Step |
Hyp |
Ref |
Expression |
1 |
|
seqf1o.1 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 ) |
2 |
|
seqf1o.2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) |
3 |
|
seqf1o.3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) |
4 |
|
seqf1o.4 |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
5 |
|
seqf1o.5 |
⊢ ( 𝜑 → 𝐶 ⊆ 𝑆 ) |
6 |
|
seqf1olem2a.1 |
⊢ ( 𝜑 → 𝐺 : 𝐴 ⟶ 𝐶 ) |
7 |
|
seqf1olem2a.3 |
⊢ ( 𝜑 → 𝐾 ∈ 𝐴 ) |
8 |
|
seqf1olem2a.4 |
⊢ ( 𝜑 → ( 𝑀 ... 𝑁 ) ⊆ 𝐴 ) |
9 |
|
eluzfz2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ( 𝑀 ... 𝑁 ) ) |
10 |
4 9
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ ( 𝑀 ... 𝑁 ) ) |
11 |
|
fveq2 |
⊢ ( 𝑚 = 𝑀 → ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) = ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑀 ) ) |
12 |
11
|
oveq2d |
⊢ ( 𝑚 = 𝑀 → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) = ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑀 ) ) ) |
13 |
11
|
oveq1d |
⊢ ( 𝑚 = 𝑀 → ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) + ( 𝐺 ‘ 𝐾 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑀 ) + ( 𝐺 ‘ 𝐾 ) ) ) |
14 |
12 13
|
eqeq12d |
⊢ ( 𝑚 = 𝑀 → ( ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) + ( 𝐺 ‘ 𝐾 ) ) ↔ ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑀 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑀 ) + ( 𝐺 ‘ 𝐾 ) ) ) ) |
15 |
14
|
imbi2d |
⊢ ( 𝑚 = 𝑀 → ( ( 𝜑 → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) + ( 𝐺 ‘ 𝐾 ) ) ) ↔ ( 𝜑 → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑀 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑀 ) + ( 𝐺 ‘ 𝐾 ) ) ) ) ) |
16 |
|
fveq2 |
⊢ ( 𝑚 = 𝑛 → ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) = ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) |
17 |
16
|
oveq2d |
⊢ ( 𝑚 = 𝑛 → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) = ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) ) |
18 |
16
|
oveq1d |
⊢ ( 𝑚 = 𝑛 → ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) + ( 𝐺 ‘ 𝐾 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ 𝐾 ) ) ) |
19 |
17 18
|
eqeq12d |
⊢ ( 𝑚 = 𝑛 → ( ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) + ( 𝐺 ‘ 𝐾 ) ) ↔ ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ 𝐾 ) ) ) ) |
20 |
19
|
imbi2d |
⊢ ( 𝑚 = 𝑛 → ( ( 𝜑 → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) + ( 𝐺 ‘ 𝐾 ) ) ) ↔ ( 𝜑 → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ 𝐾 ) ) ) ) ) |
21 |
|
fveq2 |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) = ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) ) |
22 |
21
|
oveq2d |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) = ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) ) ) |
23 |
21
|
oveq1d |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) + ( 𝐺 ‘ 𝐾 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) + ( 𝐺 ‘ 𝐾 ) ) ) |
24 |
22 23
|
eqeq12d |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) + ( 𝐺 ‘ 𝐾 ) ) ↔ ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) + ( 𝐺 ‘ 𝐾 ) ) ) ) |
25 |
24
|
imbi2d |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( ( 𝜑 → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) + ( 𝐺 ‘ 𝐾 ) ) ) ↔ ( 𝜑 → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) + ( 𝐺 ‘ 𝐾 ) ) ) ) ) |
26 |
|
fveq2 |
⊢ ( 𝑚 = 𝑁 → ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) = ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) ) |
27 |
26
|
oveq2d |
⊢ ( 𝑚 = 𝑁 → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) = ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) ) ) |
28 |
26
|
oveq1d |
⊢ ( 𝑚 = 𝑁 → ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) + ( 𝐺 ‘ 𝐾 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) + ( 𝐺 ‘ 𝐾 ) ) ) |
29 |
27 28
|
eqeq12d |
⊢ ( 𝑚 = 𝑁 → ( ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) + ( 𝐺 ‘ 𝐾 ) ) ↔ ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) + ( 𝐺 ‘ 𝐾 ) ) ) ) |
30 |
29
|
imbi2d |
⊢ ( 𝑚 = 𝑁 → ( ( 𝜑 → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) + ( 𝐺 ‘ 𝐾 ) ) ) ↔ ( 𝜑 → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) + ( 𝐺 ‘ 𝐾 ) ) ) ) ) |
31 |
6 7
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝐾 ) ∈ 𝐶 ) |
32 |
|
eluzel2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ ) |
33 |
|
seq1 |
⊢ ( 𝑀 ∈ ℤ → ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑀 ) = ( 𝐺 ‘ 𝑀 ) ) |
34 |
4 32 33
|
3syl |
⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑀 ) = ( 𝐺 ‘ 𝑀 ) ) |
35 |
|
eluzfz1 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ( 𝑀 ... 𝑁 ) ) |
36 |
4 35
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ( 𝑀 ... 𝑁 ) ) |
37 |
8 36
|
sseldd |
⊢ ( 𝜑 → 𝑀 ∈ 𝐴 ) |
38 |
6 37
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑀 ) ∈ 𝐶 ) |
39 |
34 38
|
eqeltrd |
⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑀 ) ∈ 𝐶 ) |
40 |
2 31 39
|
caovcomd |
⊢ ( 𝜑 → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑀 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑀 ) + ( 𝐺 ‘ 𝐾 ) ) ) |
41 |
40
|
a1i |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝜑 → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑀 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑀 ) + ( 𝐺 ‘ 𝐾 ) ) ) ) |
42 |
|
oveq1 |
⊢ ( ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ 𝐾 ) ) → ( ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) = ( ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ 𝐾 ) ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) |
43 |
|
elfzouz |
⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
44 |
43
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
45 |
|
seqp1 |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) → ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) |
46 |
44 45
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) |
47 |
46
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) ) = ( ( 𝐺 ‘ 𝐾 ) + ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) ) |
48 |
3
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) |
49 |
5 31
|
sseldd |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝐾 ) ∈ 𝑆 ) |
50 |
49
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐺 ‘ 𝐾 ) ∈ 𝑆 ) |
51 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝐶 ⊆ 𝑆 ) |
52 |
51
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑛 ) ) → 𝐶 ⊆ 𝑆 ) |
53 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝐺 : 𝐴 ⟶ 𝐶 ) |
54 |
53
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑛 ) ) → 𝐺 : 𝐴 ⟶ 𝐶 ) |
55 |
|
elfzouz2 |
⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑛 ) ) |
56 |
55
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑛 ) ) |
57 |
|
fzss2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑛 ) → ( 𝑀 ... 𝑛 ) ⊆ ( 𝑀 ... 𝑁 ) ) |
58 |
56 57
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝑀 ... 𝑛 ) ⊆ ( 𝑀 ... 𝑁 ) ) |
59 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝑀 ... 𝑁 ) ⊆ 𝐴 ) |
60 |
58 59
|
sstrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝑀 ... 𝑛 ) ⊆ 𝐴 ) |
61 |
60
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑛 ) ) → 𝑥 ∈ 𝐴 ) |
62 |
54 61
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑛 ) ) → ( 𝐺 ‘ 𝑥 ) ∈ 𝐶 ) |
63 |
52 62
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑛 ) ) → ( 𝐺 ‘ 𝑥 ) ∈ 𝑆 ) |
64 |
1
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 ) |
65 |
44 63 64
|
seqcl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ 𝑆 ) |
66 |
|
fzofzp1 |
⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) |
67 |
66
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) |
68 |
59 67
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝑛 + 1 ) ∈ 𝐴 ) |
69 |
53 68
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐺 ‘ ( 𝑛 + 1 ) ) ∈ 𝐶 ) |
70 |
51 69
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐺 ‘ ( 𝑛 + 1 ) ) ∈ 𝑆 ) |
71 |
48 50 65 70
|
caovassd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) = ( ( 𝐺 ‘ 𝐾 ) + ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) ) |
72 |
47 71
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) ) = ( ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) |
73 |
48 65 70 50
|
caovassd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) + ( 𝐺 ‘ 𝐾 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( ( 𝐺 ‘ ( 𝑛 + 1 ) ) + ( 𝐺 ‘ 𝐾 ) ) ) ) |
74 |
46
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) + ( 𝐺 ‘ 𝐾 ) ) = ( ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) + ( 𝐺 ‘ 𝐾 ) ) ) |
75 |
48 65 50 70
|
caovassd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ 𝐾 ) ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( ( 𝐺 ‘ 𝐾 ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) ) |
76 |
2
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) |
77 |
31
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐺 ‘ 𝐾 ) ∈ 𝐶 ) |
78 |
76 69 77
|
caovcomd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( 𝐺 ‘ ( 𝑛 + 1 ) ) + ( 𝐺 ‘ 𝐾 ) ) = ( ( 𝐺 ‘ 𝐾 ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) |
79 |
78
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( ( 𝐺 ‘ ( 𝑛 + 1 ) ) + ( 𝐺 ‘ 𝐾 ) ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( ( 𝐺 ‘ 𝐾 ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) ) |
80 |
75 79
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ 𝐾 ) ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( ( 𝐺 ‘ ( 𝑛 + 1 ) ) + ( 𝐺 ‘ 𝐾 ) ) ) ) |
81 |
73 74 80
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) + ( 𝐺 ‘ 𝐾 ) ) = ( ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ 𝐾 ) ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) |
82 |
72 81
|
eqeq12d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) + ( 𝐺 ‘ 𝐾 ) ) ↔ ( ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) = ( ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ 𝐾 ) ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) ) |
83 |
42 82
|
syl5ibr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ 𝐾 ) ) → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) + ( 𝐺 ‘ 𝐾 ) ) ) ) |
84 |
83
|
expcom |
⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → ( 𝜑 → ( ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ 𝐾 ) ) → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) + ( 𝐺 ‘ 𝐾 ) ) ) ) ) |
85 |
84
|
a2d |
⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → ( ( 𝜑 → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ 𝐾 ) ) ) → ( 𝜑 → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ ( 𝑛 + 1 ) ) + ( 𝐺 ‘ 𝐾 ) ) ) ) ) |
86 |
15 20 25 30 41 85
|
fzind2 |
⊢ ( 𝑁 ∈ ( 𝑀 ... 𝑁 ) → ( 𝜑 → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) + ( 𝐺 ‘ 𝐾 ) ) ) ) |
87 |
10 86
|
mpcom |
⊢ ( 𝜑 → ( ( 𝐺 ‘ 𝐾 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) + ( 𝐺 ‘ 𝐾 ) ) ) |