Proof of Theorem chfacfpmmul0
| Step | Hyp | Ref
| Expression |
| 1 | | eluz2 12884 |
. . . . . 6
⊢ (𝐾 ∈
(ℤ≥‘(𝑠 + 2)) ↔ ((𝑠 + 2) ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (𝑠 + 2) ≤ 𝐾)) |
| 2 | | simpll 767 |
. . . . . . . . . . . 12
⊢ (((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) ∧ (𝑠 + 2) ≤ 𝐾) → 𝐾 ∈ ℤ) |
| 3 | | nngt0 12297 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈ ℕ → 0 <
𝑠) |
| 4 | | nnre 12273 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ ℕ → 𝑠 ∈
ℝ) |
| 5 | 4 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → 𝑠 ∈
ℝ) |
| 6 | | 2rp 13039 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 2 ∈
ℝ+ |
| 7 | 6 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → 2 ∈
ℝ+) |
| 8 | 5, 7 | ltaddrpd 13110 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → 𝑠 < (𝑠 + 2)) |
| 9 | | 0red 11264 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → 0 ∈
ℝ) |
| 10 | | 2re 12340 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 2 ∈
ℝ |
| 11 | 10 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → 2 ∈
ℝ) |
| 12 | 5, 11 | readdcld 11290 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → (𝑠 + 2) ∈
ℝ) |
| 13 | | lttr 11337 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((0
∈ ℝ ∧ 𝑠
∈ ℝ ∧ (𝑠 +
2) ∈ ℝ) → ((0 < 𝑠 ∧ 𝑠 < (𝑠 + 2)) → 0 < (𝑠 + 2))) |
| 14 | 9, 5, 12, 13 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → ((0 <
𝑠 ∧ 𝑠 < (𝑠 + 2)) → 0 < (𝑠 + 2))) |
| 15 | 8, 14 | mpan2d 694 |
. . . . . . . . . . . . . . . . . 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 12617 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ ℤ → 𝐾 ∈
ℝ) |
| 21 | 20 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → 𝐾 ∈
ℝ) |
| 22 | | ltleletr 11354 |
. . . . . . . . . . . . . . 15
⊢ ((0
∈ ℝ ∧ (𝑠 +
2) ∈ ℝ ∧ 𝐾
∈ ℝ) → ((0 < (𝑠 + 2) ∧ (𝑠 + 2) ≤ 𝐾) → 0 ≤ 𝐾)) |
| 23 | 9, 12, 21, 22 | syl3anc 1373 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → ((0 <
(𝑠 + 2) ∧ (𝑠 + 2) ≤ 𝐾) → 0 ≤ 𝐾)) |
| 24 | 19, 23 | mpand 695 |
. . . . . . . . . . . . 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 12626 |
. . . . . . . . . . . 12
⊢ (𝐾 ∈ ℕ0
↔ (𝐾 ∈ ℤ
∧ 0 ≤ 𝐾)) |
| 27 | 2, 25, 26 | sylanbrc 583 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) ∧ (𝑠 + 2) ≤ 𝐾) → 𝐾 ∈
ℕ0) |
| 28 | | nncn 12274 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ ℕ → 𝑠 ∈
ℂ) |
| 29 | | add1p1 12517 |
. . . . . . . . . . . . . . . . 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 2743 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → (𝑠 + 2) = ((𝑠 + 1) + 1)) |
| 33 | 32 | breq1d 5153 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ ℤ ∧ 𝑠 ∈ ℕ) → ((𝑠 + 2) ≤ 𝐾 ↔ ((𝑠 + 1) + 1) ≤ 𝐾)) |
| 34 | | nnz 12634 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ ℕ → 𝑠 ∈
ℤ) |
| 35 | 34 | peano2zd 12725 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈
ℤ) |
| 36 | 35 | anim2i 617 |
. . . . . . . . . . . . . . 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 12667 |
. . . . . . . . . . . . . . 15
⊢ (((𝑠 + 1) ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑠 + 1) < 𝐾 ↔ ((𝑠 + 1) + 1) ≤ 𝐾)) |
| 39 | 38 | bicomd 223 |
. . . . . . . . . . . . . 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 279 |
. . . . . . . . . . . 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 1131 |
. . . . . . 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 | biimtrid 242 |
. . . . 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 11264 |
. . . . . . . . . . . . . 14
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → 0 ∈ ℝ) |
| 53 | | peano2re 11434 |
. . . . . . . . . . . . . . . . . 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 726 |
. . . . . . . . . . . . . 14
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → (𝑠 + 1) ∈ ℝ) |
| 58 | | nn0re 12535 |
. . . . . . . . . . . . . . 15
⊢ (𝐾 ∈ ℕ0
→ 𝐾 ∈
ℝ) |
| 59 | 58 | ad2antlr 727 |
. . . . . . . . . . . . . 14
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → 𝐾 ∈ ℝ) |
| 60 | | nnnn0 12533 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ ℕ → 𝑠 ∈
ℕ0) |
| 61 | 60 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑠 ∈ ℕ0) |
| 62 | 61 | ad2antlr 727 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) → 𝑠 ∈
ℕ0) |
| 63 | | nn0p1gt0 12555 |
. . . . . . . . . . . . . . . 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 11422 |
. . . . . . . . . . . . 13
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → 0 < 𝐾) |
| 68 | 67 | gt0ne0d 11827 |
. . . . . . . . . . . 12
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → 𝐾 ≠ 0) |
| 69 | 68 | neneqd 2945 |
. . . . . . . . . . 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 2741 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝐾 → (𝑛 = 0 ↔ 𝐾 = 0)) |
| 72 | 71 | notbid 318 |
. . . . . . . . . . 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 257 |
. . . . . . . . 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 4536 |
. . . . . . . 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 727 |
. . . . . . . . . . . . 13
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) → (𝑠 + 1) ∈
ℝ) |
| 77 | | ltne 11358 |
. . . . . . . . . . . . 13
⊢ (((𝑠 + 1) ∈ ℝ ∧
(𝑠 + 1) < 𝐾) → 𝐾 ≠ (𝑠 + 1)) |
| 78 | 76, 77 | sylan 580 |
. . . . . . . . . . . 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 2945 |
. . . . . . . . . . 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 2741 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝐾 → (𝑛 = (𝑠 + 1) ↔ 𝐾 = (𝑠 + 1))) |
| 82 | 81 | notbid 318 |
. . . . . . . . . . 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 257 |
. . . . . . . . 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 4536 |
. . . . . . . 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 769 |
. . . . . . . . . 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 5147 |
. . . . . . . . . . 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 257 |
. . . . . . . . 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 4533 |
. . . . . . . 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 2781 |
. . . . . . 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 769 |
. . . . . . 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 6920 |
. . . . . . . 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 7024 |
. . . . . 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 7447 |
. . . . 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 20242 |
. . . . . . . . . 10
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) |
| 99 | | cayhamlem1.p |
. . . . . . . . . . 11
⊢ 𝑃 = (Poly1‘𝑅) |
| 100 | | cayhamlem1.y |
. . . . . . . . . . 11
⊢ 𝑌 = (𝑁 Mat 𝑃) |
| 101 | 99, 100 | pmatring 22698 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring) |
| 102 | 98, 101 | sylan2 593 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring) |
| 103 | 102 | 3adant3 1133 |
. . . . . . . 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 726 |
. . . . . 6
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → 𝑌 ∈ Ring) |
| 106 | | eqid 2737 |
. . . . . . . . 9
⊢
(mulGrp‘𝑌) =
(mulGrp‘𝑌) |
| 107 | | eqid 2737 |
. . . . . . . . 9
⊢
(Base‘𝑌) =
(Base‘𝑌) |
| 108 | 106, 107 | mgpbas 20142 |
. . . . . . . 8
⊢
(Base‘𝑌) =
(Base‘(mulGrp‘𝑌)) |
| 109 | | cayhamlem1.e |
. . . . . . . 8
⊢ ↑ =
(.g‘(mulGrp‘𝑌)) |
| 110 | 106 | ringmgp 20236 |
. . . . . . . . . 10
⊢ (𝑌 ∈ Ring →
(mulGrp‘𝑌) ∈
Mnd) |
| 111 | 103, 110 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑌) ∈ Mnd) |
| 112 | 111 | ad2antrr 726 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) →
(mulGrp‘𝑌) ∈
Mnd) |
| 113 | | simpr 484 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈
ℕ0) |
| 114 | | cayhamlem1.t |
. . . . . . . . . . 11
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
| 115 | | cayhamlem1.a |
. . . . . . . . . . 11
⊢ 𝐴 = (𝑁 Mat 𝑅) |
| 116 | | cayhamlem1.b |
. . . . . . . . . . 11
⊢ 𝐵 = (Base‘𝐴) |
| 117 | 114, 115,
116, 99, 100 | mat2pmatbas 22732 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
| 118 | 98, 117 | syl3an2 1165 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
| 119 | 118 | ad2antrr 726 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
| 120 | 108, 109,
112, 113, 119 | mulgnn0cld 19113 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) → (𝐾 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌)) |
| 121 | 120 | 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)) |
| 122 | | cayhamlem1.r |
. . . . . . 7
⊢ × =
(.r‘𝑌) |
| 123 | 107, 122,
93 | ringrz 20291 |
. . . . . 6
⊢ ((𝑌 ∈ Ring ∧ (𝐾 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌)) → ((𝐾 ↑ (𝑇‘𝑀)) × 0 ) = 0 ) |
| 124 | 105, 121,
123 | syl2anc 584 |
. . . . 5
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝐾 ∈ ℕ0) ∧ (𝑠 + 1) < 𝐾) → ((𝐾 ↑ (𝑇‘𝑀)) × 0 ) = 0 ) |
| 125 | 97, 124 | eqtrd 2777 |
. . . 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 ) |
| 126 | 125 | 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 )) |
| 127 | 50, 126 | 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 )) |
| 128 | 127 | 3impia 1118 |
1
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ (ℤ≥‘(𝑠 + 2))) → ((𝐾 ↑ (𝑇‘𝑀)) × (𝐺‘𝐾)) = 0 ) |