Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvdsrabdioph Structured version   Visualization version   GIF version

Theorem dvdsrabdioph 42821
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 42812 . . . 4 ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0m (1...𝑁))𝐴 ∈ ℤ)
2 rabdiophlem1 42812 . . . 4 ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0m (1...𝑁))𝐵 ∈ ℤ)
3 divides 16292 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴𝐵 ↔ ∃𝑎 ∈ ℤ (𝑎 · 𝐴) = 𝐵))
4 oveq1 7438 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑎 · 𝐴) = (𝑏 · 𝐴))
54eqeq1d 2739 . . . . . . . 8 (𝑎 = 𝑏 → ((𝑎 · 𝐴) = 𝐵 ↔ (𝑏 · 𝐴) = 𝐵))
6 oveq1 7438 . . . . . . . . 9 (𝑎 = -𝑏 → (𝑎 · 𝐴) = (-𝑏 · 𝐴))
76eqeq1d 2739 . . . . . . . 8 (𝑎 = -𝑏 → ((𝑎 · 𝐴) = 𝐵 ↔ (-𝑏 · 𝐴) = 𝐵))
85, 7rexzrexnn0 42815 . . . . . . 7 (∃𝑎 ∈ ℤ (𝑎 · 𝐴) = 𝐵 ↔ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵))
93, 8bitrdi 287 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴𝐵 ↔ ∃𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)))
109ralimi 3083 . . . . 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 3111 . . . . 5 (∀𝑡 ∈ (ℕ0m (1...𝑁))(𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ↔ (∀𝑡 ∈ (ℕ0m (1...𝑁))𝐴 ∈ ℤ ∧ ∀𝑡 ∈ (ℕ0m (1...𝑁))𝐵 ∈ ℤ))
12 rabbi 3467 . . . . 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 291 . . . 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 596 . . 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 1131 . 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 2905 . . . 4 𝑡(ℕ0m (1...𝑁))
17 nfcv 2905 . . . 4 𝑎(ℕ0m (1...𝑁))
18 nfv 1914 . . . 4 𝑎𝑏 ∈ ℕ0 ((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵)
19 nfcv 2905 . . . . 5 𝑡0
20 nfcv 2905 . . . . . . . 8 𝑡𝑏
21 nfcv 2905 . . . . . . . 8 𝑡 ·
22 nfcsb1v 3923 . . . . . . . 8 𝑡𝑎 / 𝑡𝐴
2320, 21, 22nfov 7461 . . . . . . 7 𝑡(𝑏 · 𝑎 / 𝑡𝐴)
24 nfcsb1v 3923 . . . . . . 7 𝑡𝑎 / 𝑡𝐵
2523, 24nfeq 2919 . . . . . 6 𝑡(𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵
26 nfcv 2905 . . . . . . . 8 𝑡-𝑏
2726, 21, 22nfov 7461 . . . . . . 7 𝑡(-𝑏 · 𝑎 / 𝑡𝐴)
2827, 24nfeq 2919 . . . . . 6 𝑡(-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵
2925, 28nfor 1904 . . . . 5 𝑡((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵)
3019, 29nfrexw 3313 . . . 4 𝑡𝑏 ∈ ℕ0 ((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵)
31 csbeq1a 3913 . . . . . . . 8 (𝑡 = 𝑎𝐴 = 𝑎 / 𝑡𝐴)
3231oveq2d 7447 . . . . . . 7 (𝑡 = 𝑎 → (𝑏 · 𝐴) = (𝑏 · 𝑎 / 𝑡𝐴))
33 csbeq1a 3913 . . . . . . 7 (𝑡 = 𝑎𝐵 = 𝑎 / 𝑡𝐵)
3432, 33eqeq12d 2753 . . . . . 6 (𝑡 = 𝑎 → ((𝑏 · 𝐴) = 𝐵 ↔ (𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵))
3531oveq2d 7447 . . . . . . 7 (𝑡 = 𝑎 → (-𝑏 · 𝐴) = (-𝑏 · 𝑎 / 𝑡𝐴))
3635, 33eqeq12d 2753 . . . . . 6 (𝑡 = 𝑎 → ((-𝑏 · 𝐴) = 𝐵 ↔ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵))
3734, 36orbi12d 919 . . . . 5 (𝑡 = 𝑎 → (((𝑏 · 𝐴) = 𝐵 ∨ (-𝑏 · 𝐴) = 𝐵) ↔ ((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ∨ (-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵)))
3837rexbidv 3179 . . . 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 3473 . . 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 1137 . . . 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 12566 . . . . . . 7 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
42413ad2ant1 1134 . . . . . 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 12565 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
45 elfz1end 13594 . . . . . . . . . . 11 ((𝑁 + 1) ∈ ℕ ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
4644, 45sylib 218 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
47 mzpproj 42748 . . . . . . . . . 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 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 2737 . . . . . . . . 9 (𝑁 + 1) = (𝑁 + 1)
5150rabdiophlem2 42813 . . . . . . . 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 42753 . . . . . . . 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 1133 . . . . . 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 42813 . . . . . . 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 1132 . . . . . 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 42788 . . . . . 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 1373 . . . . 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 42755 . . . . . . . . 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 42753 . . . . . . . 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 1133 . . . . . 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 42788 . . . . . 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 1373 . . . . 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 42792 . . . . 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 7438 . . . . . . 7 (𝑏 = (𝑐‘(𝑁 + 1)) → (𝑏 · 𝑎 / 𝑡𝐴) = ((𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴))
6968eqeq1d 2739 . . . . . 6 (𝑏 = (𝑐‘(𝑁 + 1)) → ((𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ↔ ((𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵))
70 negeq 11500 . . . . . . . 8 (𝑏 = (𝑐‘(𝑁 + 1)) → -𝑏 = -(𝑐‘(𝑁 + 1)))
7170oveq1d 7446 . . . . . . 7 (𝑏 = (𝑐‘(𝑁 + 1)) → (-𝑏 · 𝑎 / 𝑡𝐴) = (-(𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴))
7271eqeq1d 2739 . . . . . 6 (𝑏 = (𝑐‘(𝑁 + 1)) → ((-𝑏 · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ↔ (-(𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵))
7369, 72orbi12d 919 . . . . 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 3902 . . . . . . . 8 (𝑎 = (𝑐 ↾ (1...𝑁)) → 𝑎 / 𝑡𝐴 = (𝑐 ↾ (1...𝑁)) / 𝑡𝐴)
7574oveq2d 7447 . . . . . . 7 (𝑎 = (𝑐 ↾ (1...𝑁)) → ((𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = ((𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴))
76 csbeq1 3902 . . . . . . 7 (𝑎 = (𝑐 ↾ (1...𝑁)) → 𝑎 / 𝑡𝐵 = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵)
7775, 76eqeq12d 2753 . . . . . 6 (𝑎 = (𝑐 ↾ (1...𝑁)) → (((𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = 𝑎 / 𝑡𝐵 ↔ ((𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴) = (𝑐 ↾ (1...𝑁)) / 𝑡𝐵))
7874oveq2d 7447 . . . . . . 7 (𝑎 = (𝑐 ↾ (1...𝑁)) → (-(𝑐‘(𝑁 + 1)) · 𝑎 / 𝑡𝐴) = (-(𝑐‘(𝑁 + 1)) · (𝑐 ↾ (1...𝑁)) / 𝑡𝐴))
7978, 76eqeq12d 2753 . . . . . 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 919 . . . . 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 42805 . . . 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 2845 . 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 2841 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 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  csb 3899   class class class wbr 5143  cmpt 5225  cres 5687  cfv 6561  (class class class)co 7431  m cmap 8866  1c1 11156   + caddc 11158   · cmul 11160  -cneg 11493  cn 12266  0cn0 12526  cz 12613  ...cfz 13547  cdvds 16290  mzPolycmzp 42733  Diophcdioph 42766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370  df-dvds 16291  df-mzpcl 42734  df-mzp 42735  df-dioph 42767
This theorem is referenced by:  rmydioph  43026  expdiophlem2  43034
  Copyright terms: Public domain W3C validator