Step | Hyp | Ref
| Expression |
1 | | fvex 6113 |
. . 3
⊢
(0g‘𝑄) ∈ V |
2 | 1 | a1i 11 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (0g‘𝑄) ∈ V) |
3 | | ovex 6577 |
. . 3
⊢ ((𝐴 Σg
(𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑙 − 𝑘))))) ∗ (𝑙 ↑ 𝑋)) ∈ V |
4 | 3 | a1i 11 |
. 2
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑙 ∈ ℕ0) → ((𝐴 Σg
(𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑙 − 𝑘))))) ∗ (𝑙 ↑ 𝑋)) ∈ V) |
5 | | oveq2 6557 |
. . . . 5
⊢ (𝑙 = 𝑛 → (0...𝑙) = (0...𝑛)) |
6 | | oveq1 6556 |
. . . . . . 7
⊢ (𝑙 = 𝑛 → (𝑙 − 𝑘) = (𝑛 − 𝑘)) |
7 | 6 | oveq2d 6565 |
. . . . . 6
⊢ (𝑙 = 𝑛 → (𝑦 decompPMat (𝑙 − 𝑘)) = (𝑦 decompPMat (𝑛 − 𝑘))) |
8 | 7 | oveq2d 6565 |
. . . . 5
⊢ (𝑙 = 𝑛 → ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑙 − 𝑘))) = ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘)))) |
9 | 5, 8 | mpteq12dv 4663 |
. . . 4
⊢ (𝑙 = 𝑛 → (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑙 − 𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))) |
10 | 9 | oveq2d 6565 |
. . 3
⊢ (𝑙 = 𝑛 → (𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑙 − 𝑘))))) = (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘)))))) |
11 | | oveq1 6556 |
. . 3
⊢ (𝑙 = 𝑛 → (𝑙 ↑ 𝑋) = (𝑛 ↑ 𝑋)) |
12 | 10, 11 | oveq12d 6567 |
. 2
⊢ (𝑙 = 𝑛 → ((𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑙 − 𝑘))))) ∗ (𝑙 ↑ 𝑋)) = ((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))) ∗ (𝑛 ↑ 𝑋))) |
13 | | simpll 786 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑁 ∈ Fin) |
14 | | simplr 788 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ Ring) |
15 | | pm2mpfo.p |
. . . . . . . . . 10
⊢ 𝑃 = (Poly1‘𝑅) |
16 | | pm2mpfo.c |
. . . . . . . . . 10
⊢ 𝐶 = (𝑁 Mat 𝑃) |
17 | 15, 16 | pmatring 20317 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring) |
18 | 17 | anim1i 590 |
. . . . . . . 8
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐶 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) |
19 | | 3anass 1035 |
. . . . . . . 8
⊢ ((𝐶 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (𝐶 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) |
20 | 18, 19 | sylibr 223 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐶 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) |
21 | | pm2mpfo.b |
. . . . . . . 8
⊢ 𝐵 = (Base‘𝐶) |
22 | | eqid 2610 |
. . . . . . . 8
⊢
(.r‘𝐶) = (.r‘𝐶) |
23 | 21, 22 | ringcl 18384 |
. . . . . . 7
⊢ ((𝐶 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝐶)𝑦) ∈ 𝐵) |
24 | 20, 23 | syl 17 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐶)𝑦) ∈ 𝐵) |
25 | | eqid 2610 |
. . . . . . 7
⊢
(0g‘𝑅) = (0g‘𝑅) |
26 | 15, 16, 21, 25 | pmatcoe1fsupp 20325 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑥(.r‘𝐶)𝑦) ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅))) |
27 | 13, 14, 24, 26 | syl3anc 1318 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅))) |
28 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑎 = 𝑖 → (𝑎(𝑥(.r‘𝐶)𝑦)𝑏) = (𝑖(𝑥(.r‘𝐶)𝑦)𝑏)) |
29 | 28 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 = 𝑖 → (coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏)) = (coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏))) |
30 | 29 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑎 = 𝑖 → ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛)) |
31 | 30 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝑖 → (((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅) ↔ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅))) |
32 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑏 = 𝑗 → (𝑖(𝑥(.r‘𝐶)𝑦)𝑏) = (𝑖(𝑥(.r‘𝐶)𝑦)𝑗)) |
33 | 32 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑏 = 𝑗 → (coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏)) = (coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))) |
34 | 33 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑏 = 𝑗 → ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) |
35 | 34 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 = 𝑗 → (((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅) ↔ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛) = (0g‘𝑅))) |
36 | 31, 35 | rspc2va 3294 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛) = (0g‘𝑅)) |
37 | 36 | expcom 450 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑎 ∈
𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅) → ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛) = (0g‘𝑅))) |
38 | 37 | adantl 481 |
. . . . . . . . . . . . . 14
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧
∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛) = (0g‘𝑅))) |
39 | 38 | 3impib 1254 |
. . . . . . . . . . . . 13
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧
∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛) = (0g‘𝑅)) |
40 | 39 | mpt2eq3dva 6617 |
. . . . . . . . . . . 12
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧
∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅))) |
41 | | pm2mpfo.a |
. . . . . . . . . . . . . 14
⊢ 𝐴 = (𝑁 Mat 𝑅) |
42 | 41, 25 | mat0op 20044 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) →
(0g‘𝐴) =
(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅))) |
43 | 42 | ad3antrrr 762 |
. . . . . . . . . . . 12
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧
∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → (0g‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅))) |
44 | 41 | matring 20068 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
45 | | pm2mpfo.q |
. . . . . . . . . . . . . . . 16
⊢ 𝑄 = (Poly1‘𝐴) |
46 | 45 | ply1sca 19444 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄)) |
47 | 44, 46 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 = (Scalar‘𝑄)) |
48 | 47 | ad3antrrr 762 |
. . . . . . . . . . . . 13
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧
∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → 𝐴 = (Scalar‘𝑄)) |
49 | 48 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧
∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → (0g‘𝐴) =
(0g‘(Scalar‘𝑄))) |
50 | 40, 43, 49 | 3eqtr2d 2650 |
. . . . . . . . . . 11
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧
∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) = (0g‘(Scalar‘𝑄))) |
51 | 50 | oveq1d 6564 |
. . . . . . . . . 10
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧
∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) ∗ (𝑛 ↑ 𝑋)) =
((0g‘(Scalar‘𝑄)) ∗ (𝑛 ↑ 𝑋))) |
52 | 45 | ply1lmod 19443 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod) |
53 | 44, 52 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod) |
54 | 53 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑄 ∈ LMod) |
55 | 54 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑄 ∈ LMod) |
56 | 44 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐴 ∈ Ring) |
57 | | pm2mpfo.x |
. . . . . . . . . . . . . 14
⊢ 𝑋 = (var1‘𝐴) |
58 | | eqid 2610 |
. . . . . . . . . . . . . 14
⊢
(mulGrp‘𝑄) =
(mulGrp‘𝑄) |
59 | | pm2mpfo.e |
. . . . . . . . . . . . . 14
⊢ ↑ =
(.g‘(mulGrp‘𝑄)) |
60 | | pm2mpfo.l |
. . . . . . . . . . . . . 14
⊢ 𝐿 = (Base‘𝑄) |
61 | 45, 57, 58, 59, 60 | ply1moncl 19462 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ Ring ∧ 𝑛 ∈ ℕ0)
→ (𝑛 ↑ 𝑋) ∈ 𝐿) |
62 | 56, 61 | sylan 487 |
. . . . . . . . . . . 12
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ 𝐿) |
63 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢
(Scalar‘𝑄) =
(Scalar‘𝑄) |
64 | | pm2mpfo.m |
. . . . . . . . . . . . 13
⊢ ∗ = (
·𝑠 ‘𝑄) |
65 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢
(0g‘(Scalar‘𝑄)) =
(0g‘(Scalar‘𝑄)) |
66 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢
(0g‘𝑄) = (0g‘𝑄) |
67 | 60, 63, 64, 65, 66 | lmod0vs 18719 |
. . . . . . . . . . . 12
⊢ ((𝑄 ∈ LMod ∧ (𝑛 ↑ 𝑋) ∈ 𝐿) →
((0g‘(Scalar‘𝑄)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)) |
68 | 55, 62, 67 | syl2anc 691 |
. . . . . . . . . . 11
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) →
((0g‘(Scalar‘𝑄)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)) |
69 | 68 | adantr 480 |
. . . . . . . . . 10
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧
∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) →
((0g‘(Scalar‘𝑄)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)) |
70 | 51, 69 | eqtrd 2644 |
. . . . . . . . 9
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧
∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)) |
71 | 70 | ex 449 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) →
(∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))) |
72 | 71 | imim2d 55 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑠 < 𝑛 → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → (𝑠 < 𝑛 → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)))) |
73 | 72 | ralimdva 2945 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)))) |
74 | 73 | reximdv 2999 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)))) |
75 | 27, 74 | mpd 15 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))) |
76 | 16, 21 | decpmatval 20389 |
. . . . . . . . . 10
⊢ (((𝑥(.r‘𝐶)𝑦) ∈ 𝐵 ∧ 𝑛 ∈ ℕ0) → ((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛))) |
77 | 24, 76 | sylan 487 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛))) |
78 | 77 | oveq1d 6564 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) ∗ (𝑛 ↑ 𝑋))) |
79 | 78 | eqeq1d 2612 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄) ↔ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))) |
80 | 79 | imbi2d 329 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑠 < 𝑛 → (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)) ↔ (𝑠 < 𝑛 → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)))) |
81 | 80 | ralbidva 2968 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)) ↔ ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)))) |
82 | 81 | rexbidv 3034 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)) ↔ ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)))) |
83 | 75, 82 | mpbird 246 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))) |
84 | | simpllr 795 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring) |
85 | | simplr 788 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) |
86 | | simpr 476 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈
ℕ0) |
87 | 15, 16, 21, 41 | decpmatmul 20396 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) = (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘)))))) |
88 | 84, 85, 86, 87 | syl3anc 1318 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) = (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘)))))) |
89 | 88 | eqcomd 2616 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝐴 Σg
(𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))) = ((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛)) |
90 | 89 | oveq1d 6564 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝐴 Σg
(𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))) ∗ (𝑛 ↑ 𝑋)) = (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋))) |
91 | 90 | eqeq1d 2612 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → (((𝐴 Σg
(𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄) ↔ (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))) |
92 | 91 | imbi2d 329 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑠 < 𝑛 → ((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)) ↔ (𝑠 < 𝑛 → (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)))) |
93 | 92 | ralbidva 2968 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)) ↔ ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)))) |
94 | 93 | rexbidv 3034 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)) ↔ ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)))) |
95 | 83, 94 | mpbird 246 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))) |
96 | 2, 4, 12, 95 | mptnn0fsuppd 12660 |
1
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑙 ∈ ℕ0 ↦ ((𝐴 Σg
(𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑙 − 𝑘))))) ∗ (𝑙 ↑ 𝑋))) finSupp (0g‘𝑄)) |