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 42291
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 42282 . . . 4 ((𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) β†’ βˆ€π‘‘ ∈ (β„•0 ↑m (1...𝑁))𝐴 ∈ β„€)
2 rabdiophlem1 42282 . . . 4 ((𝑑 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐡) ∈ (mzPolyβ€˜(1...𝑁)) β†’ βˆ€π‘‘ ∈ (β„•0 ↑m (1...𝑁))𝐡 ∈ β„€)
3 divides 16227 . . . . . . 7 ((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ (𝐴 βˆ₯ 𝐡 ↔ βˆƒπ‘Ž ∈ β„€ (π‘Ž Β· 𝐴) = 𝐡))
4 oveq1 7420 . . . . . . . . 9 (π‘Ž = 𝑏 β†’ (π‘Ž Β· 𝐴) = (𝑏 Β· 𝐴))
54eqeq1d 2727 . . . . . . . 8 (π‘Ž = 𝑏 β†’ ((π‘Ž Β· 𝐴) = 𝐡 ↔ (𝑏 Β· 𝐴) = 𝐡))
6 oveq1 7420 . . . . . . . . 9 (π‘Ž = -𝑏 β†’ (π‘Ž Β· 𝐴) = (-𝑏 Β· 𝐴))
76eqeq1d 2727 . . . . . . . 8 (π‘Ž = -𝑏 β†’ ((π‘Ž Β· 𝐴) = 𝐡 ↔ (-𝑏 Β· 𝐴) = 𝐡))
85, 7rexzrexnn0 42285 . . . . . . 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 3073 . . . . 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 3101 . . . . 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 3450 . . . . 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 1127 . 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 2892 . . . 4 Ⅎ𝑑(β„•0 ↑m (1...𝑁))
17 nfcv 2892 . . . 4 β„²π‘Ž(β„•0 ↑m (1...𝑁))
18 nfv 1909 . . . 4 β„²π‘Žβˆƒπ‘ ∈ β„•0 ((𝑏 Β· 𝐴) = 𝐡 ∨ (-𝑏 Β· 𝐴) = 𝐡)
19 nfcv 2892 . . . . 5 Ⅎ𝑑ℕ0
20 nfcv 2892 . . . . . . . 8 Ⅎ𝑑𝑏
21 nfcv 2892 . . . . . . . 8 Ⅎ𝑑 Β·
22 nfcsb1v 3911 . . . . . . . 8 β„²π‘‘β¦‹π‘Ž / π‘‘β¦Œπ΄
2320, 21, 22nfov 7443 . . . . . . 7 Ⅎ𝑑(𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄)
24 nfcsb1v 3911 . . . . . . 7 β„²π‘‘β¦‹π‘Ž / π‘‘β¦Œπ΅
2523, 24nfeq 2906 . . . . . 6 Ⅎ𝑑(𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅
26 nfcv 2892 . . . . . . . 8 Ⅎ𝑑-𝑏
2726, 21, 22nfov 7443 . . . . . . 7 Ⅎ𝑑(-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄)
2827, 24nfeq 2906 . . . . . 6 Ⅎ𝑑(-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅
2925, 28nfor 1899 . . . . 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 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 3900 . . . . . . . 8 (𝑑 = π‘Ž β†’ 𝐴 = β¦‹π‘Ž / π‘‘β¦Œπ΄)
3231oveq2d 7429 . . . . . . 7 (𝑑 = π‘Ž β†’ (𝑏 Β· 𝐴) = (𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄))
33 csbeq1a 3900 . . . . . . 7 (𝑑 = π‘Ž β†’ 𝐡 = β¦‹π‘Ž / π‘‘β¦Œπ΅)
3432, 33eqeq12d 2741 . . . . . 6 (𝑑 = π‘Ž β†’ ((𝑏 Β· 𝐴) = 𝐡 ↔ (𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄) = β¦‹π‘Ž / π‘‘β¦Œπ΅))
3531oveq2d 7429 . . . . . . 7 (𝑑 = π‘Ž β†’ (-𝑏 Β· 𝐴) = (-𝑏 Β· β¦‹π‘Ž / π‘‘β¦Œπ΄))
3635, 33eqeq12d 2741 . . . . . 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 916 . . . . 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 3169 . . . 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 3456 . . 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 1133 . . . 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 12537 . . . . . . 7 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•0)
42413ad2ant1 1130 . . . . . 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 7446 . . . . . . . . . 10 (1...(𝑁 + 1)) ∈ V
44 nn0p1nn 12536 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•)
45 elfz1end 13558 . . . . . . . . . . 11 ((𝑁 + 1) ∈ β„• ↔ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
4644, 45sylib 217 . . . . . . . . . 10 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ (1...(𝑁 + 1)))
47 mzpproj 42218 . . . . . . . . . 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 2725 . . . . . . . . 9 (𝑁 + 1) = (𝑁 + 1)
5150rabdiophlem2 42283 . . . . . . . 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 42223 . . . . . . . 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 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))) ↦ ((π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄)) ∈ (mzPolyβ€˜(1...(𝑁 + 1))))
5550rabdiophlem2 42283 . . . . . . 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 1128 . . . . . 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 42258 . . . . . 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 1368 . . . . 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 42225 . . . . . . . . 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 42223 . . . . . . . 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 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))) ↦ (-(π‘β€˜(𝑁 + 1)) Β· ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄)) ∈ (mzPolyβ€˜(1...(𝑁 + 1))))
64 eqrabdioph 42258 . . . . . 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 1368 . . . . 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 42262 . . . . 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 7420 . . . . . . 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 2727 . . . . . 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 11477 . . . . . . . 8 (𝑏 = (π‘β€˜(𝑁 + 1)) β†’ -𝑏 = -(π‘β€˜(𝑁 + 1)))
7170oveq1d 7428 . . . . . . 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 2727 . . . . . 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 916 . . . . 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 3889 . . . . . . . 8 (π‘Ž = (𝑐 β†Ύ (1...𝑁)) β†’ β¦‹π‘Ž / π‘‘β¦Œπ΄ = ⦋(𝑐 β†Ύ (1...𝑁)) / π‘‘β¦Œπ΄)
7574oveq2d 7429 . . . . . . 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 3889 . . . . . . 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 2741 . . . . . 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 7429 . . . . . . 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 2741 . . . . . 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 916 . . . . 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 42275 . . . 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 2829 . 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 2825 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 845   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  βˆƒwrex 3060  {crab 3419  Vcvv 3463  β¦‹csb 3886   class class class wbr 5144   ↦ cmpt 5227   β†Ύ cres 5675  β€˜cfv 6543  (class class class)co 7413   ↑m cmap 8838  1c1 11134   + caddc 11136   Β· cmul 11138  -cneg 11470  β„•cn 12237  β„•0cn0 12497  β„€cz 12583  ...cfz 13511   βˆ₯ cdvds 16225  mzPolycmzp 42203  Diophcdioph 42236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-inf2 9659  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7679  df-om 7866  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-dju 9919  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-n0 12498  df-z 12584  df-uz 12848  df-fz 13512  df-hash 14317  df-dvds 16226  df-mzpcl 42204  df-mzp 42205  df-dioph 42237
This theorem is referenced by:  rmydioph  42496  expdiophlem2  42504
  Copyright terms: Public domain W3C validator