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Theorem dvdsrabdioph 39285
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 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴𝐵} ∈ (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.
StepHypRef Expression
1 rabdiophlem1 39276 . . . 4 ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0m (1...𝑁))𝐴 ∈ ℤ)
2 rabdiophlem1 39276 . . . 4 ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0m (1...𝑁))𝐵 ∈ ℤ)
3 divides 15597 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴𝐵 ↔ ∃𝑎 ∈ ℤ (𝑎 · 𝐴) = 𝐵))
4 oveq1 7152 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑎 · 𝐴) = (𝑏 · 𝐴))
54eqeq1d 2820 . . . . . . . 8 (𝑎 = 𝑏 → ((𝑎 · 𝐴) = 𝐵 ↔ (𝑏 · 𝐴) = 𝐵))
6 oveq1 7152 . . . . . . . . 9 (𝑎 = -𝑏 → (𝑎 · 𝐴) = (-𝑏 · 𝐴))
76eqeq1d 2820 . . . . . . . 8 (𝑎 = -𝑏 → ((𝑎 · 𝐴) = 𝐵 ↔ (-𝑏 · 𝐴) = 𝐵))
85, 7rexzrexnn0 39279 . . . . . . 7 (∃𝑎 ∈ ℤ (𝑎 · 𝐴) = 𝐵 ↔ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵))
93, 8syl6bb 288 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴𝐵 ↔ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)))
109ralimi 3157 . . . . 5 (∀𝑡 ∈ (ℕ0m (1...𝑁))(𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ∀𝑡 ∈ (ℕ0m (1...𝑁))(𝐴𝐵 ↔ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)))
11 r19.26 3167 . . . . 5 (∀𝑡 ∈ (ℕ0m (1...𝑁))(𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ↔ (∀𝑡 ∈ (ℕ0m (1...𝑁))𝐴 ∈ ℤ ∧ ∀𝑡 ∈ (ℕ0m (1...𝑁))𝐵 ∈ ℤ))
12 rabbi 3381 . . . . 5 (∀𝑡 ∈ (ℕ0m (1...𝑁))(𝐴𝐵 ↔ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)) ↔ {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴𝐵} = {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)})
1310, 11, 123imtr3i 292 . . . 4 ((∀𝑡 ∈ (ℕ0m (1...𝑁))𝐴 ∈ ℤ ∧ ∀𝑡 ∈ (ℕ0m (1...𝑁))𝐵 ∈ ℤ) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴𝐵} = {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)})
141, 2, 13syl2an 595 . . 3 (((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴𝐵} = {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)})
15143adant1 1122 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴𝐵} = {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)})
16 nfcv 2974 . . . 4 𝑡(ℕ0m (1...𝑁))
17 nfcv 2974 . . . 4 𝑎(ℕ0m (1...𝑁))
18 nfv 1906 . . . 4 𝑎𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)
19 nfcv 2974 . . . . 5 𝑡0
20 nfcv 2974 . . . . . . . 8 𝑡𝑏
21 nfcv 2974 . . . . . . . 8 𝑡 ·
22 nfcsb1v 3904 . . . . . . . 8 𝑡𝑎 / 𝑡𝐴
2320, 21, 22nfov 7175 . . . . . . 7 𝑡(𝑏 · 𝑎 / 𝑡𝐴)
24 nfcsb1v 3904 . . . . . . 7 𝑡𝑎 / 𝑡𝐵
2523, 24nfeq 2988 . . . . . 6 𝑡(𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵
26 nfcv 2974 . . . . . . . 8 𝑡-𝑏
2726, 21, 22nfov 7175 . . . . . . 7 𝑡(-𝑏 · 𝑎 / 𝑡𝐴)
2827, 24nfeq 2988 . . . . . 6 𝑡(-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵
2925, 28nfor 1896 . . . . 5 𝑡((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵)
3019, 29nfrex 3306 . . . 4 𝑡𝑏 ∈ ℕ0 ((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵)
31 csbeq1a 3894 . . . . . . . 8 (𝑡 = 𝑎𝐴 = 𝑎 / 𝑡𝐴)
3231oveq2d 7161 . . . . . . 7 (𝑡 = 𝑎 → (𝑏 · 𝐴) = (𝑏 · 𝑎 / 𝑡𝐴))
33 csbeq1a 3894 . . . . . . 7 (𝑡 = 𝑎𝐵 = 𝑎 / 𝑡𝐵)
3432, 33eqeq12d 2834 . . . . . 6 (𝑡 = 𝑎 → ((𝑏 · 𝐴) = 𝐵 ↔ (𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵))
3531oveq2d 7161 . . . . . . 7 (𝑡 = 𝑎 → (-𝑏 · 𝐴) = (-𝑏 · 𝑎 / 𝑡𝐴))
3635, 33eqeq12d 2834 . . . . . 6 (𝑡 = 𝑎 → ((-𝑏 · 𝐴) = 𝐵 ↔ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵))
3734, 36orbi12d 912 . . . . 5 (𝑡 = 𝑎 → (((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵) ↔ ((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵)))
3837rexbidv 3294 . . . 4 (𝑡 = 𝑎 → (∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵) ↔ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵)))
3916, 17, 18, 30, 38cbvrabw 3487 . . 3 {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)} = {𝑎 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵)}
40 simp1 1128 . . . 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 11925 . . . . . . 7 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
42413ad2ant1 1125 . . . . . 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 7178 . . . . . . . . . 10 (1...(𝑁 + 1)) ∈ V
44 nn0p1nn 11924 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
45 elfz1end 12925 . . . . . . . . . . 11 ((𝑁 + 1) ∈ ℕ ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
4644, 45sylib 219 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
47 mzpproj 39212 . . . . . . . . . 10 (((1...(𝑁 + 1)) ∈ V ∧ (𝑁 + 1) ∈ (1...(𝑁 + 1))) → (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))))
4843, 46, 47sylancr 587 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))))
4948adantr 481 . . . . . . . 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 2818 . . . . . . . . 9 (𝑁 + 1) = (𝑁 + 1)
5150rabdiophlem2 39277 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) ∈ (mzPoly‘(1...(𝑁 + 1))))
52 mzpmulmpt 39217 . . . . . . . 8 (((𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))) ∧ (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) ∈ (mzPoly‘(1...(𝑁 + 1)))) → (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ ((𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴)) ∈ (mzPoly‘(1...(𝑁 + 1))))
5349, 51, 52syl2anc 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)) / 𝑡𝐴)) ∈ (mzPoly‘(1...(𝑁 + 1))))
54533adant3 1124 . . . . . 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)) / 𝑡𝐴)) ∈ (mzPoly‘(1...(𝑁 + 1))))
5550rabdiophlem2 39277 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐 ↾ (1...𝑁)) / 𝑡𝐵) ∈ (mzPoly‘(1...(𝑁 + 1))))
56553adant2 1123 . . . . . 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)) / 𝑡𝐵) ∈ (mzPoly‘(1...(𝑁 + 1))))
57 eqrabdioph 39252 . . . . . 6 (((𝑁 + 1) ∈ ℕ0 ∧ (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ ((𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴)) ∈ (mzPoly‘(1...(𝑁 + 1))) ∧ (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐 ↾ (1...𝑁)) / 𝑡𝐵) ∈ (mzPoly‘(1...(𝑁 + 1)))) → {𝑐 ∈ (ℕ0m (1...(𝑁 + 1))) ∣ ((𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵} ∈ (Dioph‘(𝑁 + 1)))
5842, 54, 56, 57syl3anc 1363 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑐 ∈ (ℕ0m (1...(𝑁 + 1))) ∣ ((𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵} ∈ (Dioph‘(𝑁 + 1)))
59 mzpnegmpt 39219 . . . . . . . . 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))))
6049, 59syl 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 39217 . . . . . . . 8 (((𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ -(𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))) ∧ (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) ∈ (mzPoly‘(1...(𝑁 + 1)))) → (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (-(𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴)) ∈ (mzPoly‘(1...(𝑁 + 1))))
6260, 51, 61syl2anc 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)) / 𝑡𝐴)) ∈ (mzPoly‘(1...(𝑁 + 1))))
63623adant3 1124 . . . . . 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)) / 𝑡𝐴)) ∈ (mzPoly‘(1...(𝑁 + 1))))
64 eqrabdioph 39252 . . . . . 6 (((𝑁 + 1) ∈ ℕ0 ∧ (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (-(𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴)) ∈ (mzPoly‘(1...(𝑁 + 1))) ∧ (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐 ↾ (1...𝑁)) / 𝑡𝐵) ∈ (mzPoly‘(1...(𝑁 + 1)))) → {𝑐 ∈ (ℕ0m (1...(𝑁 + 1))) ∣ (-(𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵} ∈ (Dioph‘(𝑁 + 1)))
6542, 63, 56, 64syl3anc 1363 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑐 ∈ (ℕ0m (1...(𝑁 + 1))) ∣ (-(𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵} ∈ (Dioph‘(𝑁 + 1)))
66 orrabdioph 39256 . . . . 5 (({𝑐 ∈ (ℕ0m (1...(𝑁 + 1))) ∣ ((𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵} ∈ (Dioph‘(𝑁 + 1)) ∧ {𝑐 ∈ (ℕ0m (1...(𝑁 + 1))) ∣ (-(𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵} ∈ (Dioph‘(𝑁 + 1))) → {𝑐 ∈ (ℕ0m (1...(𝑁 + 1))) ∣ (((𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵 ∨ (-(𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵)} ∈ (Dioph‘(𝑁 + 1)))
6758, 65, 66syl2anc 584 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑐 ∈ (ℕ0m (1...(𝑁 + 1))) ∣ (((𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵 ∨ (-(𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵)} ∈ (Dioph‘(𝑁 + 1)))
68 oveq1 7152 . . . . . . 7 (𝑏 = (𝑐‘(𝑁 + 1)) → (𝑏 · 𝑎 / 𝑡𝐴) = ((𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴))
6968eqeq1d 2820 . . . . . 6 (𝑏 = (𝑐‘(𝑁 + 1)) → ((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ↔ ((𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵))
70 negeq 10866 . . . . . . . 8 (𝑏 = (𝑐‘(𝑁 + 1)) → -𝑏 = -(𝑐‘(𝑁 + 1)))
7170oveq1d 7160 . . . . . . 7 (𝑏 = (𝑐‘(𝑁 + 1)) → (-𝑏 · 𝑎 / 𝑡𝐴) = (-(𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴))
7271eqeq1d 2820 . . . . . 6 (𝑏 = (𝑐‘(𝑁 + 1)) → ((-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ↔ (-(𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵))
7369, 72orbi12d 912 . . . . 5 (𝑏 = (𝑐‘(𝑁 + 1)) → (((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵) ↔ (((𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-(𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵)))
74 csbeq1 3883 . . . . . . . 8 (𝑎 = (𝑐 ↾ (1...𝑁)) → 𝑎 / 𝑡𝐴 = (𝑐 ↾ (1...𝑁)) / 𝑡𝐴)
7574oveq2d 7161 . . . . . . 7 (𝑎 = (𝑐 ↾ (1...𝑁)) → ((𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = ((𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴))
76 csbeq1 3883 . . . . . . 7 (𝑎 = (𝑐 ↾ (1...𝑁)) → 𝑎 / 𝑡𝐵 = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵)
7775, 76eqeq12d 2834 . . . . . 6 (𝑎 = (𝑐 ↾ (1...𝑁)) → (((𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ↔ ((𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵))
7874oveq2d 7161 . . . . . . 7 (𝑎 = (𝑐 ↾ (1...𝑁)) → (-(𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = (-(𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴))
7978, 76eqeq12d 2834 . . . . . 6 (𝑎 = (𝑐 ↾ (1...𝑁)) → ((-(𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ↔ (-(𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵))
8077, 79orbi12d 912 . . . . 5 (𝑎 = (𝑐 ↾ (1...𝑁)) → ((((𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-(𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵) ↔ (((𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵 ∨ (-(𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵)))
8150, 73, 80rexrabdioph 39269 . . . 4 ((𝑁 ∈ ℕ0 ∧ {𝑐 ∈ (ℕ0m (1...(𝑁 + 1))) ∣ (((𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵 ∨ (-(𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵)} ∈ (Dioph‘(𝑁 + 1))) → {𝑎 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵)} ∈ (Dioph‘𝑁))
8240, 67, 81syl2anc 584 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑎 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵)} ∈ (Dioph‘𝑁))
8339, 82eqeltrid 2914 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)} ∈ (Dioph‘𝑁))
8415, 83eqeltrd 2910 1 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴𝐵} ∈ (Dioph‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  csb 3880   class class class wbr 5057  cmpt 5137  cres 5550  cfv 6348  (class class class)co 7145  m cmap 8395  1c1 10526   + caddc 10528   · cmul 10530  -cneg 10859  cn 11626  0cn0 11885  cz 11969  ...cfz 12880  cdvds 15595  mzPolycmzp 39197  Diophcdioph 39230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-hash 13679  df-dvds 15596  df-mzpcl 39198  df-mzp 39199  df-dioph 39231
This theorem is referenced by:  rmydioph  39489  expdiophlem2  39497
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