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Theorem dvdsrabdioph 37893
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 37884 . . . 4 ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0𝑚 (1...𝑁))𝐴 ∈ ℤ)
2 rabdiophlem1 37884 . . . 4 ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0𝑚 (1...𝑁))𝐵 ∈ ℤ)
3 divides 15190 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴𝐵 ↔ ∃𝑎 ∈ ℤ (𝑎 · 𝐴) = 𝐵))
4 oveq1 6799 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑎 · 𝐴) = (𝑏 · 𝐴))
54eqeq1d 2772 . . . . . . . 8 (𝑎 = 𝑏 → ((𝑎 · 𝐴) = 𝐵 ↔ (𝑏 · 𝐴) = 𝐵))
6 oveq1 6799 . . . . . . . . 9 (𝑎 = -𝑏 → (𝑎 · 𝐴) = (-𝑏 · 𝐴))
76eqeq1d 2772 . . . . . . . 8 (𝑎 = -𝑏 → ((𝑎 · 𝐴) = 𝐵 ↔ (-𝑏 · 𝐴) = 𝐵))
85, 7rexzrexnn0 37887 . . . . . . 7 (∃𝑎 ∈ ℤ (𝑎 · 𝐴) = 𝐵 ↔ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵))
93, 8syl6bb 276 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴𝐵 ↔ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)))
109ralimi 3100 . . . . 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 3211 . . . . 5 (∀𝑡 ∈ (ℕ0𝑚 (1...𝑁))(𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ↔ (∀𝑡 ∈ (ℕ0𝑚 (1...𝑁))𝐴 ∈ ℤ ∧ ∀𝑡 ∈ (ℕ0𝑚 (1...𝑁))𝐵 ∈ ℤ))
12 rabbi 3268 . . . . 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 280 . . . 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 575 . . 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 1123 . 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 2912 . . . 4 𝑡(ℕ0𝑚 (1...𝑁))
17 nfcv 2912 . . . 4 𝑎(ℕ0𝑚 (1...𝑁))
18 nfv 1994 . . . 4 𝑎𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)
19 nfcv 2912 . . . . 5 𝑡0
20 nfcv 2912 . . . . . . . 8 𝑡𝑏
21 nfcv 2912 . . . . . . . 8 𝑡 ·
22 nfcsb1v 3696 . . . . . . . 8 𝑡𝑎 / 𝑡𝐴
2320, 21, 22nfov 6820 . . . . . . 7 𝑡(𝑏 · 𝑎 / 𝑡𝐴)
24 nfcsb1v 3696 . . . . . . 7 𝑡𝑎 / 𝑡𝐵
2523, 24nfeq 2924 . . . . . 6 𝑡(𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵
26 nfcv 2912 . . . . . . . 8 𝑡-𝑏
2726, 21, 22nfov 6820 . . . . . . 7 𝑡(-𝑏 · 𝑎 / 𝑡𝐴)
2827, 24nfeq 2924 . . . . . 6 𝑡(-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵
2925, 28nfor 1985 . . . . 5 𝑡((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵)
3019, 29nfrex 3154 . . . 4 𝑡𝑏 ∈ ℕ0 ((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵)
31 csbeq1a 3689 . . . . . . . 8 (𝑡 = 𝑎𝐴 = 𝑎 / 𝑡𝐴)
3231oveq2d 6808 . . . . . . 7 (𝑡 = 𝑎 → (𝑏 · 𝐴) = (𝑏 · 𝑎 / 𝑡𝐴))
33 csbeq1a 3689 . . . . . . 7 (𝑡 = 𝑎𝐵 = 𝑎 / 𝑡𝐵)
3432, 33eqeq12d 2785 . . . . . 6 (𝑡 = 𝑎 → ((𝑏 · 𝐴) = 𝐵 ↔ (𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵))
3531oveq2d 6808 . . . . . . 7 (𝑡 = 𝑎 → (-𝑏 · 𝐴) = (-𝑏 · 𝑎 / 𝑡𝐴))
3635, 33eqeq12d 2785 . . . . . 6 (𝑡 = 𝑎 → ((-𝑏 · 𝐴) = 𝐵 ↔ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵))
3734, 36orbi12d 883 . . . . 5 (𝑡 = 𝑎 → (((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵) ↔ ((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵)))
3837rexbidv 3199 . . . 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, 38cbvrab 3347 . . 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 1129 . . . 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 11534 . . . . . . 7 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
42413ad2ant1 1126 . . . . . 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 6822 . . . . . . . . . 10 (1...(𝑁 + 1)) ∈ V
44 nn0p1nn 11533 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
45 elfz1end 12577 . . . . . . . . . . 11 ((𝑁 + 1) ∈ ℕ ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
4644, 45sylib 208 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
47 mzpproj 37819 . . . . . . . . . 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 567 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑐 ∈ (ℤ ↑𝑚 (1...(𝑁 + 1))) ↦ (𝑐‘(𝑁 + 1))) ∈ (mzPoly‘(1...(𝑁 + 1))))
4948adantr 466 . . . . . . . 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 2770 . . . . . . . . 9 (𝑁 + 1) = (𝑁 + 1)
5150rabdiophlem2 37885 . . . . . . . 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 37824 . . . . . . . 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 565 . . . . . . 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 1125 . . . . . 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 37885 . . . . . . 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 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 ↾ (1...𝑁)) / 𝑡𝐵) ∈ (mzPoly‘(1...(𝑁 + 1))))
57 eqrabdioph 37860 . . . . . 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 1475 . . . . 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 37826 . . . . . . . . 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 37824 . . . . . . . 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 565 . . . . . . 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 1125 . . . . . 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 37860 . . . . . 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 1475 . . . . 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 37864 . . . . 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 565 . . . 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 6799 . . . . . . 7 (𝑏 = (𝑐‘(𝑁 + 1)) → (𝑏 · 𝑎 / 𝑡𝐴) = ((𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴))
6968eqeq1d 2772 . . . . . 6 (𝑏 = (𝑐‘(𝑁 + 1)) → ((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ↔ ((𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵))
70 negeq 10474 . . . . . . . 8 (𝑏 = (𝑐‘(𝑁 + 1)) → -𝑏 = -(𝑐‘(𝑁 + 1)))
7170oveq1d 6807 . . . . . . 7 (𝑏 = (𝑐‘(𝑁 + 1)) → (-𝑏 · 𝑎 / 𝑡𝐴) = (-(𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴))
7271eqeq1d 2772 . . . . . 6 (𝑏 = (𝑐‘(𝑁 + 1)) → ((-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ↔ (-(𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵))
7369, 72orbi12d 883 . . . . 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 3683 . . . . . . . 8 (𝑎 = (𝑐 ↾ (1...𝑁)) → 𝑎 / 𝑡𝐴 = (𝑐 ↾ (1...𝑁)) / 𝑡𝐴)
7574oveq2d 6808 . . . . . . 7 (𝑎 = (𝑐 ↾ (1...𝑁)) → ((𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = ((𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴))
76 csbeq1 3683 . . . . . . 7 (𝑎 = (𝑐 ↾ (1...𝑁)) → 𝑎 / 𝑡𝐵 = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵)
7775, 76eqeq12d 2785 . . . . . 6 (𝑎 = (𝑐 ↾ (1...𝑁)) → (((𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ↔ ((𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵))
7874oveq2d 6808 . . . . . . 7 (𝑎 = (𝑐 ↾ (1...𝑁)) → (-(𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = (-(𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴))
7978, 76eqeq12d 2785 . . . . . 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 883 . . . . 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 37877 . . . 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 565 . . 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, 82syl5eqel 2853 . 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 2849 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 196  wa 382  wo 826  w3a 1070   = wceq 1630  wcel 2144  wral 3060  wrex 3061  {crab 3064  Vcvv 3349  csb 3680   class class class wbr 4784  cmpt 4861  cres 5251  cfv 6031  (class class class)co 6792  𝑚 cmap 8008  1c1 10138   + caddc 10140   · cmul 10142  -cneg 10468  cn 11221  0cn0 11493  cz 11578  ...cfz 12532  cdvds 15188  mzPolycmzp 37804  Diophcdioph 37837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-of 7043  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-card 8964  df-cda 9191  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-n0 11494  df-z 11579  df-uz 11888  df-fz 12533  df-hash 13321  df-dvds 15189  df-mzpcl 37805  df-mzp 37806  df-dioph 37838
This theorem is referenced by:  rmydioph  38100  expdiophlem2  38108
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