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Theorem dvdsrabdioph 40668
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 40659 . . . 4 ((𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) β†’ βˆ€π‘‘ ∈ (β„•0 ↑m (1...𝑁))𝐴 ∈ β„€)
2 rabdiophlem1 40659 . . . 4 ((𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐡) ∈ (mzPolyβ€˜(1...𝑁)) β†’ βˆ€π‘‘ ∈ (β„•0 ↑m (1...𝑁))𝐡 ∈ β„€)
3 divides 16006 . . . . . . 7 ((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ (𝐴 βˆ₯ 𝐡 ↔ βˆƒπ‘Ž ∈ β„€ (π‘Ž Β· 𝐴) = 𝐡))
4 oveq1 7310 . . . . . . . . 9 (π‘Ž = 𝑏 β†’ (π‘Ž Β· 𝐴) = (𝑏 Β· 𝐴))
54eqeq1d 2738 . . . . . . . 8 (π‘Ž = 𝑏 β†’ ((π‘Ž Β· 𝐴) = 𝐡 ↔ (𝑏 Β· 𝐴) = 𝐡))
6 oveq1 7310 . . . . . . . . 9 (π‘Ž = -𝑏 β†’ (π‘Ž Β· 𝐴) = (-𝑏 Β· 𝐴))
76eqeq1d 2738 . . . . . . . 8 (π‘Ž = -𝑏 β†’ ((π‘Ž Β· 𝐴) = 𝐡 ↔ (-𝑏 Β· 𝐴) = 𝐡))
85, 7rexzrexnn0 40662 . . . . . . 7 (βˆƒπ‘Ž ∈ β„€ (π‘Ž Β· 𝐴) = 𝐡 ↔ βˆƒπ‘ ∈ β„•0 ((𝑏 Β· 𝐴) = 𝐡 ∨ (-𝑏 Β· 𝐴) = 𝐡))
93, 8bitrdi 288 . . . . . 6 ((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ (𝐴 βˆ₯ 𝐡 ↔ βˆƒπ‘ ∈ β„•0 ((𝑏 Β· 𝐴) = 𝐡 ∨ (-𝑏 Β· 𝐴) = 𝐡)))
109ralimi 3083 . . . . 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 3111 . . . . 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 3328 . . . . 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 292 . . . 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 597 . . 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 1130 . 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 2905 . . . 4 Ⅎ𝑑(β„•0 ↑m (1...𝑁))
17 nfcv 2905 . . . 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 2905 . . . . 5 Ⅎ𝑑ℕ0
20 nfcv 2905 . . . . . . . 8 Ⅎ𝑑𝑏
21 nfcv 2905 . . . . . . . 8 Ⅎ𝑑 Β·
22 nfcsb1v 3862 . . . . . . . 8 β„²π‘‘β¦‹π‘Ž / π‘‘β¦Œπ΄
2320, 21, 22nfov 7333 . . . . . . 7 Ⅎ𝑑(𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄)
24 nfcsb1v 3862 . . . . . . 7 β„²π‘‘β¦‹π‘Ž / π‘‘β¦Œπ΅
2523, 24nfeq 2918 . . . . . 6 Ⅎ𝑑(𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅
26 nfcv 2905 . . . . . . . 8 Ⅎ𝑑-𝑏
2726, 21, 22nfov 7333 . . . . . . 7 Ⅎ𝑑(-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄)
2827, 24nfeq 2918 . . . . . 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, 29nfrex 3301 . . . 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 3851 . . . . . . . 8 (𝑑 = π‘Ž β†’ 𝐴 = β¦‹π‘Ž / π‘‘β¦Œπ΄)
3231oveq2d 7319 . . . . . . 7 (𝑑 = π‘Ž β†’ (𝑏 Β· 𝐴) = (𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄))
33 csbeq1a 3851 . . . . . . 7 (𝑑 = π‘Ž β†’ 𝐡 = β¦‹π‘Ž / π‘‘β¦Œπ΅)
3432, 33eqeq12d 2752 . . . . . 6 (𝑑 = π‘Ž β†’ ((𝑏 Β· 𝐴) = 𝐡 ↔ (𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅))
3531oveq2d 7319 . . . . . . 7 (𝑑 = π‘Ž β†’ (-𝑏 Β· 𝐴) = (-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄))
3635, 33eqeq12d 2752 . . . . . 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 917 . . . . 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 3172 . . . 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 3431 . . 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 1136 . . . 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 12315 . . . . . . 7 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•0)
42413ad2ant1 1133 . . . . . 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 7336 . . . . . . . . . 10 (1...(𝑁 + 1)) ∈ V
44 nn0p1nn 12314 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•)
45 elfz1end 13328 . . . . . . . . . . 11 ((𝑁 + 1) ∈ β„• ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
4644, 45sylib 217 . . . . . . . . . 10 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
47 mzpproj 40595 . . . . . . . . . 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 588 . . . . . . . . 9 (𝑁 ∈ β„•0 β†’ (𝑐 ∈ (β„€ ↑m (1...(𝑁 + 1))) ↦ (π‘β€˜(𝑁 + 1))) ∈ (mzPolyβ€˜(1...(𝑁 + 1))))
4948adantr 482 . . . . . . . 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 2736 . . . . . . . . 9 (𝑁 + 1) = (𝑁 + 1)
5150rabdiophlem2 40660 . . . . . . . 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 40600 . . . . . . . 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 585 . . . . . . 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 1132 . . . . . 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 40660 . . . . . . 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 1131 . . . . . 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 40635 . . . . . 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 1371 . . . . 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 40602 . . . . . . . . 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 40600 . . . . . . . 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 585 . . . . . . 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 1132 . . . . . 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 40635 . . . . . 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 1371 . . . . 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 40639 . . . . 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 585 . . . 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 7310 . . . . . . 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 2738 . . . . . 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 11255 . . . . . . . 8 (𝑏 = (π‘β€˜(𝑁 + 1)) β†’ -𝑏 = -(π‘β€˜(𝑁 + 1)))
7170oveq1d 7318 . . . . . . 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 2738 . . . . . 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 917 . . . . 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 3840 . . . . . . . 8 (π‘Ž = (𝑐 β†Ύ (1...𝑁)) β†’ β¦‹π‘Ž / π‘‘β¦Œπ΄ = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄)
7574oveq2d 7319 . . . . . . 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 3840 . . . . . . 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 2752 . . . . . 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 7319 . . . . . . 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 2752 . . . . . 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 917 . . . . 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 40652 . . . 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 585 . . 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 2841 . 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 2837 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 397   ∨ wo 845   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104  βˆ€wral 3062  βˆƒwrex 3071  {crab 3284  Vcvv 3437  β¦‹csb 3837   class class class wbr 5081   ↦ cmpt 5164   β†Ύ cres 5598  β€˜cfv 6454  (class class class)co 7303   ↑m cmap 8642  1c1 10914   + caddc 10916   Β· cmul 10918  -cneg 11248  β„•cn 12015  β„•0cn0 12275  β„€cz 12361  ...cfz 13281   βˆ₯ cdvds 16004  mzPolycmzp 40580  Diophcdioph 40613
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 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7616  ax-inf2 9439  ax-cnex 10969  ax-resscn 10970  ax-1cn 10971  ax-icn 10972  ax-addcl 10973  ax-addrcl 10974  ax-mulcl 10975  ax-mulrcl 10976  ax-mulcom 10977  ax-addass 10978  ax-mulass 10979  ax-distr 10980  ax-i2m1 10981  ax-1ne0 10982  ax-1rid 10983  ax-rnegex 10984  ax-rrecex 10985  ax-cnre 10986  ax-pre-lttri 10987  ax-pre-lttrn 10988  ax-pre-ltadd 10989  ax-pre-mulgt0 10990
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5496  df-eprel 5502  df-po 5510  df-so 5511  df-fr 5551  df-we 5553  df-xp 5602  df-rel 5603  df-cnv 5604  df-co 5605  df-dm 5606  df-rn 5607  df-res 5608  df-ima 5609  df-pred 6213  df-ord 6280  df-on 6281  df-lim 6282  df-suc 6283  df-iota 6406  df-fun 6456  df-fn 6457  df-f 6458  df-f1 6459  df-fo 6460  df-f1o 6461  df-fv 6462  df-riota 7260  df-ov 7306  df-oprab 7307  df-mpo 7308  df-of 7561  df-om 7741  df-1st 7859  df-2nd 7860  df-frecs 8124  df-wrecs 8155  df-recs 8229  df-rdg 8268  df-1o 8324  df-oadd 8328  df-er 8525  df-map 8644  df-en 8761  df-dom 8762  df-sdom 8763  df-fin 8764  df-dju 9699  df-card 9737  df-pnf 11053  df-mnf 11054  df-xr 11055  df-ltxr 11056  df-le 11057  df-sub 11249  df-neg 11250  df-nn 12016  df-n0 12276  df-z 12362  df-uz 12625  df-fz 13282  df-hash 14087  df-dvds 16005  df-mzpcl 40581  df-mzp 40582  df-dioph 40614
This theorem is referenced by:  rmydioph  40873  expdiophlem2  40881
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