏 → (𝑎 · 𝐴) = (-𝑏 · 𝐴))
7	6	eqeq1d 2737	. . . . . . . 8 ⊢ (𝑎 = -𝑏 → ((𝑎 · 𝐴) = 𝐵 ↔ (-𝑏 · 𝐴) = 𝐵))
8	5, 7	rexzrexnn0 42792	. . . . . . 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 3081	. . . . 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 3109	. . . . 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 3465	. . . . 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 596	. . 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 1129	. 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 2903	. . . 4 ⊢ Ⅎ𝑡(ℕ0 ↑m (1...𝑁))
17		nfcv 2903	. . . 4 ⊢ Ⅎ𝑎(ℕ0 ↑m (1...𝑁))
18		nfv 1912	. . . 4 ⊢ Ⅎ𝑎∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)
19		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑡ℕ0
20		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑡𝑏
21		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑡 ·
22		nfcsb1v 3933	. . . . . . . 8 ⊢ Ⅎ𝑡⦋𝑎 / 𝑡⦌𝐴
23	20, 21, 22	nfov 7461	. . . . . . 7 ⊢ Ⅎ𝑡(𝑏 · ⦋𝑎 / 𝑡⦌𝐴)
24		nfcsb1v 3933	. . . . . . 7 ⊢ Ⅎ𝑡⦋𝑎 / 𝑡⦌𝐵
25	23, 24	nfeq 2917	. . . . . 6 ⊢ Ⅎ𝑡(𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵
26		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑡-𝑏
27	26, 21, 22	nfov 7461	. . . . . . 7 ⊢ Ⅎ𝑡(-𝑏 · ⦋𝑎 / 𝑡⦌𝐴)
28	27, 24	nfeq 2917	. . . . . 6 ⊢ Ⅎ𝑡(-𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵
29	25, 28	nfor 1902	. . . . 5 ⊢ Ⅎ𝑡((𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ∨ (-𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵)
30	19, 29	nfrexw 3311	. . . 4 ⊢ Ⅎ𝑡∃𝑏 ∈ ℕ0 ((𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ∨ (-𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵)
31		csbeq1a 3922	. . . . . . . 8 ⊢ (𝑡 = 𝑎 → 𝐴 = ⦋𝑎 / 𝑡⦌𝐴)
32	31	oveq2d 7447	. . . . . . 7 ⊢ (𝑡 = 𝑎 → (𝑏 · 𝐴) = (𝑏 · ⦋𝑎 / 𝑡⦌𝐴))
33		csbeq1a 3922	. . . . . . 7 ⊢ (𝑡 = 𝑎 → 𝐵 = ⦋𝑎 / 𝑡⦌𝐵)
34	32, 33	eqeq12d 2751	. . . . . 6 ⊢ (𝑡 = 𝑎 → ((𝑏 · 𝐴) = 𝐵 ↔ (𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵))
35	31	oveq2d 7447	. . . . . . 7 ⊢ (𝑡 = 𝑎 → (-𝑏 · 𝐴) = (-𝑏 · ⦋𝑎 / 𝑡⦌𝐴))
36	35, 33	eqeq12d 2751	. . . . . 6 ⊢ (𝑡 = 𝑎 → ((-𝑏 · 𝐴) = 𝐵 ↔ (-𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵))
37	34, 36	orbi12d 918	. . . . 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 3177	. . . 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 3471	. . 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 1135	. . . 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 12564	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
42	41	3ad2ant1 1132	. . . . . 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 7464	. . . . . . . . . 10 ⊢ (1...(𝑁 + 1)) ∈ V
44		nn0p1nn 12563	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
45		elfz1end 13591	. . . . . . . . . . 11 ⊢ ((𝑁 + 1) ∈ ℕ ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
46	44, 45	sylib 218	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
47		mzpproj 42725	. . . . . . . . . 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 587	. . . . . . . . 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 2735	. . . . . . . . 9 ⊢ (𝑁 + 1) = (𝑁 + 1)
51	50	rabdiophlem2 42790	. . . . . . . 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 42730	. . . . . . . 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 584	. . . . . . 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 1131	. . . . . 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 42790	. . . . . . 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 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 ↾ (1...𝑁)) / 𝑡⦌𝐵) ∈ (mzPoly‘(1...(𝑁 + 1))))
57		eqrabdioph 42765	. . . . . 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 1370	. . . . 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 42732	. . . . . . . . 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 42730	. . . . . . . 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 584	. . . . . . 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 1131	. . . . . 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 42765	. . . . . 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 1370	. . . . 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 42769	. . . . 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 584	. . . 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 7438	. . . . . . 7 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → (𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ((𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴))
69	68	eqeq1d 2737	. . . . . 6 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → ((𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵 ↔ ((𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴) = ⦋𝑎 / 𝑡⦌𝐵))
70		negeq 11498	. . . . . . . 8 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → -𝑏 = -(𝑐‘(𝑁 + 1)))
71	70	oveq1d 7446	. . . . . . 7 ⊢ (𝑏 = (𝑐‘(𝑁 + 1)) → (-𝑏 · ⦋𝑎 / 𝑡⦌𝐴) = (-(𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴))
72	71	eqeq1d 2737	. . . . . 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 918	. . . . 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 3911	. . . . . . . 8 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ⦋𝑎 / 𝑡⦌𝐴 = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴)
75	74	oveq2d 7447	. . . . . . 7 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ((𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴) = ((𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴))
76		csbeq1 3911	. . . . . . 7 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → ⦋𝑎 / 𝑡⦌𝐵 = ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐵)
77	75, 76	eqeq12d 2751	. . . . . 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 7447	. . . . . . 7 ⊢ (𝑎 = (𝑐 ↾ (1...𝑁)) → (-(𝑐‘(𝑁 + 1)) · ⦋𝑎 / 𝑡⦌𝐴) = (-(𝑐‘(𝑁 + 1)) · ⦋(𝑐 ↾ (1...𝑁)) / 𝑡⦌𝐴))
79	78, 76	eqeq12d 2751	. . . . . 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 918	. . . . 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 42782	. . . 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 584	. . 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 2843	. 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 2839	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))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  {crab 3433  Vcvv 3478  ⦋csb 3908   class class class wbr 5148   ↦ cmpt 5231   ↾ cres 5691  ‘cfv 6563  (class class class)co 7431   ↑_m cmap 8865  1c1 11154   + caddc 11156   · cmul 11158  -cneg 11491  ℕcn 12264  ℕ₀cn0 12524  ℤcz 12611  ...cfz 13544   ∥ cdvds 16287  mzPolycmzp 42710  Diophcdioph 42743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-hash 14367  df-dvds 16288  df-mzpcl 42711  df-mzp 42712  df-dioph 42744

This theorem is referenced by:  rmydioph  43003  expdiophlem2  43011