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Theorem dvdsrabdioph 39748
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 39739 . . . 4 ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0m (1...𝑁))𝐴 ∈ ℤ)
2 rabdiophlem1 39739 . . . 4 ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0m (1...𝑁))𝐵 ∈ ℤ)
3 divides 15605 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴𝐵 ↔ ∃𝑎 ∈ ℤ (𝑎 · 𝐴) = 𝐵))
4 oveq1 7146 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑎 · 𝐴) = (𝑏 · 𝐴))
54eqeq1d 2803 . . . . . . . 8 (𝑎 = 𝑏 → ((𝑎 · 𝐴) = 𝐵 ↔ (𝑏 · 𝐴) = 𝐵))
6 oveq1 7146 . . . . . . . . 9 (𝑎 = -𝑏 → (𝑎 · 𝐴) = (-𝑏 · 𝐴))
76eqeq1d 2803 . . . . . . . 8 (𝑎 = -𝑏 → ((𝑎 · 𝐴) = 𝐵 ↔ (-𝑏 · 𝐴) = 𝐵))
85, 7rexzrexnn0 39742 . . . . . . 7 (∃𝑎 ∈ ℤ (𝑎 · 𝐴) = 𝐵 ↔ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵))
93, 8syl6bb 290 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴𝐵 ↔ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)))
109ralimi 3131 . . . . 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 3140 . . . . 5 (∀𝑡 ∈ (ℕ0m (1...𝑁))(𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ↔ (∀𝑡 ∈ (ℕ0m (1...𝑁))𝐴 ∈ ℤ ∧ ∀𝑡 ∈ (ℕ0m (1...𝑁))𝐵 ∈ ℤ))
12 rabbi 3339 . . . . 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 294 . . . 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 598 . . 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 1127 . 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 2958 . . . 4 𝑡(ℕ0m (1...𝑁))
17 nfcv 2958 . . . 4 𝑎(ℕ0m (1...𝑁))
18 nfv 1915 . . . 4 𝑎𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)
19 nfcv 2958 . . . . 5 𝑡0
20 nfcv 2958 . . . . . . . 8 𝑡𝑏
21 nfcv 2958 . . . . . . . 8 𝑡 ·
22 nfcsb1v 3855 . . . . . . . 8 𝑡𝑎 / 𝑡𝐴
2320, 21, 22nfov 7169 . . . . . . 7 𝑡(𝑏 · 𝑎 / 𝑡𝐴)
24 nfcsb1v 3855 . . . . . . 7 𝑡𝑎 / 𝑡𝐵
2523, 24nfeq 2971 . . . . . 6 𝑡(𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵
26 nfcv 2958 . . . . . . . 8 𝑡-𝑏
2726, 21, 22nfov 7169 . . . . . . 7 𝑡(-𝑏 · 𝑎 / 𝑡𝐴)
2827, 24nfeq 2971 . . . . . 6 𝑡(-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵
2925, 28nfor 1905 . . . . 5 𝑡((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵)
3019, 29nfrex 3271 . . . 4 𝑡𝑏 ∈ ℕ0 ((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵)
31 csbeq1a 3845 . . . . . . . 8 (𝑡 = 𝑎𝐴 = 𝑎 / 𝑡𝐴)
3231oveq2d 7155 . . . . . . 7 (𝑡 = 𝑎 → (𝑏 · 𝐴) = (𝑏 · 𝑎 / 𝑡𝐴))
33 csbeq1a 3845 . . . . . . 7 (𝑡 = 𝑎𝐵 = 𝑎 / 𝑡𝐵)
3432, 33eqeq12d 2817 . . . . . 6 (𝑡 = 𝑎 → ((𝑏 · 𝐴) = 𝐵 ↔ (𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵))
3531oveq2d 7155 . . . . . . 7 (𝑡 = 𝑎 → (-𝑏 · 𝐴) = (-𝑏 · 𝑎 / 𝑡𝐴))
3635, 33eqeq12d 2817 . . . . . 6 (𝑡 = 𝑎 → ((-𝑏 · 𝐴) = 𝐵 ↔ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵))
3734, 36orbi12d 916 . . . . 5 (𝑡 = 𝑎 → (((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵) ↔ ((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵)))
3837rexbidv 3259 . . . 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 3440 . . 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 1133 . . . 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 11929 . . . . . . 7 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
42413ad2ant1 1130 . . . . . 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 7172 . . . . . . . . . 10 (1...(𝑁 + 1)) ∈ V
44 nn0p1nn 11928 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
45 elfz1end 12936 . . . . . . . . . . 11 ((𝑁 + 1) ∈ ℕ ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
4644, 45sylib 221 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
47 mzpproj 39675 . . . . . . . . . 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 590 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑐 ∈ (ℤ ↑m (1...(𝑁 + 1))) ↦ (𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))))
4948adantr 484 . . . . . . . 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 2801 . . . . . . . . 9 (𝑁 + 1) = (𝑁 + 1)
5150rabdiophlem2 39740 . . . . . . . 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 39680 . . . . . . . 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 587 . . . . . . 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 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‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴)) ∈ (mzPoly‘(1...(𝑁 + 1))))
5550rabdiophlem2 39740 . . . . . . 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 1128 . . . . . 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 39715 . . . . . 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 1368 . . . . 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 39682 . . . . . . . . 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 39680 . . . . . . . 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 587 . . . . . . 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 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‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴)) ∈ (mzPoly‘(1...(𝑁 + 1))))
64 eqrabdioph 39715 . . . . . 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 1368 . . . . 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 39719 . . . . 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 587 . . . 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 7146 . . . . . . 7 (𝑏 = (𝑐‘(𝑁 + 1)) → (𝑏 · 𝑎 / 𝑡𝐴) = ((𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴))
6968eqeq1d 2803 . . . . . 6 (𝑏 = (𝑐‘(𝑁 + 1)) → ((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ↔ ((𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵))
70 negeq 10871 . . . . . . . 8 (𝑏 = (𝑐‘(𝑁 + 1)) → -𝑏 = -(𝑐‘(𝑁 + 1)))
7170oveq1d 7154 . . . . . . 7 (𝑏 = (𝑐‘(𝑁 + 1)) → (-𝑏 · 𝑎 / 𝑡𝐴) = (-(𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴))
7271eqeq1d 2803 . . . . . 6 (𝑏 = (𝑐‘(𝑁 + 1)) → ((-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ↔ (-(𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵))
7369, 72orbi12d 916 . . . . 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 3834 . . . . . . . 8 (𝑎 = (𝑐 ↾ (1...𝑁)) → 𝑎 / 𝑡𝐴 = (𝑐 ↾ (1...𝑁)) / 𝑡𝐴)
7574oveq2d 7155 . . . . . . 7 (𝑎 = (𝑐 ↾ (1...𝑁)) → ((𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = ((𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴))
76 csbeq1 3834 . . . . . . 7 (𝑎 = (𝑐 ↾ (1...𝑁)) → 𝑎 / 𝑡𝐵 = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵)
7775, 76eqeq12d 2817 . . . . . 6 (𝑎 = (𝑐 ↾ (1...𝑁)) → (((𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ↔ ((𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵))
7874oveq2d 7155 . . . . . . 7 (𝑎 = (𝑐 ↾ (1...𝑁)) → (-(𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = (-(𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴))
7978, 76eqeq12d 2817 . . . . . 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 916 . . . . 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 39732 . . . 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 587 . . 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 2897 . 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 2893 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 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2112  wral 3109  wrex 3110  {crab 3113  Vcvv 3444  csb 3831   class class class wbr 5033  cmpt 5113  cres 5525  cfv 6328  (class class class)co 7139  m cmap 8393  1c1 10531   + caddc 10533   · cmul 10535  -cneg 10864  cn 11629  0cn0 11889  cz 11973  ...cfz 12889  cdvds 15603  mzPolycmzp 39660  Diophcdioph 39693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-dju 9318  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-hash 13691  df-dvds 15604  df-mzpcl 39661  df-mzp 39662  df-dioph 39694
This theorem is referenced by:  rmydioph  39952  expdiophlem2  39960
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