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 |
|
chfacfscmulcl.x |
⊢ 𝑋 = ( var1 ‘ 𝑅 ) |
11 |
|
chfacfscmulcl.m |
⊢ · = ( ·𝑠 ‘ 𝑌 ) |
12 |
|
chfacfscmulcl.e |
⊢ ↑ = ( .g ‘ ( mulGrp ‘ 𝑃 ) ) |
13 |
|
chfacfscmulgsum.p |
⊢ + = ( +g ‘ 𝑌 ) |
14 |
|
eqid |
⊢ ( Base ‘ 𝑌 ) = ( Base ‘ 𝑌 ) |
15 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
16 |
15
|
anim2i |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ) |
17 |
16
|
3adant3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ) |
18 |
3 4
|
pmatring |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑌 ∈ Ring ) |
19 |
17 18
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑌 ∈ Ring ) |
20 |
|
ringcmn |
⊢ ( 𝑌 ∈ Ring → 𝑌 ∈ CMnd ) |
21 |
19 20
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑌 ∈ CMnd ) |
22 |
21
|
adantr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝑌 ∈ CMnd ) |
23 |
|
nn0ex |
⊢ ℕ0 ∈ V |
24 |
23
|
a1i |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ℕ0 ∈ V ) |
25 |
|
simpll |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ℕ0 ) → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ) |
26 |
|
simplr |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ℕ0 ) → ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) |
27 |
|
simpr |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ℕ0 ) → 𝑖 ∈ ℕ0 ) |
28 |
25 26 27
|
3jca |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ℕ0 ) → ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑖 ∈ ℕ0 ) ) |
29 |
1 2 3 4 5 6 7 8 9 10 11 12
|
chfacfscmulcl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑖 ∈ ℕ0 ) → ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ∈ ( Base ‘ 𝑌 ) ) |
30 |
28 29
|
syl |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ℕ0 ) → ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ∈ ( Base ‘ 𝑌 ) ) |
31 |
1 2 3 4 5 6 7 8 9 10 11 12
|
chfacfscmulfsupp |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) finSupp 0 ) |
32 |
|
nn0disj |
⊢ ( ( 0 ... ( 𝑠 + 1 ) ) ∩ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ) = ∅ |
33 |
32
|
a1i |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 0 ... ( 𝑠 + 1 ) ) ∩ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ) = ∅ ) |
34 |
|
nnnn0 |
⊢ ( 𝑠 ∈ ℕ → 𝑠 ∈ ℕ0 ) |
35 |
|
peano2nn0 |
⊢ ( 𝑠 ∈ ℕ0 → ( 𝑠 + 1 ) ∈ ℕ0 ) |
36 |
34 35
|
syl |
⊢ ( 𝑠 ∈ ℕ → ( 𝑠 + 1 ) ∈ ℕ0 ) |
37 |
|
nn0split |
⊢ ( ( 𝑠 + 1 ) ∈ ℕ0 → ℕ0 = ( ( 0 ... ( 𝑠 + 1 ) ) ∪ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ) ) |
38 |
36 37
|
syl |
⊢ ( 𝑠 ∈ ℕ → ℕ0 = ( ( 0 ... ( 𝑠 + 1 ) ) ∪ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ) ) |
39 |
38
|
ad2antrl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ℕ0 = ( ( 0 ... ( 𝑠 + 1 ) ) ∪ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ) ) |
40 |
14 7 13 22 24 30 31 33 39
|
gsumsplit2 |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... ( 𝑠 + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + ( 𝑌 Σg ( 𝑖 ∈ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ) ) |
41 |
|
simpll |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ) → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ) |
42 |
|
simplr |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ) → ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) |
43 |
|
nncn |
⊢ ( 𝑠 ∈ ℕ → 𝑠 ∈ ℂ ) |
44 |
|
add1p1 |
⊢ ( 𝑠 ∈ ℂ → ( ( 𝑠 + 1 ) + 1 ) = ( 𝑠 + 2 ) ) |
45 |
43 44
|
syl |
⊢ ( 𝑠 ∈ ℕ → ( ( 𝑠 + 1 ) + 1 ) = ( 𝑠 + 2 ) ) |
46 |
45
|
ad2antrl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 𝑠 + 1 ) + 1 ) = ( 𝑠 + 2 ) ) |
47 |
46
|
fveq2d |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) = ( ℤ≥ ‘ ( 𝑠 + 2 ) ) ) |
48 |
47
|
eleq2d |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑖 ∈ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ↔ 𝑖 ∈ ( ℤ≥ ‘ ( 𝑠 + 2 ) ) ) ) |
49 |
48
|
biimpa |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ) → 𝑖 ∈ ( ℤ≥ ‘ ( 𝑠 + 2 ) ) ) |
50 |
1 2 3 4 5 6 7 8 9 10 11 12
|
chfacfscmul0 |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ ( 𝑠 + 2 ) ) ) → ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) = 0 ) |
51 |
41 42 49 50
|
syl3anc |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ) → ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) = 0 ) |
52 |
51
|
mpteq2dva |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑖 ∈ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) = ( 𝑖 ∈ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ↦ 0 ) ) |
53 |
52
|
oveq2d |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) = ( 𝑌 Σg ( 𝑖 ∈ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ↦ 0 ) ) ) |
54 |
15 18
|
sylan2 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑌 ∈ Ring ) |
55 |
|
ringmnd |
⊢ ( 𝑌 ∈ Ring → 𝑌 ∈ Mnd ) |
56 |
54 55
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑌 ∈ Mnd ) |
57 |
56
|
3adant3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑌 ∈ Mnd ) |
58 |
|
fvex |
⊢ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ∈ V |
59 |
57 58
|
jctir |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 𝑌 ∈ Mnd ∧ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ∈ V ) ) |
60 |
59
|
adantr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑌 ∈ Mnd ∧ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ∈ V ) ) |
61 |
7
|
gsumz |
⊢ ( ( 𝑌 ∈ Mnd ∧ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ∈ V ) → ( 𝑌 Σg ( 𝑖 ∈ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ↦ 0 ) ) = 0 ) |
62 |
60 61
|
syl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ↦ 0 ) ) = 0 ) |
63 |
53 62
|
eqtrd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) = 0 ) |
64 |
63
|
oveq2d |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... ( 𝑠 + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + ( 𝑌 Σg ( 𝑖 ∈ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... ( 𝑠 + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + 0 ) ) |
65 |
|
fzfid |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 0 ... ( 𝑠 + 1 ) ) ∈ Fin ) |
66 |
|
elfznn0 |
⊢ ( 𝑖 ∈ ( 0 ... ( 𝑠 + 1 ) ) → 𝑖 ∈ ℕ0 ) |
67 |
66 28
|
sylan2 |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... ( 𝑠 + 1 ) ) ) → ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑖 ∈ ℕ0 ) ) |
68 |
67 29
|
syl |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... ( 𝑠 + 1 ) ) ) → ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ∈ ( Base ‘ 𝑌 ) ) |
69 |
68
|
ralrimiva |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ∀ 𝑖 ∈ ( 0 ... ( 𝑠 + 1 ) ) ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ∈ ( Base ‘ 𝑌 ) ) |
70 |
14 22 65 69
|
gsummptcl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... ( 𝑠 + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ∈ ( Base ‘ 𝑌 ) ) |
71 |
14 13 7
|
mndrid |
⊢ ( ( 𝑌 ∈ Mnd ∧ ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... ( 𝑠 + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ∈ ( Base ‘ 𝑌 ) ) → ( ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... ( 𝑠 + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + 0 ) = ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... ( 𝑠 + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ) |
72 |
57 70 71
|
syl2an2r |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... ( 𝑠 + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + 0 ) = ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... ( 𝑠 + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ) |
73 |
64 72
|
eqtrd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... ( 𝑠 + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + ( 𝑌 Σg ( 𝑖 ∈ ( ℤ≥ ‘ ( ( 𝑠 + 1 ) + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ) = ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... ( 𝑠 + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ) |
74 |
34
|
ad2antrl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝑠 ∈ ℕ0 ) |
75 |
14 13 22 74 68
|
gsummptfzsplit |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... ( 𝑠 + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + ( 𝑌 Σg ( 𝑖 ∈ { ( 𝑠 + 1 ) } ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ) ) |
76 |
|
elfznn0 |
⊢ ( 𝑖 ∈ ( 0 ... 𝑠 ) → 𝑖 ∈ ℕ0 ) |
77 |
76 30
|
sylan2 |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑠 ) ) → ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ∈ ( Base ‘ 𝑌 ) ) |
78 |
14 13 22 74 77
|
gsummptfzsplitl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + ( 𝑌 Σg ( 𝑖 ∈ { 0 } ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ) ) |
79 |
57
|
adantr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝑌 ∈ Mnd ) |
80 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
81 |
80
|
a1i |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 0 ∈ ℕ0 ) |
82 |
1 2 3 4 5 6 7 8 9 10 11 12
|
chfacfscmulcl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 0 ∈ ℕ0 ) → ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) ∈ ( Base ‘ 𝑌 ) ) |
83 |
81 82
|
mpd3an3 |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) ∈ ( Base ‘ 𝑌 ) ) |
84 |
|
oveq1 |
⊢ ( 𝑖 = 0 → ( 𝑖 ↑ 𝑋 ) = ( 0 ↑ 𝑋 ) ) |
85 |
|
fveq2 |
⊢ ( 𝑖 = 0 → ( 𝐺 ‘ 𝑖 ) = ( 𝐺 ‘ 0 ) ) |
86 |
84 85
|
oveq12d |
⊢ ( 𝑖 = 0 → ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) = ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) ) |
87 |
14 86
|
gsumsn |
⊢ ( ( 𝑌 ∈ Mnd ∧ 0 ∈ ℕ0 ∧ ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) ∈ ( Base ‘ 𝑌 ) ) → ( 𝑌 Σg ( 𝑖 ∈ { 0 } ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) = ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) ) |
88 |
79 81 83 87
|
syl3anc |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ { 0 } ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) = ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) ) |
89 |
88
|
oveq2d |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + ( 𝑌 Σg ( 𝑖 ∈ { 0 } ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) ) ) |
90 |
78 89
|
eqtrd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) ) ) |
91 |
|
ovexd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑠 + 1 ) ∈ V ) |
92 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
93 |
92
|
a1i |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 1 ∈ ℕ0 ) |
94 |
74 93
|
nn0addcld |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑠 + 1 ) ∈ ℕ0 ) |
95 |
1 2 3 4 5 6 7 8 9 10 11 12
|
chfacfscmulcl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ ( 𝑠 + 1 ) ∈ ℕ0 ) → ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) ∈ ( Base ‘ 𝑌 ) ) |
96 |
94 95
|
mpd3an3 |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) ∈ ( Base ‘ 𝑌 ) ) |
97 |
|
oveq1 |
⊢ ( 𝑖 = ( 𝑠 + 1 ) → ( 𝑖 ↑ 𝑋 ) = ( ( 𝑠 + 1 ) ↑ 𝑋 ) ) |
98 |
|
fveq2 |
⊢ ( 𝑖 = ( 𝑠 + 1 ) → ( 𝐺 ‘ 𝑖 ) = ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) |
99 |
97 98
|
oveq12d |
⊢ ( 𝑖 = ( 𝑠 + 1 ) → ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) = ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) ) |
100 |
14 99
|
gsumsn |
⊢ ( ( 𝑌 ∈ Mnd ∧ ( 𝑠 + 1 ) ∈ V ∧ ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) ∈ ( Base ‘ 𝑌 ) ) → ( 𝑌 Σg ( 𝑖 ∈ { ( 𝑠 + 1 ) } ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) = ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) ) |
101 |
79 91 96 100
|
syl3anc |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ { ( 𝑠 + 1 ) } ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) = ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) ) |
102 |
90 101
|
oveq12d |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + ( 𝑌 Σg ( 𝑖 ∈ { ( 𝑠 + 1 ) } ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ) = ( ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) ) + ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) ) ) |
103 |
|
fzfid |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 1 ... 𝑠 ) ∈ Fin ) |
104 |
|
simpll |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ) |
105 |
|
simplr |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) → ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) |
106 |
|
elfznn |
⊢ ( 𝑖 ∈ ( 1 ... 𝑠 ) → 𝑖 ∈ ℕ ) |
107 |
106
|
nnnn0d |
⊢ ( 𝑖 ∈ ( 1 ... 𝑠 ) → 𝑖 ∈ ℕ0 ) |
108 |
107
|
adantl |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) → 𝑖 ∈ ℕ0 ) |
109 |
104 105 108 29
|
syl3anc |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) → ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ∈ ( Base ‘ 𝑌 ) ) |
110 |
109
|
ralrimiva |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ∀ 𝑖 ∈ ( 1 ... 𝑠 ) ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ∈ ( Base ‘ 𝑌 ) ) |
111 |
14 22 103 110
|
gsummptcl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ∈ ( Base ‘ 𝑌 ) ) |
112 |
14 13
|
mndass |
⊢ ( ( 𝑌 ∈ Mnd ∧ ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ∈ ( Base ‘ 𝑌 ) ∧ ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) ∈ ( Base ‘ 𝑌 ) ∧ ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) ∈ ( Base ‘ 𝑌 ) ) ) → ( ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) ) + ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + ( ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) + ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) ) ) ) |
113 |
79 111 83 96 112
|
syl13anc |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) ) + ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + ( ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) + ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) ) ) ) |
114 |
106
|
nnne0d |
⊢ ( 𝑖 ∈ ( 1 ... 𝑠 ) → 𝑖 ≠ 0 ) |
115 |
114
|
ad2antlr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) → 𝑖 ≠ 0 ) |
116 |
|
neeq1 |
⊢ ( 𝑛 = 𝑖 → ( 𝑛 ≠ 0 ↔ 𝑖 ≠ 0 ) ) |
117 |
116
|
adantl |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) → ( 𝑛 ≠ 0 ↔ 𝑖 ≠ 0 ) ) |
118 |
115 117
|
mpbird |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) → 𝑛 ≠ 0 ) |
119 |
|
eqneqall |
⊢ ( 𝑛 = 0 → ( 𝑛 ≠ 0 → 0 = ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) ) ) |
120 |
118 119
|
mpan9 |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ 𝑛 = 0 ) → 0 = ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) ) |
121 |
|
simplr |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ 𝑛 = 0 ) → 𝑛 = 𝑖 ) |
122 |
|
eqeq1 |
⊢ ( 0 = 𝑛 → ( 0 = 𝑖 ↔ 𝑛 = 𝑖 ) ) |
123 |
122
|
eqcoms |
⊢ ( 𝑛 = 0 → ( 0 = 𝑖 ↔ 𝑛 = 𝑖 ) ) |
124 |
123
|
adantl |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ 𝑛 = 0 ) → ( 0 = 𝑖 ↔ 𝑛 = 𝑖 ) ) |
125 |
121 124
|
mpbird |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ 𝑛 = 0 ) → 0 = 𝑖 ) |
126 |
125
|
fveq2d |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ 𝑛 = 0 ) → ( 𝑏 ‘ 0 ) = ( 𝑏 ‘ 𝑖 ) ) |
127 |
126
|
fveq2d |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ 𝑛 = 0 ) → ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) = ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) |
128 |
127
|
oveq2d |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ 𝑛 = 0 ) → ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) = ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) |
129 |
120 128
|
oveq12d |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ 𝑛 = 0 ) → ( 0 − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) = ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) |
130 |
|
elfz2 |
⊢ ( 𝑖 ∈ ( 1 ... 𝑠 ) ↔ ( ( 1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ ) ∧ ( 1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠 ) ) ) |
131 |
|
zleltp1 |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑠 ∈ ℤ ) → ( 𝑖 ≤ 𝑠 ↔ 𝑖 < ( 𝑠 + 1 ) ) ) |
132 |
131
|
ancoms |
⊢ ( ( 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ ) → ( 𝑖 ≤ 𝑠 ↔ 𝑖 < ( 𝑠 + 1 ) ) ) |
133 |
132
|
3adant1 |
⊢ ( ( 1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ ) → ( 𝑖 ≤ 𝑠 ↔ 𝑖 < ( 𝑠 + 1 ) ) ) |
134 |
133
|
biimpcd |
⊢ ( 𝑖 ≤ 𝑠 → ( ( 1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ ) → 𝑖 < ( 𝑠 + 1 ) ) ) |
135 |
134
|
adantl |
⊢ ( ( 1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠 ) → ( ( 1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ ) → 𝑖 < ( 𝑠 + 1 ) ) ) |
136 |
135
|
impcom |
⊢ ( ( ( 1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ ) ∧ ( 1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠 ) ) → 𝑖 < ( 𝑠 + 1 ) ) |
137 |
136
|
orcd |
⊢ ( ( ( 1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ ) ∧ ( 1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠 ) ) → ( 𝑖 < ( 𝑠 + 1 ) ∨ ( 𝑠 + 1 ) < 𝑖 ) ) |
138 |
|
zre |
⊢ ( 𝑠 ∈ ℤ → 𝑠 ∈ ℝ ) |
139 |
|
1red |
⊢ ( 𝑠 ∈ ℤ → 1 ∈ ℝ ) |
140 |
138 139
|
readdcld |
⊢ ( 𝑠 ∈ ℤ → ( 𝑠 + 1 ) ∈ ℝ ) |
141 |
|
zre |
⊢ ( 𝑖 ∈ ℤ → 𝑖 ∈ ℝ ) |
142 |
140 141
|
anim12ci |
⊢ ( ( 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ ) → ( 𝑖 ∈ ℝ ∧ ( 𝑠 + 1 ) ∈ ℝ ) ) |
143 |
142
|
3adant1 |
⊢ ( ( 1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ ) → ( 𝑖 ∈ ℝ ∧ ( 𝑠 + 1 ) ∈ ℝ ) ) |
144 |
|
lttri2 |
⊢ ( ( 𝑖 ∈ ℝ ∧ ( 𝑠 + 1 ) ∈ ℝ ) → ( 𝑖 ≠ ( 𝑠 + 1 ) ↔ ( 𝑖 < ( 𝑠 + 1 ) ∨ ( 𝑠 + 1 ) < 𝑖 ) ) ) |
145 |
143 144
|
syl |
⊢ ( ( 1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ ) → ( 𝑖 ≠ ( 𝑠 + 1 ) ↔ ( 𝑖 < ( 𝑠 + 1 ) ∨ ( 𝑠 + 1 ) < 𝑖 ) ) ) |
146 |
145
|
adantr |
⊢ ( ( ( 1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ ) ∧ ( 1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠 ) ) → ( 𝑖 ≠ ( 𝑠 + 1 ) ↔ ( 𝑖 < ( 𝑠 + 1 ) ∨ ( 𝑠 + 1 ) < 𝑖 ) ) ) |
147 |
137 146
|
mpbird |
⊢ ( ( ( 1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ ) ∧ ( 1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠 ) ) → 𝑖 ≠ ( 𝑠 + 1 ) ) |
148 |
130 147
|
sylbi |
⊢ ( 𝑖 ∈ ( 1 ... 𝑠 ) → 𝑖 ≠ ( 𝑠 + 1 ) ) |
149 |
148
|
ad2antlr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) → 𝑖 ≠ ( 𝑠 + 1 ) ) |
150 |
|
neeq1 |
⊢ ( 𝑛 = 𝑖 → ( 𝑛 ≠ ( 𝑠 + 1 ) ↔ 𝑖 ≠ ( 𝑠 + 1 ) ) ) |
151 |
150
|
adantl |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) → ( 𝑛 ≠ ( 𝑠 + 1 ) ↔ 𝑖 ≠ ( 𝑠 + 1 ) ) ) |
152 |
149 151
|
mpbird |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) → 𝑛 ≠ ( 𝑠 + 1 ) ) |
153 |
152
|
adantr |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ ¬ 𝑛 = 0 ) → 𝑛 ≠ ( 𝑠 + 1 ) ) |
154 |
153
|
neneqd |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ ¬ 𝑛 = 0 ) → ¬ 𝑛 = ( 𝑠 + 1 ) ) |
155 |
154
|
pm2.21d |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ ¬ 𝑛 = 0 ) → ( 𝑛 = ( 𝑠 + 1 ) → ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) = ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) |
156 |
155
|
imp |
⊢ ( ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ ¬ 𝑛 = 0 ) ∧ 𝑛 = ( 𝑠 + 1 ) ) → ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) = ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) |
157 |
106
|
nnred |
⊢ ( 𝑖 ∈ ( 1 ... 𝑠 ) → 𝑖 ∈ ℝ ) |
158 |
|
eleq1w |
⊢ ( 𝑛 = 𝑖 → ( 𝑛 ∈ ℝ ↔ 𝑖 ∈ ℝ ) ) |
159 |
157 158
|
syl5ibrcom |
⊢ ( 𝑖 ∈ ( 1 ... 𝑠 ) → ( 𝑛 = 𝑖 → 𝑛 ∈ ℝ ) ) |
160 |
159
|
adantl |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) → ( 𝑛 = 𝑖 → 𝑛 ∈ ℝ ) ) |
161 |
160
|
imp |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) → 𝑛 ∈ ℝ ) |
162 |
74
|
nn0red |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝑠 ∈ ℝ ) |
163 |
162
|
ad2antrr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) → 𝑠 ∈ ℝ ) |
164 |
|
1red |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) → 1 ∈ ℝ ) |
165 |
163 164
|
readdcld |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) → ( 𝑠 + 1 ) ∈ ℝ ) |
166 |
130 136
|
sylbi |
⊢ ( 𝑖 ∈ ( 1 ... 𝑠 ) → 𝑖 < ( 𝑠 + 1 ) ) |
167 |
166
|
ad2antlr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) → 𝑖 < ( 𝑠 + 1 ) ) |
168 |
|
breq1 |
⊢ ( 𝑛 = 𝑖 → ( 𝑛 < ( 𝑠 + 1 ) ↔ 𝑖 < ( 𝑠 + 1 ) ) ) |
169 |
168
|
adantl |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) → ( 𝑛 < ( 𝑠 + 1 ) ↔ 𝑖 < ( 𝑠 + 1 ) ) ) |
170 |
167 169
|
mpbird |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) → 𝑛 < ( 𝑠 + 1 ) ) |
171 |
161 165 170
|
ltnsymd |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) → ¬ ( 𝑠 + 1 ) < 𝑛 ) |
172 |
171
|
pm2.21d |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) → ( ( 𝑠 + 1 ) < 𝑛 → 0 = ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) |
173 |
172
|
ad2antrr |
⊢ ( ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ ¬ 𝑛 = 0 ) ∧ ¬ 𝑛 = ( 𝑠 + 1 ) ) → ( ( 𝑠 + 1 ) < 𝑛 → 0 = ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) |
174 |
173
|
imp |
⊢ ( ( ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ ¬ 𝑛 = 0 ) ∧ ¬ 𝑛 = ( 𝑠 + 1 ) ) ∧ ( 𝑠 + 1 ) < 𝑛 ) → 0 = ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) |
175 |
|
simp-4r |
⊢ ( ( ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ ¬ 𝑛 = 0 ) ∧ ¬ 𝑛 = ( 𝑠 + 1 ) ) ∧ ¬ ( 𝑠 + 1 ) < 𝑛 ) → 𝑛 = 𝑖 ) |
176 |
175
|
fvoveq1d |
⊢ ( ( ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ ¬ 𝑛 = 0 ) ∧ ¬ 𝑛 = ( 𝑠 + 1 ) ) ∧ ¬ ( 𝑠 + 1 ) < 𝑛 ) → ( 𝑏 ‘ ( 𝑛 − 1 ) ) = ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) |
177 |
176
|
fveq2d |
⊢ ( ( ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ ¬ 𝑛 = 0 ) ∧ ¬ 𝑛 = ( 𝑠 + 1 ) ) ∧ ¬ ( 𝑠 + 1 ) < 𝑛 ) → ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑛 − 1 ) ) ) = ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) ) |
178 |
175
|
fveq2d |
⊢ ( ( ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ ¬ 𝑛 = 0 ) ∧ ¬ 𝑛 = ( 𝑠 + 1 ) ) ∧ ¬ ( 𝑠 + 1 ) < 𝑛 ) → ( 𝑏 ‘ 𝑛 ) = ( 𝑏 ‘ 𝑖 ) ) |
179 |
178
|
fveq2d |
⊢ ( ( ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ ¬ 𝑛 = 0 ) ∧ ¬ 𝑛 = ( 𝑠 + 1 ) ) ∧ ¬ ( 𝑠 + 1 ) < 𝑛 ) → ( 𝑇 ‘ ( 𝑏 ‘ 𝑛 ) ) = ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) |
180 |
179
|
oveq2d |
⊢ ( ( ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ ¬ 𝑛 = 0 ) ∧ ¬ 𝑛 = ( 𝑠 + 1 ) ) ∧ ¬ ( 𝑠 + 1 ) < 𝑛 ) → ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑛 ) ) ) = ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) |
181 |
177 180
|
oveq12d |
⊢ ( ( ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ ¬ 𝑛 = 0 ) ∧ ¬ 𝑛 = ( 𝑠 + 1 ) ) ∧ ¬ ( 𝑠 + 1 ) < 𝑛 ) → ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑛 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑛 ) ) ) ) = ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) |
182 |
174 181
|
ifeqda |
⊢ ( ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ ¬ 𝑛 = 0 ) ∧ ¬ 𝑛 = ( 𝑠 + 1 ) ) → if ( ( 𝑠 + 1 ) < 𝑛 , 0 , ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑛 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑛 ) ) ) ) ) = ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) |
183 |
156 182
|
ifeqda |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) ∧ ¬ 𝑛 = 0 ) → if ( 𝑛 = ( 𝑠 + 1 ) , ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) , if ( ( 𝑠 + 1 ) < 𝑛 , 0 , ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑛 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑛 ) ) ) ) ) ) = ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) |
184 |
129 183
|
ifeqda |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) ∧ 𝑛 = 𝑖 ) → 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 ) ) ) ) ) ) ) = ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) |
185 |
|
ovexd |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) → ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ∈ V ) |
186 |
9 184 108 185
|
fvmptd2 |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) → ( 𝐺 ‘ 𝑖 ) = ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) |
187 |
186
|
oveq2d |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑠 ) ) → ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) = ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) |
188 |
187
|
mpteq2dva |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) = ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) ) |
189 |
188
|
oveq2d |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) = ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) ) ) |
190 |
|
nn0p1gt0 |
⊢ ( 𝑠 ∈ ℕ0 → 0 < ( 𝑠 + 1 ) ) |
191 |
|
0red |
⊢ ( 𝑠 ∈ ℕ0 → 0 ∈ ℝ ) |
192 |
|
ltne |
⊢ ( ( 0 ∈ ℝ ∧ 0 < ( 𝑠 + 1 ) ) → ( 𝑠 + 1 ) ≠ 0 ) |
193 |
191 192
|
sylan |
⊢ ( ( 𝑠 ∈ ℕ0 ∧ 0 < ( 𝑠 + 1 ) ) → ( 𝑠 + 1 ) ≠ 0 ) |
194 |
|
neeq1 |
⊢ ( 𝑛 = ( 𝑠 + 1 ) → ( 𝑛 ≠ 0 ↔ ( 𝑠 + 1 ) ≠ 0 ) ) |
195 |
193 194
|
syl5ibrcom |
⊢ ( ( 𝑠 ∈ ℕ0 ∧ 0 < ( 𝑠 + 1 ) ) → ( 𝑛 = ( 𝑠 + 1 ) → 𝑛 ≠ 0 ) ) |
196 |
34 190 195
|
syl2anc2 |
⊢ ( 𝑠 ∈ ℕ → ( 𝑛 = ( 𝑠 + 1 ) → 𝑛 ≠ 0 ) ) |
197 |
196
|
ad2antrl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑛 = ( 𝑠 + 1 ) → 𝑛 ≠ 0 ) ) |
198 |
197
|
imp |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 = ( 𝑠 + 1 ) ) → 𝑛 ≠ 0 ) |
199 |
|
eqneqall |
⊢ ( 𝑛 = 0 → ( 𝑛 ≠ 0 → ( 0 − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) = ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) ) |
200 |
198 199
|
mpan9 |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 = ( 𝑠 + 1 ) ) ∧ 𝑛 = 0 ) → ( 0 − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) = ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) |
201 |
|
iftrue |
⊢ ( 𝑛 = ( 𝑠 + 1 ) → if ( 𝑛 = ( 𝑠 + 1 ) , ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) , if ( ( 𝑠 + 1 ) < 𝑛 , 0 , ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑛 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑛 ) ) ) ) ) ) = ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) |
202 |
201
|
ad2antlr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 = ( 𝑠 + 1 ) ) ∧ ¬ 𝑛 = 0 ) → if ( 𝑛 = ( 𝑠 + 1 ) , ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) , if ( ( 𝑠 + 1 ) < 𝑛 , 0 , ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑛 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑛 ) ) ) ) ) ) = ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) |
203 |
200 202
|
ifeqda |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ 𝑛 = ( 𝑠 + 1 ) ) → 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 ) ) ) ) ) ) ) = ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) |
204 |
74 35
|
syl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑠 + 1 ) ∈ ℕ0 ) |
205 |
|
fvexd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ∈ V ) |
206 |
9 203 204 205
|
fvmptd2 |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝐺 ‘ ( 𝑠 + 1 ) ) = ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) |
207 |
206
|
oveq2d |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) = ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) ) |
208 |
15
|
3ad2ant2 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑅 ∈ Ring ) |
209 |
|
eqid |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) |
210 |
10 3 209
|
vr1cl |
⊢ ( 𝑅 ∈ Ring → 𝑋 ∈ ( Base ‘ 𝑃 ) ) |
211 |
208 210
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑋 ∈ ( Base ‘ 𝑃 ) ) |
212 |
|
eqid |
⊢ ( mulGrp ‘ 𝑃 ) = ( mulGrp ‘ 𝑃 ) |
213 |
212 209
|
mgpbas |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ ( mulGrp ‘ 𝑃 ) ) |
214 |
|
eqid |
⊢ ( 1r ‘ 𝑃 ) = ( 1r ‘ 𝑃 ) |
215 |
212 214
|
ringidval |
⊢ ( 1r ‘ 𝑃 ) = ( 0g ‘ ( mulGrp ‘ 𝑃 ) ) |
216 |
213 215 12
|
mulg0 |
⊢ ( 𝑋 ∈ ( Base ‘ 𝑃 ) → ( 0 ↑ 𝑋 ) = ( 1r ‘ 𝑃 ) ) |
217 |
211 216
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 0 ↑ 𝑋 ) = ( 1r ‘ 𝑃 ) ) |
218 |
3
|
ply1crng |
⊢ ( 𝑅 ∈ CRing → 𝑃 ∈ CRing ) |
219 |
218
|
anim2i |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ( 𝑁 ∈ Fin ∧ 𝑃 ∈ CRing ) ) |
220 |
219
|
3adant3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 𝑁 ∈ Fin ∧ 𝑃 ∈ CRing ) ) |
221 |
4
|
matsca2 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑃 ∈ CRing ) → 𝑃 = ( Scalar ‘ 𝑌 ) ) |
222 |
220 221
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑃 = ( Scalar ‘ 𝑌 ) ) |
223 |
222
|
fveq2d |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 1r ‘ 𝑃 ) = ( 1r ‘ ( Scalar ‘ 𝑌 ) ) ) |
224 |
217 223
|
eqtrd |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 0 ↑ 𝑋 ) = ( 1r ‘ ( Scalar ‘ 𝑌 ) ) ) |
225 |
224
|
adantr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 0 ↑ 𝑋 ) = ( 1r ‘ ( Scalar ‘ 𝑌 ) ) ) |
226 |
225
|
oveq1d |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) = ( ( 1r ‘ ( Scalar ‘ 𝑌 ) ) · ( 𝐺 ‘ 0 ) ) ) |
227 |
3 4
|
pmatlmod |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑌 ∈ LMod ) |
228 |
15 227
|
sylan2 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑌 ∈ LMod ) |
229 |
228
|
3adant3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑌 ∈ LMod ) |
230 |
1 2 3 4 5 6 7 8 9
|
chfacfisf |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝐺 : ℕ0 ⟶ ( Base ‘ 𝑌 ) ) |
231 |
15 230
|
syl3anl2 |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝐺 : ℕ0 ⟶ ( Base ‘ 𝑌 ) ) |
232 |
231 81
|
ffvelrnd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝐺 ‘ 0 ) ∈ ( Base ‘ 𝑌 ) ) |
233 |
|
eqid |
⊢ ( Scalar ‘ 𝑌 ) = ( Scalar ‘ 𝑌 ) |
234 |
|
eqid |
⊢ ( 1r ‘ ( Scalar ‘ 𝑌 ) ) = ( 1r ‘ ( Scalar ‘ 𝑌 ) ) |
235 |
14 233 11 234
|
lmodvs1 |
⊢ ( ( 𝑌 ∈ LMod ∧ ( 𝐺 ‘ 0 ) ∈ ( Base ‘ 𝑌 ) ) → ( ( 1r ‘ ( Scalar ‘ 𝑌 ) ) · ( 𝐺 ‘ 0 ) ) = ( 𝐺 ‘ 0 ) ) |
236 |
229 232 235
|
syl2an2r |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 1r ‘ ( Scalar ‘ 𝑌 ) ) · ( 𝐺 ‘ 0 ) ) = ( 𝐺 ‘ 0 ) ) |
237 |
|
iftrue |
⊢ ( 𝑛 = 0 → 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 − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) |
238 |
|
ovexd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 0 − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ∈ V ) |
239 |
9 237 81 238
|
fvmptd3 |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝐺 ‘ 0 ) = ( 0 − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) |
240 |
226 236 239
|
3eqtrd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) = ( 0 − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) |
241 |
207 240
|
oveq12d |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) + ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) ) = ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) + ( 0 − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) ) |
242 |
14 13
|
cmncom |
⊢ ( ( 𝑌 ∈ CMnd ∧ ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) ∈ ( Base ‘ 𝑌 ) ∧ ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) ∈ ( Base ‘ 𝑌 ) ) → ( ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) + ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) ) = ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) + ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) ) ) |
243 |
22 83 96 242
|
syl3anc |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) + ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) ) = ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) + ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) ) ) |
244 |
|
ringgrp |
⊢ ( 𝑌 ∈ Ring → 𝑌 ∈ Grp ) |
245 |
19 244
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑌 ∈ Grp ) |
246 |
245
|
adantr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝑌 ∈ Grp ) |
247 |
207 96
|
eqeltrrd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) ∈ ( Base ‘ 𝑌 ) ) |
248 |
19
|
adantr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝑌 ∈ Ring ) |
249 |
8 1 2 3 4
|
mat2pmatbas |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ( 𝑇 ‘ 𝑀 ) ∈ ( Base ‘ 𝑌 ) ) |
250 |
15 249
|
syl3an2 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 𝑇 ‘ 𝑀 ) ∈ ( Base ‘ 𝑌 ) ) |
251 |
250
|
adantr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑇 ‘ 𝑀 ) ∈ ( Base ‘ 𝑌 ) ) |
252 |
|
simpl1 |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝑁 ∈ Fin ) |
253 |
208
|
adantr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝑅 ∈ Ring ) |
254 |
|
elmapi |
⊢ ( 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) → 𝑏 : ( 0 ... 𝑠 ) ⟶ 𝐵 ) |
255 |
254
|
adantl |
⊢ ( ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → 𝑏 : ( 0 ... 𝑠 ) ⟶ 𝐵 ) |
256 |
255
|
adantl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝑏 : ( 0 ... 𝑠 ) ⟶ 𝐵 ) |
257 |
|
0elfz |
⊢ ( 𝑠 ∈ ℕ0 → 0 ∈ ( 0 ... 𝑠 ) ) |
258 |
34 257
|
syl |
⊢ ( 𝑠 ∈ ℕ → 0 ∈ ( 0 ... 𝑠 ) ) |
259 |
258
|
ad2antrl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 0 ∈ ( 0 ... 𝑠 ) ) |
260 |
256 259
|
ffvelrnd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑏 ‘ 0 ) ∈ 𝐵 ) |
261 |
8 1 2 3 4
|
mat2pmatbas |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ ( 𝑏 ‘ 0 ) ∈ 𝐵 ) → ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ∈ ( Base ‘ 𝑌 ) ) |
262 |
252 253 260 261
|
syl3anc |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ∈ ( Base ‘ 𝑌 ) ) |
263 |
14 5
|
ringcl |
⊢ ( ( 𝑌 ∈ Ring ∧ ( 𝑇 ‘ 𝑀 ) ∈ ( Base ‘ 𝑌 ) ∧ ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ∈ ( Base ‘ 𝑌 ) ) → ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ∈ ( Base ‘ 𝑌 ) ) |
264 |
248 251 262 263
|
syl3anc |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ∈ ( Base ‘ 𝑌 ) ) |
265 |
14 7 6 13
|
grpsubadd0sub |
⊢ ( ( 𝑌 ∈ Grp ∧ ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) ∈ ( Base ‘ 𝑌 ) ∧ ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ∈ ( Base ‘ 𝑌 ) ) → ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) = ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) + ( 0 − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) ) |
266 |
246 247 264 265
|
syl3anc |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) = ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) + ( 0 − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) ) |
267 |
241 243 266
|
3eqtr4d |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) + ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) ) = ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) |
268 |
189 267
|
oveq12d |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + ( ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) + ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) ) + ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) ) |
269 |
113 268
|
eqtrd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) + ( ( 0 ↑ 𝑋 ) · ( 𝐺 ‘ 0 ) ) ) + ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝐺 ‘ ( 𝑠 + 1 ) ) ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) ) + ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) ) |
270 |
75 102 269
|
3eqtrd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ ( 0 ... ( 𝑠 + 1 ) ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) ) + ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) ) |
271 |
40 73 270
|
3eqtrd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) ) + ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) ) |