Step |
Hyp |
Ref |
Expression |
1 |
|
chfacfisf.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
2 |
|
chfacfisf.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
3 |
|
chfacfisf.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
4 |
|
chfacfisf.y |
⊢ 𝑌 = ( 𝑁 Mat 𝑃 ) |
5 |
|
chfacfisf.r |
⊢ × = ( .r ‘ 𝑌 ) |
6 |
|
chfacfisf.s |
⊢ − = ( -g ‘ 𝑌 ) |
7 |
|
chfacfisf.0 |
⊢ 0 = ( 0g ‘ 𝑌 ) |
8 |
|
chfacfisf.t |
⊢ 𝑇 = ( 𝑁 matToPolyMat 𝑅 ) |
9 |
|
chfacfisf.g |
⊢ 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 ) ) ) ) ) ) ) ) |
10 |
3 4
|
pmatring |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑌 ∈ Ring ) |
11 |
10
|
3adant3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → 𝑌 ∈ Ring ) |
12 |
|
ringgrp |
⊢ ( 𝑌 ∈ Ring → 𝑌 ∈ Grp ) |
13 |
11 12
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → 𝑌 ∈ Grp ) |
14 |
13
|
adantr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝑌 ∈ Grp ) |
15 |
|
eqid |
⊢ ( Base ‘ 𝑌 ) = ( Base ‘ 𝑌 ) |
16 |
15 7
|
ring0cl |
⊢ ( 𝑌 ∈ Ring → 0 ∈ ( Base ‘ 𝑌 ) ) |
17 |
11 16
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → 0 ∈ ( Base ‘ 𝑌 ) ) |
18 |
17
|
adantr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 0 ∈ ( Base ‘ 𝑌 ) ) |
19 |
11
|
adantr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝑌 ∈ Ring ) |
20 |
8 1 2 3 4
|
mat2pmatbas |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ( 𝑇 ‘ 𝑀 ) ∈ ( Base ‘ 𝑌 ) ) |
21 |
20
|
adantr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑇 ‘ 𝑀 ) ∈ ( Base ‘ 𝑌 ) ) |
22 |
|
3simpa |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ) |
23 |
|
elmapi |
⊢ ( 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) → 𝑏 : ( 0 ... 𝑠 ) ⟶ 𝐵 ) |
24 |
23
|
adantl |
⊢ ( ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → 𝑏 : ( 0 ... 𝑠 ) ⟶ 𝐵 ) |
25 |
|
nnnn0 |
⊢ ( 𝑠 ∈ ℕ → 𝑠 ∈ ℕ0 ) |
26 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
27 |
25 26
|
eleqtrdi |
⊢ ( 𝑠 ∈ ℕ → 𝑠 ∈ ( ℤ≥ ‘ 0 ) ) |
28 |
|
eluzfz1 |
⊢ ( 𝑠 ∈ ( ℤ≥ ‘ 0 ) → 0 ∈ ( 0 ... 𝑠 ) ) |
29 |
27 28
|
syl |
⊢ ( 𝑠 ∈ ℕ → 0 ∈ ( 0 ... 𝑠 ) ) |
30 |
29
|
adantr |
⊢ ( ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → 0 ∈ ( 0 ... 𝑠 ) ) |
31 |
24 30
|
ffvelrnd |
⊢ ( ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → ( 𝑏 ‘ 0 ) ∈ 𝐵 ) |
32 |
22 31
|
anim12i |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑏 ‘ 0 ) ∈ 𝐵 ) ) |
33 |
|
df-3an |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ ( 𝑏 ‘ 0 ) ∈ 𝐵 ) ↔ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑏 ‘ 0 ) ∈ 𝐵 ) ) |
34 |
32 33
|
sylibr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ ( 𝑏 ‘ 0 ) ∈ 𝐵 ) ) |
35 |
8 1 2 3 4
|
mat2pmatbas |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ ( 𝑏 ‘ 0 ) ∈ 𝐵 ) → ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ∈ ( Base ‘ 𝑌 ) ) |
36 |
34 35
|
syl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ∈ ( Base ‘ 𝑌 ) ) |
37 |
15 5
|
ringcl |
⊢ ( ( 𝑌 ∈ Ring ∧ ( 𝑇 ‘ 𝑀 ) ∈ ( Base ‘ 𝑌 ) ∧ ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ∈ ( Base ‘ 𝑌 ) ) → ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ∈ ( Base ‘ 𝑌 ) ) |
38 |
19 21 36 37
|
syl3anc |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ∈ ( Base ‘ 𝑌 ) ) |
39 |
15 6
|
grpsubcl |
⊢ ( ( 𝑌 ∈ Grp ∧ 0 ∈ ( Base ‘ 𝑌 ) ∧ ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ∈ ( Base ‘ 𝑌 ) ) → ( 0 − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ∈ ( Base ‘ 𝑌 ) ) |
40 |
14 18 38 39
|
syl3anc |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 0 − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ∈ ( Base ‘ 𝑌 ) ) |
41 |
40
|
ad2antrr |
⊢ ( ( ( ( ( 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 ) ) |
42 |
25
|
adantr |
⊢ ( ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → 𝑠 ∈ ℕ0 ) |
43 |
22 42
|
anim12i |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑠 ∈ ℕ0 ) ) |
44 |
|
df-3an |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0 ) ↔ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑠 ∈ ℕ0 ) ) |
45 |
43 44
|
sylibr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0 ) ) |
46 |
|
eluzfz2 |
⊢ ( 𝑠 ∈ ( ℤ≥ ‘ 0 ) → 𝑠 ∈ ( 0 ... 𝑠 ) ) |
47 |
27 46
|
syl |
⊢ ( 𝑠 ∈ ℕ → 𝑠 ∈ ( 0 ... 𝑠 ) ) |
48 |
47
|
anim1ci |
⊢ ( ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → ( 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ∧ 𝑠 ∈ ( 0 ... 𝑠 ) ) ) |
49 |
48
|
adantl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ∧ 𝑠 ∈ ( 0 ... 𝑠 ) ) ) |
50 |
1 2 3 4 8
|
m2pmfzmap |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0 ) ∧ ( 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ∧ 𝑠 ∈ ( 0 ... 𝑠 ) ) ) → ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ∈ ( Base ‘ 𝑌 ) ) |
51 |
45 49 50
|
syl2anc |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ∈ ( Base ‘ 𝑌 ) ) |
52 |
51
|
adantr |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ∈ ( Base ‘ 𝑌 ) ) |
53 |
52
|
ad2antrr |
⊢ ( ( ( ( ( ( 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 ) ) |
54 |
18
|
ad4antr |
⊢ ( ( ( ( ( ( ( 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 ) ) |
55 |
|
nn0re |
⊢ ( 𝑛 ∈ ℕ0 → 𝑛 ∈ ℝ ) |
56 |
55
|
adantl |
⊢ ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) → 𝑛 ∈ ℝ ) |
57 |
|
peano2nn |
⊢ ( 𝑠 ∈ ℕ → ( 𝑠 + 1 ) ∈ ℕ ) |
58 |
57
|
nnred |
⊢ ( 𝑠 ∈ ℕ → ( 𝑠 + 1 ) ∈ ℝ ) |
59 |
58
|
adantr |
⊢ ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) → ( 𝑠 + 1 ) ∈ ℝ ) |
60 |
56 59
|
lenltd |
⊢ ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) → ( 𝑛 ≤ ( 𝑠 + 1 ) ↔ ¬ ( 𝑠 + 1 ) < 𝑛 ) ) |
61 |
|
nesym |
⊢ ( ( 𝑠 + 1 ) ≠ 𝑛 ↔ ¬ 𝑛 = ( 𝑠 + 1 ) ) |
62 |
|
ltlen |
⊢ ( ( 𝑛 ∈ ℝ ∧ ( 𝑠 + 1 ) ∈ ℝ ) → ( 𝑛 < ( 𝑠 + 1 ) ↔ ( 𝑛 ≤ ( 𝑠 + 1 ) ∧ ( 𝑠 + 1 ) ≠ 𝑛 ) ) ) |
63 |
55 58 62
|
syl2anr |
⊢ ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) → ( 𝑛 < ( 𝑠 + 1 ) ↔ ( 𝑛 ≤ ( 𝑠 + 1 ) ∧ ( 𝑠 + 1 ) ≠ 𝑛 ) ) ) |
64 |
63
|
biimprd |
⊢ ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) → ( ( 𝑛 ≤ ( 𝑠 + 1 ) ∧ ( 𝑠 + 1 ) ≠ 𝑛 ) → 𝑛 < ( 𝑠 + 1 ) ) ) |
65 |
64
|
expcomd |
⊢ ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) → ( ( 𝑠 + 1 ) ≠ 𝑛 → ( 𝑛 ≤ ( 𝑠 + 1 ) → 𝑛 < ( 𝑠 + 1 ) ) ) ) |
66 |
61 65
|
syl5bir |
⊢ ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) → ( ¬ 𝑛 = ( 𝑠 + 1 ) → ( 𝑛 ≤ ( 𝑠 + 1 ) → 𝑛 < ( 𝑠 + 1 ) ) ) ) |
67 |
66
|
com23 |
⊢ ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) → ( 𝑛 ≤ ( 𝑠 + 1 ) → ( ¬ 𝑛 = ( 𝑠 + 1 ) → 𝑛 < ( 𝑠 + 1 ) ) ) ) |
68 |
60 67
|
sylbird |
⊢ ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) → ( ¬ ( 𝑠 + 1 ) < 𝑛 → ( ¬ 𝑛 = ( 𝑠 + 1 ) → 𝑛 < ( 𝑠 + 1 ) ) ) ) |
69 |
68
|
impcomd |
⊢ ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) → ( ( ¬ 𝑛 = ( 𝑠 + 1 ) ∧ ¬ ( 𝑠 + 1 ) < 𝑛 ) → 𝑛 < ( 𝑠 + 1 ) ) ) |
70 |
69
|
ex |
⊢ ( 𝑠 ∈ ℕ → ( 𝑛 ∈ ℕ0 → ( ( ¬ 𝑛 = ( 𝑠 + 1 ) ∧ ¬ ( 𝑠 + 1 ) < 𝑛 ) → 𝑛 < ( 𝑠 + 1 ) ) ) ) |
71 |
70
|
ad2antrl |
⊢ ( ( ( 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
|
imp |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 ∈ ℕ0 ) → ( ( ¬ 𝑛 = ( 𝑠 + 1 ) ∧ ¬ ( 𝑠 + 1 ) < 𝑛 ) → 𝑛 < ( 𝑠 + 1 ) ) ) |
73 |
72
|
adantr |
⊢ ( ( ( ( ( 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 ) ) ) |
74 |
10 12
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑌 ∈ Grp ) |
75 |
74
|
3adant3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → 𝑌 ∈ Grp ) |
76 |
75
|
ad4antr |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → 𝑌 ∈ Grp ) |
77 |
22
|
ad4antr |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ) |
78 |
24
|
ad4antlr |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → 𝑏 : ( 0 ... 𝑠 ) ⟶ 𝐵 ) |
79 |
|
neqne |
⊢ ( ¬ 𝑛 = 0 → 𝑛 ≠ 0 ) |
80 |
79
|
anim2i |
⊢ ( ( 𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0 ) → ( 𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0 ) ) |
81 |
|
elnnne0 |
⊢ ( 𝑛 ∈ ℕ ↔ ( 𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0 ) ) |
82 |
80 81
|
sylibr |
⊢ ( ( 𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0 ) → 𝑛 ∈ ℕ ) |
83 |
|
nnm1nn0 |
⊢ ( 𝑛 ∈ ℕ → ( 𝑛 − 1 ) ∈ ℕ0 ) |
84 |
82 83
|
syl |
⊢ ( ( 𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0 ) → ( 𝑛 − 1 ) ∈ ℕ0 ) |
85 |
84
|
ad4ant23 |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → ( 𝑛 − 1 ) ∈ ℕ0 ) |
86 |
42
|
ad4antlr |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → 𝑠 ∈ ℕ0 ) |
87 |
63
|
simprbda |
⊢ ( ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → 𝑛 ≤ ( 𝑠 + 1 ) ) |
88 |
56
|
adantr |
⊢ ( ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → 𝑛 ∈ ℝ ) |
89 |
|
1red |
⊢ ( ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → 1 ∈ ℝ ) |
90 |
|
nnre |
⊢ ( 𝑠 ∈ ℕ → 𝑠 ∈ ℝ ) |
91 |
90
|
ad2antrr |
⊢ ( ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → 𝑠 ∈ ℝ ) |
92 |
88 89 91
|
lesubaddd |
⊢ ( ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → ( ( 𝑛 − 1 ) ≤ 𝑠 ↔ 𝑛 ≤ ( 𝑠 + 1 ) ) ) |
93 |
87 92
|
mpbird |
⊢ ( ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → ( 𝑛 − 1 ) ≤ 𝑠 ) |
94 |
93
|
exp31 |
⊢ ( 𝑠 ∈ ℕ → ( 𝑛 ∈ ℕ0 → ( 𝑛 < ( 𝑠 + 1 ) → ( 𝑛 − 1 ) ≤ 𝑠 ) ) ) |
95 |
94
|
ad2antrl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑛 ∈ ℕ0 → ( 𝑛 < ( 𝑠 + 1 ) → ( 𝑛 − 1 ) ≤ 𝑠 ) ) ) |
96 |
95
|
imp |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝑛 < ( 𝑠 + 1 ) → ( 𝑛 − 1 ) ≤ 𝑠 ) ) |
97 |
96
|
adantr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( 𝑛 < ( 𝑠 + 1 ) → ( 𝑛 − 1 ) ≤ 𝑠 ) ) |
98 |
97
|
imp |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → ( 𝑛 − 1 ) ≤ 𝑠 ) |
99 |
|
elfz2nn0 |
⊢ ( ( 𝑛 − 1 ) ∈ ( 0 ... 𝑠 ) ↔ ( ( 𝑛 − 1 ) ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ ( 𝑛 − 1 ) ≤ 𝑠 ) ) |
100 |
85 86 98 99
|
syl3anbrc |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → ( 𝑛 − 1 ) ∈ ( 0 ... 𝑠 ) ) |
101 |
78 100
|
ffvelrnd |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → ( 𝑏 ‘ ( 𝑛 − 1 ) ) ∈ 𝐵 ) |
102 |
|
df-3an |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ ( 𝑏 ‘ ( 𝑛 − 1 ) ) ∈ 𝐵 ) ↔ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑏 ‘ ( 𝑛 − 1 ) ) ∈ 𝐵 ) ) |
103 |
77 101 102
|
sylanbrc |
⊢ ( ( ( ( ( ( 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 ) ) |
104 |
8 1 2 3 4
|
mat2pmatbas |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ ( 𝑏 ‘ ( 𝑛 − 1 ) ) ∈ 𝐵 ) → ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑛 − 1 ) ) ) ∈ ( Base ‘ 𝑌 ) ) |
105 |
103 104
|
syl |
⊢ ( ( ( ( ( ( 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 ) ) |
106 |
19
|
ad2antrr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → 𝑌 ∈ Ring ) |
107 |
21
|
ad2antrr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → ( 𝑇 ‘ 𝑀 ) ∈ ( Base ‘ 𝑌 ) ) |
108 |
45
|
ad2antrr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0 ) ) |
109 |
|
simprr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) |
110 |
109
|
ad2antrr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) |
111 |
|
simplr |
⊢ ( ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → 𝑛 ∈ ℕ0 ) |
112 |
25
|
ad2antrr |
⊢ ( ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → 𝑠 ∈ ℕ0 ) |
113 |
|
nn0z |
⊢ ( 𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ ) |
114 |
|
nnz |
⊢ ( 𝑠 ∈ ℕ → 𝑠 ∈ ℤ ) |
115 |
|
zleltp1 |
⊢ ( ( 𝑛 ∈ ℤ ∧ 𝑠 ∈ ℤ ) → ( 𝑛 ≤ 𝑠 ↔ 𝑛 < ( 𝑠 + 1 ) ) ) |
116 |
113 114 115
|
syl2anr |
⊢ ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) → ( 𝑛 ≤ 𝑠 ↔ 𝑛 < ( 𝑠 + 1 ) ) ) |
117 |
116
|
biimpar |
⊢ ( ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → 𝑛 ≤ 𝑠 ) |
118 |
|
elfz2nn0 |
⊢ ( 𝑛 ∈ ( 0 ... 𝑠 ) ↔ ( 𝑛 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑛 ≤ 𝑠 ) ) |
119 |
111 112 117 118
|
syl3anbrc |
⊢ ( ( ( 𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → 𝑛 ∈ ( 0 ... 𝑠 ) ) |
120 |
119
|
exp31 |
⊢ ( 𝑠 ∈ ℕ → ( 𝑛 ∈ ℕ0 → ( 𝑛 < ( 𝑠 + 1 ) → 𝑛 ∈ ( 0 ... 𝑠 ) ) ) ) |
121 |
120
|
ad2antrl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑛 ∈ ℕ0 → ( 𝑛 < ( 𝑠 + 1 ) → 𝑛 ∈ ( 0 ... 𝑠 ) ) ) ) |
122 |
121
|
imp31 |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → 𝑛 ∈ ( 0 ... 𝑠 ) ) |
123 |
1 2 3 4 8
|
m2pmfzmap |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0 ) ∧ ( 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ∧ 𝑛 ∈ ( 0 ... 𝑠 ) ) ) → ( 𝑇 ‘ ( 𝑏 ‘ 𝑛 ) ) ∈ ( Base ‘ 𝑌 ) ) |
124 |
108 110 122 123
|
syl12anc |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 < ( 𝑠 + 1 ) ) → ( 𝑇 ‘ ( 𝑏 ‘ 𝑛 ) ) ∈ ( Base ‘ 𝑌 ) ) |
125 |
15 5
|
ringcl |
⊢ ( ( 𝑌 ∈ Ring ∧ ( 𝑇 ‘ 𝑀 ) ∈ ( Base ‘ 𝑌 ) ∧ ( 𝑇 ‘ ( 𝑏 ‘ 𝑛 ) ) ∈ ( Base ‘ 𝑌 ) ) → ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑛 ) ) ) ∈ ( Base ‘ 𝑌 ) ) |
126 |
106 107 124 125
|
syl3anc |
⊢ ( ( ( ( ( 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 ) ) |
127 |
126
|
adantlr |
⊢ ( ( ( ( ( ( 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 ) ) |
128 |
15 6
|
grpsubcl |
⊢ ( ( 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 ) ) |
129 |
76 105 127 128
|
syl3anc |
⊢ ( ( ( ( ( ( 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 |
129
|
ex |
⊢ ( ( ( ( ( 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 ) ) ) |
131 |
73 130
|
syld |
⊢ ( ( ( ( ( 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 |
131
|
impl |
⊢ ( ( ( ( ( ( ( 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 ) ) |
133 |
54 132
|
ifclda |
⊢ ( ( ( ( ( ( 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 ) ) |
134 |
53 133
|
ifclda |
⊢ ( ( ( ( ( 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 ) ) |
135 |
41 134
|
ifclda |
⊢ ( ( ( ( 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 ) ) |
136 |
135 9
|
fmptd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝐺 : ℕ0 ⟶ ( Base ‘ 𝑌 ) ) |