Proof of Theorem chfacfpmmul0
Step | Hyp | Ref
| Expression |
1 | | eluz2 12570 |
. . . . . 6
⊢ (𝐾 ∈
(ℤ≥‘(𝑠 + 2)) ↔ ((𝑠 + 2) ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (𝑠 + 2) ≤ 𝐾)) |
2 | | simpll 763 |
. . . . . . . . . . . 12
⊢ (((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) ∧ (𝑠 + 2) ≤ 𝐾) → 𝐾 ∈ ℤ) |
3 | | nngt0 11987 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈ ℕ → 0 <
𝑠) |
4 | | nnre 11963 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ ℕ → 𝑠 ∈
ℝ) |
5 | 4 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → 𝑠 ∈
ℝ) |
6 | | 2rp 12717 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 2 ∈
ℝ+ |
7 | 6 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → 2 ∈
ℝ+) |
8 | 5, 7 | ltaddrpd 12787 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → 𝑠 < (𝑠 + 2)) |
9 | | 0red 10962 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → 0 ∈
ℝ) |
10 | | 2re 12030 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 2 ∈
ℝ |
11 | 10 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → 2 ∈
ℝ) |
12 | 5, 11 | readdcld 10988 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → (𝑠 + 2) ∈
ℝ) |
13 | | lttr 11035 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((0
∈ ℝ ∧ 𝑠
∈ ℝ ∧ (𝑠 +
2) ∈ ℝ) → ((0 < 𝑠 ∧ 𝑠 < (𝑠 + 2)) → 0 < (𝑠 + 2))) |
14 | 9, 5, 12, 13 | syl3anc 1369 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → ((0 <
𝑠 ∧ 𝑠 < (𝑠 + 2)) → 0 < (𝑠 + 2))) |
15 | 8, 14 | mpan2d 690 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → (0 <
𝑠 → 0 < (𝑠 + 2))) |
16 | 15 | ex 412 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐾 ∈ ℤ → (𝑠 ∈ ℕ → (0 <
𝑠 → 0 < (𝑠 + 2)))) |
17 | 16 | com13 88 |
. . . . . . . . . . . . . . . 16
⊢ (0 <
𝑠 → (𝑠 ∈ ℕ → (𝐾 ∈ ℤ → 0 <
(𝑠 + 2)))) |
18 | 3, 17 | mpcom 38 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 ∈ ℕ → (𝐾 ∈ ℤ → 0 <
(𝑠 + 2))) |
19 | 18 | impcom 407 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → 0 <
(𝑠 + 2)) |
20 | | zre 12306 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ ℤ → 𝐾 ∈
ℝ) |
21 | 20 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → 𝐾 ∈
ℝ) |
22 | | ltleletr 11051 |
. . . . . . . . . . . . . . 15
⊢ ((0
∈ ℝ ∧ (𝑠 +
2) ∈ ℝ ∧ 𝐾
∈ ℝ) → ((0 < (𝑠 + 2) ∧ (𝑠 + 2) ≤ 𝐾) → 0 ≤ 𝐾)) |
23 | 9, 12, 21, 22 | syl3anc 1369 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → ((0 <
(𝑠 + 2) ∧ (𝑠 + 2) ≤ 𝐾) → 0 ≤ 𝐾)) |
24 | 19, 23 | mpand 691 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → ((𝑠 + 2) ≤ 𝐾 → 0 ≤ 𝐾)) |
25 | 24 | imp 406 |
. . . . . . . . . . . 12
⊢ (((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) ∧ (𝑠 + 2) ≤ 𝐾) → 0 ≤ 𝐾) |
26 | | elnn0z 12315 |
. . . . . . . . . . . 12
⊢ (𝐾 ∈ ℕ0
↔ (𝐾 ∈ ℤ
∧ 0 ≤ 𝐾)) |
27 | 2, 25, 26 | sylanbrc 582 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) ∧ (𝑠 + 2) ≤ 𝐾) → 𝐾 ∈
ℕ0) |
28 | | nncn 11964 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ ℕ → 𝑠 ∈
ℂ) |
29 | | add1p1 12207 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ ℂ → ((𝑠 + 1) + 1) = (𝑠 + 2)) |
30 | 28, 29 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈ ℕ → ((𝑠 + 1) + 1) = (𝑠 + 2)) |
31 | 30 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → ((𝑠 + 1) + 1) = (𝑠 + 2)) |
32 | 31 | eqcomd 2745 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → (𝑠 + 2) = ((𝑠 + 1) + 1)) |
33 | 32 | breq1d 5088 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → ((𝑠 + 2) ≤ 𝐾 ↔ ((𝑠 + 1) + 1) ≤ 𝐾)) |
34 | | nnz 12325 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ ℕ → 𝑠 ∈
ℤ) |
35 | 34 | peano2zd 12411 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈
ℤ) |
36 | 35 | anim2i 616 |
. . . . . . . . . . . . . . 15
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → (𝐾 ∈ ℤ ∧ (𝑠 + 1) ∈
ℤ)) |
37 | 36 | ancomd 461 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → ((𝑠 + 1) ∈ ℤ ∧ 𝐾 ∈
ℤ)) |
38 | | zltp1le 12353 |
. . . . . . . . . . . . . . 15
⊢ (((𝑠 + 1) ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑠 + 1) < 𝐾 ↔ ((𝑠 + 1) + 1) ≤ 𝐾)) |
39 | 38 | bicomd 222 |
. . . . . . . . . . . . . 14
⊢ (((𝑠 + 1) ∈ ℤ ∧ 𝐾 ∈ ℤ) → (((𝑠 + 1) + 1) ≤ 𝐾 ↔ (𝑠 + 1) < 𝐾)) |
40 | 37, 39 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → (((𝑠 + 1) + 1) ≤ 𝐾 ↔ (𝑠 + 1) < 𝐾)) |
41 | 33, 40 | bitrd 278 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → ((𝑠 + 2) ≤ 𝐾 ↔ (𝑠 + 1) < 𝐾)) |
42 | 41 | biimpa 476 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) ∧ (𝑠 + 2) ≤ 𝐾) → (𝑠 + 1) < 𝐾) |
43 | 27, 42 | jca 511 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) ∧ (𝑠 + 2) ≤ 𝐾) → (𝐾 ∈ ℕ0 ∧ (𝑠 + 1) < 𝐾)) |
44 | 43 | ex 412 |
. . . . . . . . 9
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → ((𝑠 + 2) ≤ 𝐾 → (𝐾 ∈ ℕ0 ∧ (𝑠 + 1) < 𝐾))) |
45 | 44 | impancom 451 |
. . . . . . . 8
⊢ ((𝐾 ∈ ℤ ∧ (𝑠 + 2) ≤ 𝐾) → (𝑠 ∈ ℕ → (𝐾 ∈ ℕ0 ∧ (𝑠 + 1) < 𝐾))) |
46 | 45 | 3adant1 1128 |
. . . . . . 7
⊢ (((𝑠 + 2) ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (𝑠 + 2) ≤ 𝐾) → (𝑠 ∈ ℕ → (𝐾 ∈ ℕ0 ∧ (𝑠 + 1) < 𝐾))) |
47 | 46 | com12 32 |
. . . . . 6
⊢ (𝑠 ∈ ℕ → (((𝑠 + 2) ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (𝑠 + 2) ≤ 𝐾) → (𝐾 ∈ ℕ0 ∧ (𝑠 + 1) < 𝐾))) |
48 | 1, 47 | syl5bi 241 |
. . . . 5
⊢ (𝑠 ∈ ℕ → (𝐾 ∈
(ℤ≥‘(𝑠 + 2)) → (𝐾 ∈ ℕ0 ∧ (𝑠 + 1) < 𝐾))) |
49 | 48 | adantr 480 |
. . . 4
⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝐾 ∈ (ℤ≥‘(𝑠 + 2)) → (𝐾 ∈ ℕ0 ∧ (𝑠 + 1) < 𝐾))) |
50 | 49 | adantl 481 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝐾 ∈ (ℤ≥‘(𝑠 + 2)) → (𝐾 ∈ ℕ0 ∧ (𝑠 + 1) < 𝐾))) |
51 | | cayhamlem1.g |
. . . . . . 7
⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) |
52 | | 0red 10962 |
. . . . . . . . . . . . . 14
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → 0 ∈ ℝ) |
53 | | peano2re 11131 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ ℝ → (𝑠 + 1) ∈
ℝ) |
54 | 4, 53 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈
ℝ) |
55 | 54 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑠 + 1) ∈ ℝ) |
56 | 55 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑠 + 1) ∈ ℝ) |
57 | 56 | ad2antrr 722 |
. . . . . . . . . . . . . 14
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → (𝑠 + 1) ∈ ℝ) |
58 | | nn0re 12225 |
. . . . . . . . . . . . . . 15
⊢ (𝐾 ∈ ℕ0
→ 𝐾 ∈
ℝ) |
59 | 58 | ad2antlr 723 |
. . . . . . . . . . . . . 14
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → 𝐾 ∈ ℝ) |
60 | | nnnn0 12223 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ ℕ → 𝑠 ∈
ℕ0) |
61 | 60 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑠 ∈ ℕ0) |
62 | 61 | ad2antlr 723 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) → 𝑠 ∈
ℕ0) |
63 | | nn0p1gt0 12245 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈ ℕ0
→ 0 < (𝑠 +
1)) |
64 | 62, 63 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) → 0 <
(𝑠 + 1)) |
65 | 64 | adantr 480 |
. . . . . . . . . . . . . 14
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → 0 < (𝑠 + 1)) |
66 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → (𝑠 + 1) < 𝐾) |
67 | 52, 57, 59, 65, 66 | lttrd 11119 |
. . . . . . . . . . . . 13
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → 0 < 𝐾) |
68 | 67 | gt0ne0d 11522 |
. . . . . . . . . . . 12
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → 𝐾 ≠ 0) |
69 | 68 | neneqd 2949 |
. . . . . . . . . . 11
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → ¬ 𝐾 = 0) |
70 | 69 | adantr 480 |
. . . . . . . . . 10
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) ∧ 𝑛 = 𝐾) → ¬ 𝐾 = 0) |
71 | | eqeq1 2743 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝐾 → (𝑛 = 0 ↔ 𝐾 = 0)) |
72 | 71 | notbid 317 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝐾 → (¬ 𝑛 = 0 ↔ ¬ 𝐾 = 0)) |
73 | 72 | adantl 481 |
. . . . . . . . . 10
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) ∧ 𝑛 = 𝐾) → (¬ 𝑛 = 0 ↔ ¬ 𝐾 = 0)) |
74 | 70, 73 | mpbird 256 |
. . . . . . . . 9
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) ∧ 𝑛 = 𝐾) → ¬ 𝑛 = 0) |
75 | 74 | iffalsed 4475 |
. . . . . . . 8
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) ∧ 𝑛 = 𝐾) → if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) |
76 | 55 | ad2antlr 723 |
. . . . . . . . . . . . 13
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) → (𝑠 + 1) ∈
ℝ) |
77 | | ltne 11055 |
. . . . . . . . . . . . 13
⊢ (((𝑠 + 1) ∈ ℝ ∧
(𝑠 + 1) < 𝐾) → 𝐾 ≠ (𝑠 + 1)) |
78 | 76, 77 | sylan 579 |
. . . . . . . . . . . 12
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → 𝐾 ≠ (𝑠 + 1)) |
79 | 78 | neneqd 2949 |
. . . . . . . . . . 11
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → ¬ 𝐾 = (𝑠 + 1)) |
80 | 79 | adantr 480 |
. . . . . . . . . 10
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) ∧ 𝑛 = 𝐾) → ¬ 𝐾 = (𝑠 + 1)) |
81 | | eqeq1 2743 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝐾 → (𝑛 = (𝑠 + 1) ↔ 𝐾 = (𝑠 + 1))) |
82 | 81 | notbid 317 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝐾 → (¬ 𝑛 = (𝑠 + 1) ↔ ¬ 𝐾 = (𝑠 + 1))) |
83 | 82 | adantl 481 |
. . . . . . . . . 10
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) ∧ 𝑛 = 𝐾) → (¬ 𝑛 = (𝑠 + 1) ↔ ¬ 𝐾 = (𝑠 + 1))) |
84 | 80, 83 | mpbird 256 |
. . . . . . . . 9
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) ∧ 𝑛 = 𝐾) → ¬ 𝑛 = (𝑠 + 1)) |
85 | 84 | iffalsed 4475 |
. . . . . . . 8
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) ∧ 𝑛 = 𝐾) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) = if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) |
86 | | simplr 765 |
. . . . . . . . . 10
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) ∧ 𝑛 = 𝐾) → (𝑠 + 1) < 𝐾) |
87 | | breq2 5082 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝐾 → ((𝑠 + 1) < 𝑛 ↔ (𝑠 + 1) < 𝐾)) |
88 | 87 | adantl 481 |
. . . . . . . . . 10
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) ∧ 𝑛 = 𝐾) → ((𝑠 + 1) < 𝑛 ↔ (𝑠 + 1) < 𝐾)) |
89 | 86, 88 | mpbird 256 |
. . . . . . . . 9
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) ∧ 𝑛 = 𝐾) → (𝑠 + 1) < 𝑛) |
90 | 89 | iftrued 4472 |
. . . . . . . 8
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) ∧ 𝑛 = 𝐾) → if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))) = 0 ) |
91 | 75, 85, 90 | 3eqtrd 2783 |
. . . . . . 7
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) ∧ 𝑛 = 𝐾) → if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = 0 ) |
92 | | simplr 765 |
. . . . . . 7
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → 𝐾 ∈
ℕ0) |
93 | | cayhamlem1.0 |
. . . . . . . . 9
⊢ 0 =
(0g‘𝑌) |
94 | 93 | fvexi 6782 |
. . . . . . . 8
⊢ 0 ∈
V |
95 | 94 | a1i 11 |
. . . . . . 7
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → 0 ∈ V) |
96 | 51, 91, 92, 95 | fvmptd2 6877 |
. . . . . 6
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → (𝐺‘𝐾) = 0 ) |
97 | 96 | oveq2d 7284 |
. . . . 5
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → ((𝐾 ↑ (𝑇‘𝑀)) × (𝐺‘𝐾)) = ((𝐾 ↑ (𝑇‘𝑀)) × 0 )) |
98 | | crngring 19776 |
. . . . . . . . . 10
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) |
99 | | cayhamlem1.p |
. . . . . . . . . . 11
⊢ 𝑃 = (Poly1‘𝑅) |
100 | | cayhamlem1.y |
. . . . . . . . . . 11
⊢ 𝑌 = (𝑁 Mat 𝑃) |
101 | 99, 100 | pmatring 21822 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring) |
102 | 98, 101 | sylan2 592 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring) |
103 | 102 | 3adant3 1130 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring) |
104 | 103 | adantr 480 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑌 ∈ Ring) |
105 | 104 | ad2antrr 722 |
. . . . . 6
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → 𝑌 ∈ Ring) |
106 | | eqid 2739 |
. . . . . . . . . . 11
⊢
(mulGrp‘𝑌) =
(mulGrp‘𝑌) |
107 | 106 | ringmgp 19770 |
. . . . . . . . . 10
⊢ (𝑌 ∈ Ring →
(mulGrp‘𝑌) ∈
Mnd) |
108 | 103, 107 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑌) ∈ Mnd) |
109 | 108 | ad2antrr 722 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) →
(mulGrp‘𝑌) ∈
Mnd) |
110 | | simpr 484 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈
ℕ0) |
111 | | cayhamlem1.t |
. . . . . . . . . . 11
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
112 | | cayhamlem1.a |
. . . . . . . . . . 11
⊢ 𝐴 = (𝑁 Mat 𝑅) |
113 | | cayhamlem1.b |
. . . . . . . . . . 11
⊢ 𝐵 = (Base‘𝐴) |
114 | 111, 112,
113, 99, 100 | mat2pmatbas 21856 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
115 | 98, 114 | syl3an2 1162 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
116 | 115 | ad2antrr 722 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
117 | | eqid 2739 |
. . . . . . . . . 10
⊢
(Base‘𝑌) =
(Base‘𝑌) |
118 | 106, 117 | mgpbas 19707 |
. . . . . . . . 9
⊢
(Base‘𝑌) =
(Base‘(mulGrp‘𝑌)) |
119 | | cayhamlem1.e |
. . . . . . . . 9
⊢ ↑ =
(.g‘(mulGrp‘𝑌)) |
120 | 118, 119 | mulgnn0cl 18701 |
. . . . . . . 8
⊢
(((mulGrp‘𝑌)
∈ Mnd ∧ 𝐾 ∈
ℕ0 ∧ (𝑇‘𝑀) ∈ (Base‘𝑌)) → (𝐾 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌)) |
121 | 109, 110,
116, 120 | syl3anc 1369 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) → (𝐾 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌)) |
122 | 121 | adantr 480 |
. . . . . 6
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → (𝐾 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌)) |
123 | | cayhamlem1.r |
. . . . . . 7
⊢ × =
(.r‘𝑌) |
124 | 117, 123,
93 | ringrz 19808 |
. . . . . 6
⊢ ((𝑌 ∈ Ring ∧ (𝐾 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌)) → ((𝐾 ↑ (𝑇‘𝑀)) × 0 ) = 0 ) |
125 | 105, 122,
124 | syl2anc 583 |
. . . . 5
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → ((𝐾 ↑ (𝑇‘𝑀)) × 0 ) = 0 ) |
126 | 97, 125 | eqtrd 2779 |
. . . 4
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → ((𝐾 ↑ (𝑇‘𝑀)) × (𝐺‘𝐾)) = 0 ) |
127 | 126 | expl 457 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝐾 ∈ ℕ0 ∧ (𝑠 + 1) < 𝐾) → ((𝐾 ↑ (𝑇‘𝑀)) × (𝐺‘𝐾)) = 0 )) |
128 | 50, 127 | syld 47 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝐾 ∈ (ℤ≥‘(𝑠 + 2)) → ((𝐾 ↑ (𝑇‘𝑀)) × (𝐺‘𝐾)) = 0 )) |
129 | 128 | 3impia 1115 |
1
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ (ℤ≥‘(𝑠 + 2))) → ((𝐾 ↑ (𝑇‘𝑀)) × (𝐺‘𝐾)) = 0 ) |