Proof of Theorem chfacfisf
Step | Hyp | Ref
| Expression |
1 | | chfacfisf.p |
. . . . . . . . 9
⊢ 𝑃 = (Poly1‘𝑅) |
2 | | chfacfisf.y |
. . . . . . . . 9
⊢ 𝑌 = (𝑁 Mat 𝑃) |
3 | 1, 2 | pmatring 21839 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring) |
4 | 3 | 3adant3 1131 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring) |
5 | | ringgrp 19786 |
. . . . . . 7
⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp) |
6 | 4, 5 | syl 17 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Grp) |
7 | 6 | adantr 481 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑌 ∈ Grp) |
8 | | eqid 2738 |
. . . . . . . 8
⊢
(Base‘𝑌) =
(Base‘𝑌) |
9 | | chfacfisf.0 |
. . . . . . . 8
⊢ 0 =
(0g‘𝑌) |
10 | 8, 9 | ring0cl 19806 |
. . . . . . 7
⊢ (𝑌 ∈ Ring → 0 ∈
(Base‘𝑌)) |
11 | 4, 10 | syl 17 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 0 ∈ (Base‘𝑌)) |
12 | 11 | adantr 481 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 0 ∈ (Base‘𝑌)) |
13 | 4 | adantr 481 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑌 ∈ Ring) |
14 | | chfacfisf.t |
. . . . . . . 8
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
15 | | chfacfisf.a |
. . . . . . . 8
⊢ 𝐴 = (𝑁 Mat 𝑅) |
16 | | chfacfisf.b |
. . . . . . . 8
⊢ 𝐵 = (Base‘𝐴) |
17 | 14, 15, 16, 1, 2 | mat2pmatbas 21873 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
18 | 17 | adantr 481 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
19 | | 3simpa 1147 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
20 | | elmapi 8635 |
. . . . . . . . . . 11
⊢ (𝑏 ∈ (𝐵 ↑m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵) |
21 | 20 | adantl 482 |
. . . . . . . . . 10
⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵) |
22 | | nnnn0 12238 |
. . . . . . . . . . . . 13
⊢ (𝑠 ∈ ℕ → 𝑠 ∈
ℕ0) |
23 | | nn0uz 12618 |
. . . . . . . . . . . . 13
⊢
ℕ0 = (ℤ≥‘0) |
24 | 22, 23 | eleqtrdi 2849 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ ℕ → 𝑠 ∈
(ℤ≥‘0)) |
25 | | eluzfz1 13261 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈
(ℤ≥‘0) → 0 ∈ (0...𝑠)) |
26 | 24, 25 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ ℕ → 0 ∈
(0...𝑠)) |
27 | 26 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 0 ∈ (0...𝑠)) |
28 | 21, 27 | ffvelrnd 6964 |
. . . . . . . . 9
⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑏‘0) ∈ 𝐵) |
29 | 19, 28 | anim12i 613 |
. . . . . . . 8
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑏‘0) ∈ 𝐵)) |
30 | | df-3an 1088 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑏‘0) ∈ 𝐵)) |
31 | 29, 30 | sylibr 233 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵)) |
32 | 14, 15, 16, 1, 2 | mat2pmatbas 21873 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌)) |
33 | 31, 32 | syl 17 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌)) |
34 | | chfacfisf.r |
. . . . . . 7
⊢ × =
(.r‘𝑌) |
35 | 8, 34 | ringcl 19798 |
. . . . . 6
⊢ ((𝑌 ∈ Ring ∧ (𝑇‘𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌)) |
36 | 13, 18, 33, 35 | syl3anc 1370 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌)) |
37 | | chfacfisf.s |
. . . . . 6
⊢ − =
(-g‘𝑌) |
38 | 8, 37 | grpsubcl 18653 |
. . . . 5
⊢ ((𝑌 ∈ Grp ∧ 0 ∈
(Base‘𝑌) ∧
((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌)) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) ∈ (Base‘𝑌)) |
39 | 7, 12, 36, 38 | syl3anc 1370 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) ∈ (Base‘𝑌)) |
40 | 39 | ad2antrr 723 |
. . 3
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) ∈ (Base‘𝑌)) |
41 | 22 | adantr 481 |
. . . . . . . . 9
⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑠 ∈ ℕ0) |
42 | 19, 41 | anim12i 613 |
. . . . . . . 8
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑠 ∈
ℕ0)) |
43 | | df-3an 1088 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0)
↔ ((𝑁 ∈ Fin ∧
𝑅 ∈ Ring) ∧ 𝑠 ∈
ℕ0)) |
44 | 42, 43 | sylibr 233 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈
ℕ0)) |
45 | | eluzfz2 13262 |
. . . . . . . . . 10
⊢ (𝑠 ∈
(ℤ≥‘0) → 𝑠 ∈ (0...𝑠)) |
46 | 24, 45 | syl 17 |
. . . . . . . . 9
⊢ (𝑠 ∈ ℕ → 𝑠 ∈ (0...𝑠)) |
47 | 46 | anim1ci 616 |
. . . . . . . 8
⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑏 ∈ (𝐵 ↑m (0...𝑠)) ∧ 𝑠 ∈ (0...𝑠))) |
48 | 47 | adantl 482 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑏 ∈ (𝐵 ↑m (0...𝑠)) ∧ 𝑠 ∈ (0...𝑠))) |
49 | 15, 16, 1, 2, 14 | m2pmfzmap 21894 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0)
∧ (𝑏 ∈ (𝐵 ↑m (0...𝑠)) ∧ 𝑠 ∈ (0...𝑠))) → (𝑇‘(𝑏‘𝑠)) ∈ (Base‘𝑌)) |
50 | 44, 48, 49 | syl2anc 584 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘(𝑏‘𝑠)) ∈ (Base‘𝑌)) |
51 | 50 | adantr 481 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑏‘𝑠)) ∈ (Base‘𝑌)) |
52 | 51 | ad2antrr 723 |
. . . 4
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) ∧ 𝑛 = (𝑠 + 1)) → (𝑇‘(𝑏‘𝑠)) ∈ (Base‘𝑌)) |
53 | 12 | ad4antr 729 |
. . . . 5
⊢
(((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ (𝑠 + 1) < 𝑛) → 0 ∈ (Base‘𝑌)) |
54 | | nn0re 12240 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ0
→ 𝑛 ∈
ℝ) |
55 | 54 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
→ 𝑛 ∈
ℝ) |
56 | | peano2nn 11983 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈
ℕ) |
57 | 56 | nnred 11986 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈
ℝ) |
58 | 57 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
→ (𝑠 + 1) ∈
ℝ) |
59 | 55, 58 | lenltd 11119 |
. . . . . . . . . . . . 13
⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
→ (𝑛 ≤ (𝑠 + 1) ↔ ¬ (𝑠 + 1) < 𝑛)) |
60 | | nesym 3000 |
. . . . . . . . . . . . . . 15
⊢ ((𝑠 + 1) ≠ 𝑛 ↔ ¬ 𝑛 = (𝑠 + 1)) |
61 | | ltlen 11074 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ) →
(𝑛 < (𝑠 + 1) ↔ (𝑛 ≤ (𝑠 + 1) ∧ (𝑠 + 1) ≠ 𝑛))) |
62 | 54, 57, 61 | syl2anr 597 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
→ (𝑛 < (𝑠 + 1) ↔ (𝑛 ≤ (𝑠 + 1) ∧ (𝑠 + 1) ≠ 𝑛))) |
63 | 62 | biimprd 247 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
→ ((𝑛 ≤ (𝑠 + 1) ∧ (𝑠 + 1) ≠ 𝑛) → 𝑛 < (𝑠 + 1))) |
64 | 63 | expcomd 417 |
. . . . . . . . . . . . . . 15
⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
→ ((𝑠 + 1) ≠ 𝑛 → (𝑛 ≤ (𝑠 + 1) → 𝑛 < (𝑠 + 1)))) |
65 | 60, 64 | syl5bir 242 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
→ (¬ 𝑛 = (𝑠 + 1) → (𝑛 ≤ (𝑠 + 1) → 𝑛 < (𝑠 + 1)))) |
66 | 65 | com23 86 |
. . . . . . . . . . . . 13
⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
→ (𝑛 ≤ (𝑠 + 1) → (¬ 𝑛 = (𝑠 + 1) → 𝑛 < (𝑠 + 1)))) |
67 | 59, 66 | sylbird 259 |
. . . . . . . . . . . 12
⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
→ (¬ (𝑠 + 1) <
𝑛 → (¬ 𝑛 = (𝑠 + 1) → 𝑛 < (𝑠 + 1)))) |
68 | 67 | impcomd 412 |
. . . . . . . . . . 11
⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
→ ((¬ 𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 < (𝑠 + 1))) |
69 | 68 | ex 413 |
. . . . . . . . . 10
⊢ (𝑠 ∈ ℕ → (𝑛 ∈ ℕ0
→ ((¬ 𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 < (𝑠 + 1)))) |
70 | 69 | ad2antrl 725 |
. . . . . . . . 9
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑛 ∈ ℕ0 → ((¬
𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 < (𝑠 + 1)))) |
71 | 70 | imp 407 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → ((¬
𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 < (𝑠 + 1))) |
72 | 71 | adantr 481 |
. . . . . . 7
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) → ((¬ 𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 < (𝑠 + 1))) |
73 | 3, 5 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Grp) |
74 | 73 | 3adant3 1131 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Grp) |
75 | 74 | ad4antr 729 |
. . . . . . . . 9
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → 𝑌 ∈ Grp) |
76 | 19 | ad4antr 729 |
. . . . . . . . . . 11
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
77 | 21 | ad4antlr 730 |
. . . . . . . . . . . 12
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → 𝑏:(0...𝑠)⟶𝐵) |
78 | | neqne 2951 |
. . . . . . . . . . . . . . . . 17
⊢ (¬
𝑛 = 0 → 𝑛 ≠ 0) |
79 | 78 | anim2i 617 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 ∈ ℕ0
∧ ¬ 𝑛 = 0) →
(𝑛 ∈
ℕ0 ∧ 𝑛
≠ 0)) |
80 | | elnnne0 12245 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0
∧ 𝑛 ≠
0)) |
81 | 79, 80 | sylibr 233 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ ℕ0
∧ ¬ 𝑛 = 0) →
𝑛 ∈
ℕ) |
82 | | nnm1nn0 12272 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈
ℕ0) |
83 | 81, 82 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ ℕ0
∧ ¬ 𝑛 = 0) →
(𝑛 − 1) ∈
ℕ0) |
84 | 83 | ad4ant23 750 |
. . . . . . . . . . . . 13
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑛 − 1) ∈
ℕ0) |
85 | 41 | ad4antlr 730 |
. . . . . . . . . . . . 13
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → 𝑠 ∈ ℕ0) |
86 | 62 | simprbda 499 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
∧ 𝑛 < (𝑠 + 1)) → 𝑛 ≤ (𝑠 + 1)) |
87 | 55 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
∧ 𝑛 < (𝑠 + 1)) → 𝑛 ∈ ℝ) |
88 | | 1red 10974 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
∧ 𝑛 < (𝑠 + 1)) → 1 ∈
ℝ) |
89 | | nnre 11978 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ ℕ → 𝑠 ∈
ℝ) |
90 | 89 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
∧ 𝑛 < (𝑠 + 1)) → 𝑠 ∈ ℝ) |
91 | 87, 88, 90 | lesubaddd 11570 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
∧ 𝑛 < (𝑠 + 1)) → ((𝑛 − 1) ≤ 𝑠 ↔ 𝑛 ≤ (𝑠 + 1))) |
92 | 86, 91 | mpbird 256 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
∧ 𝑛 < (𝑠 + 1)) → (𝑛 − 1) ≤ 𝑠) |
93 | 92 | exp31 420 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ ℕ → (𝑛 ∈ ℕ0
→ (𝑛 < (𝑠 + 1) → (𝑛 − 1) ≤ 𝑠))) |
94 | 93 | ad2antrl 725 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑛 ∈ ℕ0 → (𝑛 < (𝑠 + 1) → (𝑛 − 1) ≤ 𝑠))) |
95 | 94 | imp 407 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (𝑛 < (𝑠 + 1) → (𝑛 − 1) ≤ 𝑠)) |
96 | 95 | adantr 481 |
. . . . . . . . . . . . . 14
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) → (𝑛 < (𝑠 + 1) → (𝑛 − 1) ≤ 𝑠)) |
97 | 96 | imp 407 |
. . . . . . . . . . . . 13
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑛 − 1) ≤ 𝑠) |
98 | | elfz2nn0 13345 |
. . . . . . . . . . . . 13
⊢ ((𝑛 − 1) ∈ (0...𝑠) ↔ ((𝑛 − 1) ∈ ℕ0 ∧
𝑠 ∈
ℕ0 ∧ (𝑛 − 1) ≤ 𝑠)) |
99 | 84, 85, 97, 98 | syl3anbrc 1342 |
. . . . . . . . . . . 12
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑛 − 1) ∈ (0...𝑠)) |
100 | 77, 99 | ffvelrnd 6964 |
. . . . . . . . . . 11
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑏‘(𝑛 − 1)) ∈ 𝐵) |
101 | | df-3an 1088 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑛 − 1)) ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑏‘(𝑛 − 1)) ∈ 𝐵)) |
102 | 76, 100, 101 | sylanbrc 583 |
. . . . . . . . . 10
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑛 − 1)) ∈ 𝐵)) |
103 | 14, 15, 16, 1, 2 | mat2pmatbas 21873 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑛 − 1)) ∈ 𝐵) → (𝑇‘(𝑏‘(𝑛 − 1))) ∈ (Base‘𝑌)) |
104 | 102, 103 | syl 17 |
. . . . . . . . 9
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑇‘(𝑏‘(𝑛 − 1))) ∈ (Base‘𝑌)) |
105 | 13 | ad2antrr 723 |
. . . . . . . . . . 11
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑌 ∈ Ring) |
106 | 18 | ad2antrr 723 |
. . . . . . . . . . 11
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
107 | 44 | ad2antrr 723 |
. . . . . . . . . . . 12
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈
ℕ0)) |
108 | | simprr 770 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑏 ∈ (𝐵 ↑m (0...𝑠))) |
109 | 108 | ad2antrr 723 |
. . . . . . . . . . . 12
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑏 ∈ (𝐵 ↑m (0...𝑠))) |
110 | | simplr 766 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
∧ 𝑛 < (𝑠 + 1)) → 𝑛 ∈ ℕ0) |
111 | 22 | ad2antrr 723 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
∧ 𝑛 < (𝑠 + 1)) → 𝑠 ∈ ℕ0) |
112 | | nn0z 12341 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ0
→ 𝑛 ∈
ℤ) |
113 | | nnz 12340 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ ℕ → 𝑠 ∈
ℤ) |
114 | | zleltp1 12369 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∈ ℤ ∧ 𝑠 ∈ ℤ) → (𝑛 ≤ 𝑠 ↔ 𝑛 < (𝑠 + 1))) |
115 | 112, 113,
114 | syl2anr 597 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
→ (𝑛 ≤ 𝑠 ↔ 𝑛 < (𝑠 + 1))) |
116 | 115 | biimpar 478 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
∧ 𝑛 < (𝑠 + 1)) → 𝑛 ≤ 𝑠) |
117 | | elfz2nn0 13345 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ (0...𝑠) ↔ (𝑛 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0
∧ 𝑛 ≤ 𝑠)) |
118 | 110, 111,
116, 117 | syl3anbrc 1342 |
. . . . . . . . . . . . . . 15
⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
∧ 𝑛 < (𝑠 + 1)) → 𝑛 ∈ (0...𝑠)) |
119 | 118 | exp31 420 |
. . . . . . . . . . . . . 14
⊢ (𝑠 ∈ ℕ → (𝑛 ∈ ℕ0
→ (𝑛 < (𝑠 + 1) → 𝑛 ∈ (0...𝑠)))) |
120 | 119 | ad2antrl 725 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑛 ∈ ℕ0 → (𝑛 < (𝑠 + 1) → 𝑛 ∈ (0...𝑠)))) |
121 | 120 | imp31 418 |
. . . . . . . . . . . 12
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛 ∈ (0...𝑠)) |
122 | 15, 16, 1, 2, 14 | m2pmfzmap 21894 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0)
∧ (𝑏 ∈ (𝐵 ↑m (0...𝑠)) ∧ 𝑛 ∈ (0...𝑠))) → (𝑇‘(𝑏‘𝑛)) ∈ (Base‘𝑌)) |
123 | 107, 109,
121, 122 | syl12anc 834 |
. . . . . . . . . . 11
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → (𝑇‘(𝑏‘𝑛)) ∈ (Base‘𝑌)) |
124 | 8, 34 | ringcl 19798 |
. . . . . . . . . . 11
⊢ ((𝑌 ∈ Ring ∧ (𝑇‘𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘𝑛)) ∈ (Base‘𝑌)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))) ∈ (Base‘𝑌)) |
125 | 105, 106,
123, 124 | syl3anc 1370 |
. . . . . . . . . 10
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))) ∈ (Base‘𝑌)) |
126 | 125 | adantlr 712 |
. . . . . . . . 9
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))) ∈ (Base‘𝑌)) |
127 | 8, 37 | grpsubcl 18653 |
. . . . . . . . 9
⊢ ((𝑌 ∈ Grp ∧ (𝑇‘(𝑏‘(𝑛 − 1))) ∈ (Base‘𝑌) ∧ ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))) ∈ (Base‘𝑌)) → ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) ∈ (Base‘𝑌)) |
128 | 75, 104, 126, 127 | syl3anc 1370 |
. . . . . . . 8
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) ∈ (Base‘𝑌)) |
129 | 128 | ex 413 |
. . . . . . 7
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) → (𝑛 < (𝑠 + 1) → ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) ∈ (Base‘𝑌))) |
130 | 72, 129 | syld 47 |
. . . . . 6
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) → ((¬ 𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) ∈ (Base‘𝑌))) |
131 | 130 | impl 456 |
. . . . 5
⊢
(((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) ∈ (Base‘𝑌)) |
132 | 53, 131 | ifclda 4496 |
. . . 4
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) → if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))) ∈ (Base‘𝑌)) |
133 | 52, 132 | ifclda 4496 |
. . 3
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬
𝑛 = 0) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) ∈ (Base‘𝑌)) |
134 | 40, 133 | ifclda 4496 |
. 2
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 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))))))) ∈ (Base‘𝑌)) |
135 | | chfacfisf.g |
. 2
⊢ 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)))))))) |
136 | 134, 135 | fmptd 6990 |
1
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌)) |