Step |
Hyp |
Ref |
Expression |
1 |
|
telgsumfzs.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
telgsumfzs.g |
⊢ ( 𝜑 → 𝐺 ∈ Abel ) |
3 |
|
telgsumfzs.m |
⊢ − = ( -g ‘ 𝐺 ) |
4 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
5 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) → 𝐺 ∈ Abel ) |
6 |
|
ablcmn |
⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ CMnd ) |
7 |
5 6
|
syl |
⊢ ( ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) → 𝐺 ∈ CMnd ) |
8 |
7
|
adantl |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) → 𝐺 ∈ CMnd ) |
9 |
|
fzfid |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) → ( 𝑀 ... ( 𝑦 + 1 ) ) ∈ Fin ) |
10 |
|
ablgrp |
⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp ) |
11 |
2 10
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
12 |
11
|
ad2antrl |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) → 𝐺 ∈ Grp ) |
13 |
12
|
adantr |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) ∧ 𝑖 ∈ ( 𝑀 ... ( 𝑦 + 1 ) ) ) → 𝐺 ∈ Grp ) |
14 |
|
fzelp1 |
⊢ ( 𝑖 ∈ ( 𝑀 ... ( 𝑦 + 1 ) ) → 𝑖 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) ) |
15 |
|
simpr |
⊢ ( ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) → ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) |
16 |
15
|
adantl |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) → ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) |
17 |
|
rspcsbela |
⊢ ( ( 𝑖 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) → ⦋ 𝑖 / 𝑘 ⦌ 𝐶 ∈ 𝐵 ) |
18 |
14 16 17
|
syl2anr |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) ∧ 𝑖 ∈ ( 𝑀 ... ( 𝑦 + 1 ) ) ) → ⦋ 𝑖 / 𝑘 ⦌ 𝐶 ∈ 𝐵 ) |
19 |
|
fzp1elp1 |
⊢ ( 𝑖 ∈ ( 𝑀 ... ( 𝑦 + 1 ) ) → ( 𝑖 + 1 ) ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) ) |
20 |
|
rspcsbela |
⊢ ( ( ( 𝑖 + 1 ) ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) → ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ∈ 𝐵 ) |
21 |
19 16 20
|
syl2anr |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) ∧ 𝑖 ∈ ( 𝑀 ... ( 𝑦 + 1 ) ) ) → ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ∈ 𝐵 ) |
22 |
1 3
|
grpsubcl |
⊢ ( ( 𝐺 ∈ Grp ∧ ⦋ 𝑖 / 𝑘 ⦌ 𝐶 ∈ 𝐵 ∧ ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ∈ 𝐵 ) → ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ∈ 𝐵 ) |
23 |
13 18 21 22
|
syl3anc |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) ∧ 𝑖 ∈ ( 𝑀 ... ( 𝑦 + 1 ) ) ) → ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ∈ 𝐵 ) |
24 |
|
fzp1disj |
⊢ ( ( 𝑀 ... 𝑦 ) ∩ { ( 𝑦 + 1 ) } ) = ∅ |
25 |
24
|
a1i |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) → ( ( 𝑀 ... 𝑦 ) ∩ { ( 𝑦 + 1 ) } ) = ∅ ) |
26 |
|
fzsuc |
⊢ ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑀 ... ( 𝑦 + 1 ) ) = ( ( 𝑀 ... 𝑦 ) ∪ { ( 𝑦 + 1 ) } ) ) |
27 |
26
|
adantr |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) → ( 𝑀 ... ( 𝑦 + 1 ) ) = ( ( 𝑀 ... 𝑦 ) ∪ { ( 𝑦 + 1 ) } ) ) |
28 |
1 4 8 9 23 25 27
|
gsummptfidmsplit |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) → ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... ( 𝑦 + 1 ) ) ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) = ( ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... 𝑦 ) ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg ( 𝑖 ∈ { ( 𝑦 + 1 ) } ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) ) ) |
29 |
28
|
adantr |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) ∧ ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... 𝑦 ) ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) = ( ⦋ 𝑀 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 ) ) → ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... ( 𝑦 + 1 ) ) ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) = ( ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... 𝑦 ) ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg ( 𝑖 ∈ { ( 𝑦 + 1 ) } ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) ) ) |
30 |
|
simpr |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) ∧ ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... 𝑦 ) ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) = ( ⦋ 𝑀 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 ) ) → ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... 𝑦 ) ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) = ( ⦋ 𝑀 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 ) ) |
31 |
|
grpmnd |
⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd ) |
32 |
11 31
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ Mnd ) |
33 |
32
|
ad2antrl |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) → 𝐺 ∈ Mnd ) |
34 |
|
ovexd |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) → ( 𝑦 + 1 ) ∈ V ) |
35 |
|
peano2uz |
⊢ ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑦 + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) ) |
36 |
|
eluzfz2 |
⊢ ( ( 𝑦 + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑦 + 1 ) ∈ ( 𝑀 ... ( 𝑦 + 1 ) ) ) |
37 |
35 36
|
syl |
⊢ ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑦 + 1 ) ∈ ( 𝑀 ... ( 𝑦 + 1 ) ) ) |
38 |
|
fzelp1 |
⊢ ( ( 𝑦 + 1 ) ∈ ( 𝑀 ... ( 𝑦 + 1 ) ) → ( 𝑦 + 1 ) ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) ) |
39 |
37 38
|
syl |
⊢ ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑦 + 1 ) ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) ) |
40 |
|
rspcsbela |
⊢ ( ( ( 𝑦 + 1 ) ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) → ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 ∈ 𝐵 ) |
41 |
39 15 40
|
syl2an |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) → ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 ∈ 𝐵 ) |
42 |
|
peano2uz |
⊢ ( ( 𝑦 + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) ) |
43 |
35 42
|
syl |
⊢ ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) ) |
44 |
|
eluzfz2 |
⊢ ( ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) ) |
45 |
43 44
|
syl |
⊢ ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) ) |
46 |
|
rspcsbela |
⊢ ( ( ( ( 𝑦 + 1 ) + 1 ) ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) → ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ∈ 𝐵 ) |
47 |
45 15 46
|
syl2an |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) → ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ∈ 𝐵 ) |
48 |
1 3
|
grpsubcl |
⊢ ( ( 𝐺 ∈ Grp ∧ ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 ∈ 𝐵 ∧ ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ∈ 𝐵 ) → ( ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 − ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ) ∈ 𝐵 ) |
49 |
12 41 47 48
|
syl3anc |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) → ( ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 − ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ) ∈ 𝐵 ) |
50 |
|
csbeq1 |
⊢ ( 𝑖 = ( 𝑦 + 1 ) → ⦋ 𝑖 / 𝑘 ⦌ 𝐶 = ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 ) |
51 |
|
oveq1 |
⊢ ( 𝑖 = ( 𝑦 + 1 ) → ( 𝑖 + 1 ) = ( ( 𝑦 + 1 ) + 1 ) ) |
52 |
51
|
csbeq1d |
⊢ ( 𝑖 = ( 𝑦 + 1 ) → ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 = ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ) |
53 |
50 52
|
oveq12d |
⊢ ( 𝑖 = ( 𝑦 + 1 ) → ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) = ( ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 − ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ) ) |
54 |
53
|
adantl |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) ∧ 𝑖 = ( 𝑦 + 1 ) ) → ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) = ( ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 − ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ) ) |
55 |
1 33 34 49 54
|
gsumsnd |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) → ( 𝐺 Σg ( 𝑖 ∈ { ( 𝑦 + 1 ) } ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) = ( ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 − ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ) ) |
56 |
55
|
adantr |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) ∧ ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... 𝑦 ) ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) = ( ⦋ 𝑀 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 ) ) → ( 𝐺 Σg ( 𝑖 ∈ { ( 𝑦 + 1 ) } ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) = ( ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 − ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ) ) |
57 |
30 56
|
oveq12d |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) ∧ ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... 𝑦 ) ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) = ( ⦋ 𝑀 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 ) ) → ( ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... 𝑦 ) ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg ( 𝑖 ∈ { ( 𝑦 + 1 ) } ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) ) = ( ( ⦋ 𝑀 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 ) ( +g ‘ 𝐺 ) ( ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 − ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) |
58 |
|
eluzfz1 |
⊢ ( ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) ) |
59 |
43 58
|
syl |
⊢ ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) ) |
60 |
|
rspcsbela |
⊢ ( ( 𝑀 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) → ⦋ 𝑀 / 𝑘 ⦌ 𝐶 ∈ 𝐵 ) |
61 |
59 15 60
|
syl2an |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) → ⦋ 𝑀 / 𝑘 ⦌ 𝐶 ∈ 𝐵 ) |
62 |
1 4 3
|
grpnpncan |
⊢ ( ( 𝐺 ∈ Grp ∧ ( ⦋ 𝑀 / 𝑘 ⦌ 𝐶 ∈ 𝐵 ∧ ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 ∈ 𝐵 ∧ ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ∈ 𝐵 ) ) → ( ( ⦋ 𝑀 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 ) ( +g ‘ 𝐺 ) ( ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 − ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ) ) = ( ⦋ 𝑀 / 𝑘 ⦌ 𝐶 − ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ) ) |
63 |
12 61 41 47 62
|
syl13anc |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) → ( ( ⦋ 𝑀 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 ) ( +g ‘ 𝐺 ) ( ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 − ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ) ) = ( ⦋ 𝑀 / 𝑘 ⦌ 𝐶 − ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ) ) |
64 |
63
|
adantr |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) ∧ ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... 𝑦 ) ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) = ( ⦋ 𝑀 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 ) ) → ( ( ⦋ 𝑀 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 ) ( +g ‘ 𝐺 ) ( ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 − ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ) ) = ( ⦋ 𝑀 / 𝑘 ⦌ 𝐶 − ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ) ) |
65 |
29 57 64
|
3eqtrd |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) ∧ ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... 𝑦 ) ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) = ( ⦋ 𝑀 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 ) ) → ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... ( 𝑦 + 1 ) ) ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) = ( ⦋ 𝑀 / 𝑘 ⦌ 𝐶 − ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ) ) |
66 |
65
|
ex |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝜑 ∧ ∀ 𝑘 ∈ ( 𝑀 ... ( ( 𝑦 + 1 ) + 1 ) ) 𝐶 ∈ 𝐵 ) ) → ( ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... 𝑦 ) ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) = ( ⦋ 𝑀 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑦 + 1 ) / 𝑘 ⦌ 𝐶 ) → ( 𝐺 Σg ( 𝑖 ∈ ( 𝑀 ... ( 𝑦 + 1 ) ) ↦ ( ⦋ 𝑖 / 𝑘 ⦌ 𝐶 − ⦋ ( 𝑖 + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) = ( ⦋ 𝑀 / 𝑘 ⦌ 𝐶 − ⦋ ( ( 𝑦 + 1 ) + 1 ) / 𝑘 ⦌ 𝐶 ) ) ) |