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Theorem dvdsrabdioph 41850
Description: Divisibility is a Diophantine relation. (Contributed by Stefan O'Rear, 11-Oct-2014.)
Assertion
Ref Expression
dvdsrabdioph ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐡) ∈ (mzPolyβ€˜(1...𝑁))) β†’ {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ 𝐴 βˆ₯ 𝐡} ∈ (Diophβ€˜π‘))
Distinct variable group:   𝑑,𝑁
Allowed substitution hints:   𝐴(𝑑)   𝐡(𝑑)

Proof of Theorem dvdsrabdioph
Dummy variables π‘Ž 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rabdiophlem1 41841 . . . 4 ((𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) β†’ βˆ€π‘‘ ∈ (β„•0 ↑m (1...𝑁))𝐴 ∈ β„€)
2 rabdiophlem1 41841 . . . 4 ((𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐡) ∈ (mzPolyβ€˜(1...𝑁)) β†’ βˆ€π‘‘ ∈ (β„•0 ↑m (1...𝑁))𝐡 ∈ β„€)
3 divides 16203 . . . . . . 7 ((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ (𝐴 βˆ₯ 𝐡 ↔ βˆƒπ‘Ž ∈ β„€ (π‘Ž Β· 𝐴) = 𝐡))
4 oveq1 7418 . . . . . . . . 9 (π‘Ž = 𝑏 β†’ (π‘Ž Β· 𝐴) = (𝑏 Β· 𝐴))
54eqeq1d 2732 . . . . . . . 8 (π‘Ž = 𝑏 β†’ ((π‘Ž Β· 𝐴) = 𝐡 ↔ (𝑏 Β· 𝐴) = 𝐡))
6 oveq1 7418 . . . . . . . . 9 (π‘Ž = -𝑏 β†’ (π‘Ž Β· 𝐴) = (-𝑏 Β· 𝐴))
76eqeq1d 2732 . . . . . . . 8 (π‘Ž = -𝑏 β†’ ((π‘Ž Β· 𝐴) = 𝐡 ↔ (-𝑏 Β· 𝐴) = 𝐡))
85, 7rexzrexnn0 41844 . . . . . . 7 (βˆƒπ‘Ž ∈ β„€ (π‘Ž Β· 𝐴) = 𝐡 ↔ βˆƒπ‘ ∈ β„•0 ((𝑏 Β· 𝐴) = 𝐡 ∨ (-𝑏 Β· 𝐴) = 𝐡))
93, 8bitrdi 286 . . . . . 6 ((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ (𝐴 βˆ₯ 𝐡 ↔ βˆƒπ‘ ∈ β„•0 ((𝑏 Β· 𝐴) = 𝐡 ∨ (-𝑏 Β· 𝐴) = 𝐡)))
109ralimi 3081 . . . . 5 (βˆ€π‘‘ ∈ (β„•0 ↑m (1...𝑁))(𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ βˆ€π‘‘ ∈ (β„•0 ↑m (1...𝑁))(𝐴 βˆ₯ 𝐡 ↔ βˆƒπ‘ ∈ β„•0 ((𝑏 Β· 𝐴) = 𝐡 ∨ (-𝑏 Β· 𝐴) = 𝐡)))
11 r19.26 3109 . . . . 5 (βˆ€π‘‘ ∈ (β„•0 ↑m (1...𝑁))(𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) ↔ (βˆ€π‘‘ ∈ (β„•0 ↑m (1...𝑁))𝐴 ∈ β„€ ∧ βˆ€π‘‘ ∈ (β„•0 ↑m (1...𝑁))𝐡 ∈ β„€))
12 rabbi 3460 . . . . 5 (βˆ€π‘‘ ∈ (β„•0 ↑m (1...𝑁))(𝐴 βˆ₯ 𝐡 ↔ βˆƒπ‘ ∈ β„•0 ((𝑏 Β· 𝐴) = 𝐡 ∨ (-𝑏 Β· 𝐴) = 𝐡)) ↔ {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ 𝐴 βˆ₯ 𝐡} = {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘ ∈ β„•0 ((𝑏 Β· 𝐴) = 𝐡 ∨ (-𝑏 Β· 𝐴) = 𝐡)})
1310, 11, 123imtr3i 290 . . . 4 ((βˆ€π‘‘ ∈ (β„•0 ↑m (1...𝑁))𝐴 ∈ β„€ ∧ βˆ€π‘‘ ∈ (β„•0 ↑m (1...𝑁))𝐡 ∈ β„€) β†’ {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ 𝐴 βˆ₯ 𝐡} = {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘ ∈ β„•0 ((𝑏 Β· 𝐴) = 𝐡 ∨ (-𝑏 Β· 𝐴) = 𝐡)})
141, 2, 13syl2an 594 . . 3 (((𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐡) ∈ (mzPolyβ€˜(1...𝑁))) β†’ {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ 𝐴 βˆ₯ 𝐡} = {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘ ∈ β„•0 ((𝑏 Β· 𝐴) = 𝐡 ∨ (-𝑏 Β· 𝐴) = 𝐡)})
15143adant1 1128 . 2 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐡) ∈ (mzPolyβ€˜(1...𝑁))) β†’ {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ 𝐴 βˆ₯ 𝐡} = {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘ ∈ β„•0 ((𝑏 Β· 𝐴) = 𝐡 ∨ (-𝑏 Β· 𝐴) = 𝐡)})
16 nfcv 2901 . . . 4 Ⅎ𝑑(β„•0 ↑m (1...𝑁))
17 nfcv 2901 . . . 4 β„²π‘Ž(β„•0 ↑m (1...𝑁))
18 nfv 1915 . . . 4 β„²π‘Žβˆƒπ‘ ∈ β„•0 ((𝑏 Β· 𝐴) = 𝐡 ∨ (-𝑏 Β· 𝐴) = 𝐡)
19 nfcv 2901 . . . . 5 Ⅎ𝑑ℕ0
20 nfcv 2901 . . . . . . . 8 Ⅎ𝑑𝑏
21 nfcv 2901 . . . . . . . 8 Ⅎ𝑑 Β·
22 nfcsb1v 3917 . . . . . . . 8 β„²π‘‘β¦‹π‘Ž / π‘‘β¦Œπ΄
2320, 21, 22nfov 7441 . . . . . . 7 Ⅎ𝑑(𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄)
24 nfcsb1v 3917 . . . . . . 7 β„²π‘‘β¦‹π‘Ž / π‘‘β¦Œπ΅
2523, 24nfeq 2914 . . . . . 6 Ⅎ𝑑(𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅
26 nfcv 2901 . . . . . . . 8 Ⅎ𝑑-𝑏
2726, 21, 22nfov 7441 . . . . . . 7 Ⅎ𝑑(-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄)
2827, 24nfeq 2914 . . . . . 6 Ⅎ𝑑(-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅
2925, 28nfor 1905 . . . . 5 Ⅎ𝑑((𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅ ∨ (-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅)
3019, 29nfrexw 3308 . . . 4 β„²π‘‘βˆƒπ‘ ∈ β„•0 ((𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅ ∨ (-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅)
31 csbeq1a 3906 . . . . . . . 8 (𝑑 = π‘Ž β†’ 𝐴 = β¦‹π‘Ž / π‘‘β¦Œπ΄)
3231oveq2d 7427 . . . . . . 7 (𝑑 = π‘Ž β†’ (𝑏 Β· 𝐴) = (𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄))
33 csbeq1a 3906 . . . . . . 7 (𝑑 = π‘Ž β†’ 𝐡 = β¦‹π‘Ž / π‘‘β¦Œπ΅)
3432, 33eqeq12d 2746 . . . . . 6 (𝑑 = π‘Ž β†’ ((𝑏 Β· 𝐴) = 𝐡 ↔ (𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅))
3531oveq2d 7427 . . . . . . 7 (𝑑 = π‘Ž β†’ (-𝑏 Β· 𝐴) = (-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄))
3635, 33eqeq12d 2746 . . . . . 6 (𝑑 = π‘Ž β†’ ((-𝑏 Β· 𝐴) = 𝐡 ↔ (-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅))
3734, 36orbi12d 915 . . . . 5 (𝑑 = π‘Ž β†’ (((𝑏 Β· 𝐴) = 𝐡 ∨ (-𝑏 Β· 𝐴) = 𝐡) ↔ ((𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅ ∨ (-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅)))
3837rexbidv 3176 . . . 4 (𝑑 = π‘Ž β†’ (βˆƒπ‘ ∈ β„•0 ((𝑏 Β· 𝐴) = 𝐡 ∨ (-𝑏 Β· 𝐴) = 𝐡) ↔ βˆƒπ‘ ∈ β„•0 ((𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅ ∨ (-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅)))
3916, 17, 18, 30, 38cbvrabw 3465 . . 3 {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘ ∈ β„•0 ((𝑏 Β· 𝐴) = 𝐡 ∨ (-𝑏 Β· 𝐴) = 𝐡)} = {π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘ ∈ β„•0 ((𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅ ∨ (-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅)}
40 simp1 1134 . . . 4 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐡) ∈ (mzPolyβ€˜(1...𝑁))) β†’ 𝑁 ∈ β„•0)
41 peano2nn0 12516 . . . . . . 7 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•0)
42413ad2ant1 1131 . . . . . 6 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐡) ∈ (mzPolyβ€˜(1...𝑁))) β†’ (𝑁 + 1) ∈ β„•0)
43 ovex 7444 . . . . . . . . . 10 (1...(𝑁 + 1)) ∈ V
44 nn0p1nn 12515 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•)
45 elfz1end 13535 . . . . . . . . . . 11 ((𝑁 + 1) ∈ β„• ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
4644, 45sylib 217 . . . . . . . . . 10 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
47 mzpproj 41777 . . . . . . . . . 10 (((1...(𝑁 + 1)) ∈ V ∧ (𝑁 + 1) ∈ (1...(𝑁 + 1))) β†’ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ (π‘β€˜(𝑁 + 1))) ∈ (mzPolyβ€˜(1...(𝑁 + 1))))
4843, 46, 47sylancr 585 . . . . . . . . 9 (𝑁 ∈ β„•0 β†’ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ (π‘β€˜(𝑁 + 1))) ∈ (mzPolyβ€˜(1...(𝑁 + 1))))
4948adantr 479 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁))) β†’ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ (π‘β€˜(𝑁 + 1))) ∈ (mzPolyβ€˜(1...(𝑁 + 1))))
50 eqid 2730 . . . . . . . . 9 (𝑁 + 1) = (𝑁 + 1)
5150rabdiophlem2 41842 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁))) β†’ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) ∈ (mzPolyβ€˜(1...(𝑁 + 1))))
52 mzpmulmpt 41782 . . . . . . . 8 (((𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ (π‘β€˜(𝑁 + 1))) ∈ (mzPolyβ€˜(1...(𝑁 + 1))) ∧ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) ∈ (mzPolyβ€˜(1...(𝑁 + 1)))) β†’ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ ((π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄)) ∈ (mzPolyβ€˜(1...(𝑁 + 1))))
5349, 51, 52syl2anc 582 . . . . . . 7 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁))) β†’ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ ((π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄)) ∈ (mzPolyβ€˜(1...(𝑁 + 1))))
54533adant3 1130 . . . . . 6 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐡) ∈ (mzPolyβ€˜(1...𝑁))) β†’ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ ((π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄)) ∈ (mzPolyβ€˜(1...(𝑁 + 1))))
5550rabdiophlem2 41842 . . . . . . 7 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐡) ∈ (mzPolyβ€˜(1...𝑁))) β†’ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅) ∈ (mzPolyβ€˜(1...(𝑁 + 1))))
56553adant2 1129 . . . . . 6 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐡) ∈ (mzPolyβ€˜(1...𝑁))) β†’ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅) ∈ (mzPolyβ€˜(1...(𝑁 + 1))))
57 eqrabdioph 41817 . . . . . 6 (((𝑁 + 1) ∈ β„•0 ∧ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ ((π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄)) ∈ (mzPolyβ€˜(1...(𝑁 + 1))) ∧ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅) ∈ (mzPolyβ€˜(1...(𝑁 + 1)))) β†’ {𝑐 ∈ (β„•0 ↑m (1...(𝑁 + 1))) ∣ ((π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅} ∈ (Diophβ€˜(𝑁 + 1)))
5842, 54, 56, 57syl3anc 1369 . . . . 5 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐡) ∈ (mzPolyβ€˜(1...𝑁))) β†’ {𝑐 ∈ (β„•0 ↑m (1...(𝑁 + 1))) ∣ ((π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅} ∈ (Diophβ€˜(𝑁 + 1)))
59 mzpnegmpt 41784 . . . . . . . . 9 ((𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ (π‘β€˜(𝑁 + 1))) ∈ (mzPolyβ€˜(1...(𝑁 + 1))) β†’ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ -(π‘β€˜(𝑁 + 1))) ∈ (mzPolyβ€˜(1...(𝑁 + 1))))
6049, 59syl 17 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁))) β†’ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ -(π‘β€˜(𝑁 + 1))) ∈ (mzPolyβ€˜(1...(𝑁 + 1))))
61 mzpmulmpt 41782 . . . . . . . 8 (((𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ -(π‘β€˜(𝑁 + 1))) ∈ (mzPolyβ€˜(1...(𝑁 + 1))) ∧ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) ∈ (mzPolyβ€˜(1...(𝑁 + 1)))) β†’ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ (-(π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄)) ∈ (mzPolyβ€˜(1...(𝑁 + 1))))
6260, 51, 61syl2anc 582 . . . . . . 7 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁))) β†’ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ (-(π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄)) ∈ (mzPolyβ€˜(1...(𝑁 + 1))))
63623adant3 1130 . . . . . 6 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐡) ∈ (mzPolyβ€˜(1...𝑁))) β†’ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ (-(π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄)) ∈ (mzPolyβ€˜(1...(𝑁 + 1))))
64 eqrabdioph 41817 . . . . . 6 (((𝑁 + 1) ∈ β„•0 ∧ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ (-(π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄)) ∈ (mzPolyβ€˜(1...(𝑁 + 1))) ∧ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅) ∈ (mzPolyβ€˜(1...(𝑁 + 1)))) β†’ {𝑐 ∈ (β„•0 ↑m (1...(𝑁 + 1))) ∣ (-(π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅} ∈ (Diophβ€˜(𝑁 + 1)))
6542, 63, 56, 64syl3anc 1369 . . . . 5 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐡) ∈ (mzPolyβ€˜(1...𝑁))) β†’ {𝑐 ∈ (β„•0 ↑m (1...(𝑁 + 1))) ∣ (-(π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅} ∈ (Diophβ€˜(𝑁 + 1)))
66 orrabdioph 41821 . . . . 5 (({𝑐 ∈ (β„•0 ↑m (1...(𝑁 + 1))) ∣ ((π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅} ∈ (Diophβ€˜(𝑁 + 1)) ∧ {𝑐 ∈ (β„•0 ↑m (1...(𝑁 + 1))) ∣ (-(π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅} ∈ (Diophβ€˜(𝑁 + 1))) β†’ {𝑐 ∈ (β„•0 ↑m (1...(𝑁 + 1))) ∣ (((π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅ ∨ (-(π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅)} ∈ (Diophβ€˜(𝑁 + 1)))
6758, 65, 66syl2anc 582 . . . 4 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐡) ∈ (mzPolyβ€˜(1...𝑁))) β†’ {𝑐 ∈ (β„•0 ↑m (1...(𝑁 + 1))) ∣ (((π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅ ∨ (-(π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅)} ∈ (Diophβ€˜(𝑁 + 1)))
68 oveq1 7418 . . . . . . 7 (𝑏 = (π‘β€˜(𝑁 + 1)) β†’ (𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = ((π‘β€˜(𝑁 + 1)) Β· β¦‹π‘Ž / π‘‘β¦Œπ΄))
6968eqeq1d 2732 . . . . . 6 (𝑏 = (π‘β€˜(𝑁 + 1)) β†’ ((𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅ ↔ ((π‘β€˜(𝑁 + 1)) Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅))
70 negeq 11456 . . . . . . . 8 (𝑏 = (π‘β€˜(𝑁 + 1)) β†’ -𝑏 = -(π‘β€˜(𝑁 + 1)))
7170oveq1d 7426 . . . . . . 7 (𝑏 = (π‘β€˜(𝑁 + 1)) β†’ (-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = (-(π‘β€˜(𝑁 + 1)) Β· β¦‹π‘Ž / π‘‘β¦Œπ΄))
7271eqeq1d 2732 . . . . . 6 (𝑏 = (π‘β€˜(𝑁 + 1)) β†’ ((-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅ ↔ (-(π‘β€˜(𝑁 + 1)) Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅))
7369, 72orbi12d 915 . . . . 5 (𝑏 = (π‘β€˜(𝑁 + 1)) β†’ (((𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅ ∨ (-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅) ↔ (((π‘β€˜(𝑁 + 1)) Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅ ∨ (-(π‘β€˜(𝑁 + 1)) Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅)))
74 csbeq1 3895 . . . . . . . 8 (π‘Ž = (𝑐 β†Ύ (1...𝑁)) β†’ β¦‹π‘Ž / π‘‘β¦Œπ΄ = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄)
7574oveq2d 7427 . . . . . . 7 (π‘Ž = (𝑐 β†Ύ (1...𝑁)) β†’ ((π‘β€˜(𝑁 + 1)) Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = ((π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄))
76 csbeq1 3895 . . . . . . 7 (π‘Ž = (𝑐 β†Ύ (1...𝑁)) β†’ β¦‹π‘Ž / π‘‘β¦Œπ΅ = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅)
7775, 76eqeq12d 2746 . . . . . 6 (π‘Ž = (𝑐 β†Ύ (1...𝑁)) β†’ (((π‘β€˜(𝑁 + 1)) Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅ ↔ ((π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅))
7874oveq2d 7427 . . . . . . 7 (π‘Ž = (𝑐 β†Ύ (1...𝑁)) β†’ (-(π‘β€˜(𝑁 + 1)) Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = (-(π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄))
7978, 76eqeq12d 2746 . . . . . 6 (π‘Ž = (𝑐 β†Ύ (1...𝑁)) β†’ ((-(π‘β€˜(𝑁 + 1)) Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅ ↔ (-(π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅))
8077, 79orbi12d 915 . . . . 5 (π‘Ž = (𝑐 β†Ύ (1...𝑁)) β†’ ((((π‘β€˜(𝑁 + 1)) Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅ ∨ (-(π‘β€˜(𝑁 + 1)) Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅) ↔ (((π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅ ∨ (-(π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅)))
8150, 73, 80rexrabdioph 41834 . . . 4 ((𝑁 ∈ β„•0 ∧ {𝑐 ∈ (β„•0 ↑m (1...(𝑁 + 1))) ∣ (((π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅ ∨ (-(π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄) = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΅)} ∈ (Diophβ€˜(𝑁 + 1))) β†’ {π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘ ∈ β„•0 ((𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅ ∨ (-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅)} ∈ (Diophβ€˜π‘))
8240, 67, 81syl2anc 582 . . 3 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐡) ∈ (mzPolyβ€˜(1...𝑁))) β†’ {π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘ ∈ β„•0 ((𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅ ∨ (-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅)} ∈ (Diophβ€˜π‘))
8339, 82eqeltrid 2835 . 2 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐡) ∈ (mzPolyβ€˜(1...𝑁))) β†’ {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘ ∈ β„•0 ((𝑏 Β· 𝐴) = 𝐡 ∨ (-𝑏 Β· 𝐴) = 𝐡)} ∈ (Diophβ€˜π‘))
8415, 83eqeltrd 2831 1 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) ∧ (𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐡) ∈ (mzPolyβ€˜(1...𝑁))) β†’ {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ 𝐴 βˆ₯ 𝐡} ∈ (Diophβ€˜π‘))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068  {crab 3430  Vcvv 3472  β¦‹csb 3892   class class class wbr 5147   ↦ cmpt 5230   β†Ύ cres 5677  β€˜cfv 6542  (class class class)co 7411   ↑m cmap 8822  1c1 11113   + caddc 11115   Β· cmul 11117  -cneg 11449  β„•cn 12216  β„•0cn0 12476  β„€cz 12562  ...cfz 13488   βˆ₯ cdvds 16201  mzPolycmzp 41762  Diophcdioph 41795
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-hash 14295  df-dvds 16202  df-mzpcl 41763  df-mzp 41764  df-dioph 41796
This theorem is referenced by:  rmydioph  42055  expdiophlem2  42063
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