Step |
Hyp |
Ref |
Expression |
1 |
|
seqsplit.1 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 ) |
2 |
|
seqsplit.2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) |
3 |
|
seqsplit.3 |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) |
4 |
|
seqsplit.4 |
⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ 𝐾 ) ) |
5 |
|
seqsplit.5 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐾 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 ) |
6 |
|
eluzfz2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) → 𝑁 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) |
7 |
3 6
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) |
8 |
|
eleq1 |
⊢ ( 𝑥 = ( 𝑀 + 1 ) → ( 𝑥 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ↔ ( 𝑀 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
9 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝑀 + 1 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑥 ) = ( seq 𝐾 ( + , 𝐹 ) ‘ ( 𝑀 + 1 ) ) ) |
10 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝑀 + 1 ) → ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) = ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑀 + 1 ) ) ) |
11 |
10
|
oveq2d |
⊢ ( 𝑥 = ( 𝑀 + 1 ) → ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑀 + 1 ) ) ) ) |
12 |
9 11
|
eqeq12d |
⊢ ( 𝑥 = ( 𝑀 + 1 ) → ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑥 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) ) ↔ ( seq 𝐾 ( + , 𝐹 ) ‘ ( 𝑀 + 1 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑀 + 1 ) ) ) ) ) |
13 |
8 12
|
imbi12d |
⊢ ( 𝑥 = ( 𝑀 + 1 ) → ( ( 𝑥 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑥 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) ) ) ↔ ( ( 𝑀 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ ( 𝑀 + 1 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑀 + 1 ) ) ) ) ) ) |
14 |
13
|
imbi2d |
⊢ ( 𝑥 = ( 𝑀 + 1 ) → ( ( 𝜑 → ( 𝑥 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑥 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) ) ) ) ↔ ( 𝜑 → ( ( 𝑀 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ ( 𝑀 + 1 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑀 + 1 ) ) ) ) ) ) ) |
15 |
|
eleq1 |
⊢ ( 𝑥 = 𝑛 → ( 𝑥 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ↔ 𝑛 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
16 |
|
fveq2 |
⊢ ( 𝑥 = 𝑛 → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑥 ) = ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑛 ) ) |
17 |
|
fveq2 |
⊢ ( 𝑥 = 𝑛 → ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) = ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) ) |
18 |
17
|
oveq2d |
⊢ ( 𝑥 = 𝑛 → ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) ) ) |
19 |
16 18
|
eqeq12d |
⊢ ( 𝑥 = 𝑛 → ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑥 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) ) ↔ ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑛 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) ) ) ) |
20 |
15 19
|
imbi12d |
⊢ ( 𝑥 = 𝑛 → ( ( 𝑥 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑥 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) ) ) ↔ ( 𝑛 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑛 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) ) ) ) ) |
21 |
20
|
imbi2d |
⊢ ( 𝑥 = 𝑛 → ( ( 𝜑 → ( 𝑥 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑥 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) ) ) ) ↔ ( 𝜑 → ( 𝑛 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑛 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) ) ) ) ) ) |
22 |
|
eleq1 |
⊢ ( 𝑥 = ( 𝑛 + 1 ) → ( 𝑥 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ↔ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
23 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝑛 + 1 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑥 ) = ( seq 𝐾 ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) |
24 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝑛 + 1 ) → ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) = ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) |
25 |
24
|
oveq2d |
⊢ ( 𝑥 = ( 𝑛 + 1 ) → ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) ) |
26 |
23 25
|
eqeq12d |
⊢ ( 𝑥 = ( 𝑛 + 1 ) → ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑥 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) ) ↔ ( seq 𝐾 ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) ) ) |
27 |
22 26
|
imbi12d |
⊢ ( 𝑥 = ( 𝑛 + 1 ) → ( ( 𝑥 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑥 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) ) ) ↔ ( ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) ) ) ) |
28 |
27
|
imbi2d |
⊢ ( 𝑥 = ( 𝑛 + 1 ) → ( ( 𝜑 → ( 𝑥 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑥 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) ) ) ) ↔ ( 𝜑 → ( ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) ) ) ) ) |
29 |
|
eleq1 |
⊢ ( 𝑥 = 𝑁 → ( 𝑥 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ↔ 𝑁 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
30 |
|
fveq2 |
⊢ ( 𝑥 = 𝑁 → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑥 ) = ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑁 ) ) |
31 |
|
fveq2 |
⊢ ( 𝑥 = 𝑁 → ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) = ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑁 ) ) |
32 |
31
|
oveq2d |
⊢ ( 𝑥 = 𝑁 → ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑁 ) ) ) |
33 |
30 32
|
eqeq12d |
⊢ ( 𝑥 = 𝑁 → ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑥 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) ) ↔ ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑁 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑁 ) ) ) ) |
34 |
29 33
|
imbi12d |
⊢ ( 𝑥 = 𝑁 → ( ( 𝑥 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑥 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) ) ) ↔ ( 𝑁 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑁 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑁 ) ) ) ) ) |
35 |
34
|
imbi2d |
⊢ ( 𝑥 = 𝑁 → ( ( 𝜑 → ( 𝑥 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑥 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑥 ) ) ) ) ↔ ( 𝜑 → ( 𝑁 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑁 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑁 ) ) ) ) ) ) |
36 |
|
seqp1 |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 𝐾 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ ( 𝑀 + 1 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( 𝐹 ‘ ( 𝑀 + 1 ) ) ) ) |
37 |
4 36
|
syl |
⊢ ( 𝜑 → ( seq 𝐾 ( + , 𝐹 ) ‘ ( 𝑀 + 1 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( 𝐹 ‘ ( 𝑀 + 1 ) ) ) ) |
38 |
|
eluzel2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) → ( 𝑀 + 1 ) ∈ ℤ ) |
39 |
|
seq1 |
⊢ ( ( 𝑀 + 1 ) ∈ ℤ → ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑀 + 1 ) ) = ( 𝐹 ‘ ( 𝑀 + 1 ) ) ) |
40 |
3 38 39
|
3syl |
⊢ ( 𝜑 → ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑀 + 1 ) ) = ( 𝐹 ‘ ( 𝑀 + 1 ) ) ) |
41 |
40
|
oveq2d |
⊢ ( 𝜑 → ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑀 + 1 ) ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( 𝐹 ‘ ( 𝑀 + 1 ) ) ) ) |
42 |
37 41
|
eqtr4d |
⊢ ( 𝜑 → ( seq 𝐾 ( + , 𝐹 ) ‘ ( 𝑀 + 1 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑀 + 1 ) ) ) ) |
43 |
42
|
a1i13 |
⊢ ( ( 𝑀 + 1 ) ∈ ℤ → ( 𝜑 → ( ( 𝑀 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ ( 𝑀 + 1 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑀 + 1 ) ) ) ) ) ) |
44 |
|
peano2fzr |
⊢ ( ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) → 𝑛 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) |
45 |
44
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → 𝑛 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) |
46 |
45
|
expr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → 𝑛 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
47 |
46
|
imim1d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( ( 𝑛 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑛 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) ) ) → ( ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑛 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) ) ) ) ) |
48 |
|
oveq1 |
⊢ ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑛 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) ) → ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) = ( ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) |
49 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) |
50 |
|
peano2uz |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 𝐾 ) → ( 𝑀 + 1 ) ∈ ( ℤ≥ ‘ 𝐾 ) ) |
51 |
4 50
|
syl |
⊢ ( 𝜑 → ( 𝑀 + 1 ) ∈ ( ℤ≥ ‘ 𝐾 ) ) |
52 |
51
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( 𝑀 + 1 ) ∈ ( ℤ≥ ‘ 𝐾 ) ) |
53 |
|
uztrn |
⊢ ( ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑀 + 1 ) ∈ ( ℤ≥ ‘ 𝐾 ) ) → 𝑛 ∈ ( ℤ≥ ‘ 𝐾 ) ) |
54 |
49 52 53
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → 𝑛 ∈ ( ℤ≥ ‘ 𝐾 ) ) |
55 |
|
seqp1 |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝐾 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) |
56 |
54 55
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( seq 𝐾 ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) |
57 |
|
seqp1 |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) → ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) = ( ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) |
58 |
49 57
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) = ( ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) |
59 |
58
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) |
60 |
|
simpl |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → 𝜑 ) |
61 |
|
eluzelz |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 𝐾 ) → 𝑀 ∈ ℤ ) |
62 |
4 61
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
63 |
|
peano2uzr |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
64 |
62 3 63
|
syl2anc |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
65 |
|
fzss2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝐾 ... 𝑀 ) ⊆ ( 𝐾 ... 𝑁 ) ) |
66 |
64 65
|
syl |
⊢ ( 𝜑 → ( 𝐾 ... 𝑀 ) ⊆ ( 𝐾 ... 𝑁 ) ) |
67 |
66
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐾 ... 𝑀 ) ) → 𝑥 ∈ ( 𝐾 ... 𝑁 ) ) |
68 |
67 5
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐾 ... 𝑀 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 ) |
69 |
4 68 1
|
seqcl |
⊢ ( 𝜑 → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) ∈ 𝑆 ) |
70 |
69
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) ∈ 𝑆 ) |
71 |
|
elfzuz3 |
⊢ ( 𝑛 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑛 ) ) |
72 |
|
fzss2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑛 ) → ( ( 𝑀 + 1 ) ... 𝑛 ) ⊆ ( ( 𝑀 + 1 ) ... 𝑁 ) ) |
73 |
45 71 72
|
3syl |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( ( 𝑀 + 1 ) ... 𝑛 ) ⊆ ( ( 𝑀 + 1 ) ... 𝑁 ) ) |
74 |
|
fzss1 |
⊢ ( ( 𝑀 + 1 ) ∈ ( ℤ≥ ‘ 𝐾 ) → ( ( 𝑀 + 1 ) ... 𝑁 ) ⊆ ( 𝐾 ... 𝑁 ) ) |
75 |
4 50 74
|
3syl |
⊢ ( 𝜑 → ( ( 𝑀 + 1 ) ... 𝑁 ) ⊆ ( 𝐾 ... 𝑁 ) ) |
76 |
75
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( ( 𝑀 + 1 ) ... 𝑁 ) ⊆ ( 𝐾 ... 𝑁 ) ) |
77 |
73 76
|
sstrd |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( ( 𝑀 + 1 ) ... 𝑛 ) ⊆ ( 𝐾 ... 𝑁 ) ) |
78 |
77
|
sselda |
⊢ ( ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ∧ 𝑥 ∈ ( ( 𝑀 + 1 ) ... 𝑛 ) ) → 𝑥 ∈ ( 𝐾 ... 𝑁 ) ) |
79 |
5
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ∧ 𝑥 ∈ ( 𝐾 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 ) |
80 |
78 79
|
syldan |
⊢ ( ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ∧ 𝑥 ∈ ( ( 𝑀 + 1 ) ... 𝑛 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 ) |
81 |
1
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 ) |
82 |
49 80 81
|
seqcl |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) ∈ 𝑆 ) |
83 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝑛 + 1 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) |
84 |
83
|
eleq1d |
⊢ ( 𝑥 = ( 𝑛 + 1 ) → ( ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 ↔ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∈ 𝑆 ) ) |
85 |
5
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ( 𝐾 ... 𝑁 ) ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 ) |
86 |
85
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ∀ 𝑥 ∈ ( 𝐾 ... 𝑁 ) ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 ) |
87 |
|
simpr |
⊢ ( ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) → ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) |
88 |
|
ssel2 |
⊢ ( ( ( ( 𝑀 + 1 ) ... 𝑁 ) ⊆ ( 𝐾 ... 𝑁 ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) → ( 𝑛 + 1 ) ∈ ( 𝐾 ... 𝑁 ) ) |
89 |
75 87 88
|
syl2an |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( 𝑛 + 1 ) ∈ ( 𝐾 ... 𝑁 ) ) |
90 |
84 86 89
|
rspcdva |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∈ 𝑆 ) |
91 |
2
|
caovassg |
⊢ ( ( 𝜑 ∧ ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) ∈ 𝑆 ∧ ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) ∈ 𝑆 ∧ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∈ 𝑆 ) ) → ( ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) |
92 |
60 70 82 90 91
|
syl13anc |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) |
93 |
59 92
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) = ( ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) |
94 |
56 93
|
eqeq12d |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( ( seq 𝐾 ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) ↔ ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) = ( ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) |
95 |
48 94
|
syl5ibr |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑛 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) ) → ( seq 𝐾 ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) ) ) |
96 |
47 95
|
animpimp2impd |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) → ( ( 𝜑 → ( 𝑛 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑛 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑛 ) ) ) ) → ( 𝜑 → ( ( 𝑛 + 1 ) ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) ) ) ) ) |
97 |
14 21 28 35 43 96
|
uzind4 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) → ( 𝜑 → ( 𝑁 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑁 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑁 ) ) ) ) ) |
98 |
3 97
|
mpcom |
⊢ ( 𝜑 → ( 𝑁 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑁 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑁 ) ) ) ) |
99 |
7 98
|
mpd |
⊢ ( 𝜑 → ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑁 ) = ( ( seq 𝐾 ( + , 𝐹 ) ‘ 𝑀 ) + ( seq ( 𝑀 + 1 ) ( + , 𝐹 ) ‘ 𝑁 ) ) ) |