Step | Hyp | Ref
| Expression |
1 | | seqsplit.3 |
. . 3
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) |
2 | | eluzfz2 12220 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘(𝑀 + 1)) → 𝑁 ∈ ((𝑀 + 1)...𝑁)) |
3 | 1, 2 | syl 17 |
. 2
⊢ (𝜑 → 𝑁 ∈ ((𝑀 + 1)...𝑁)) |
4 | | eleq1 2676 |
. . . . . 6
⊢ (𝑥 = (𝑀 + 1) → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ (𝑀 + 1) ∈ ((𝑀 + 1)...𝑁))) |
5 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑥 = (𝑀 + 1) → (seq𝐾( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐹)‘(𝑀 + 1))) |
6 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑥 = (𝑀 + 1) → (seq(𝑀 + 1)( + , 𝐹)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1))) |
7 | 6 | oveq2d 6565 |
. . . . . . 7
⊢ (𝑥 = (𝑀 + 1) → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)))) |
8 | 5, 7 | eqeq12d 2625 |
. . . . . 6
⊢ (𝑥 = (𝑀 + 1) → ((seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) ↔ (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1))))) |
9 | 4, 8 | imbi12d 333 |
. . . . 5
⊢ (𝑥 = (𝑀 + 1) → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥))) ↔ ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)))))) |
10 | 9 | imbi2d 329 |
. . . 4
⊢ (𝑥 = (𝑀 + 1) → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)))) ↔ (𝜑 → ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1))))))) |
11 | | eleq1 2676 |
. . . . . 6
⊢ (𝑥 = 𝑛 → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ 𝑛 ∈ ((𝑀 + 1)...𝑁))) |
12 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑥 = 𝑛 → (seq𝐾( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐹)‘𝑛)) |
13 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑥 = 𝑛 → (seq(𝑀 + 1)( + , 𝐹)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) |
14 | 13 | oveq2d 6565 |
. . . . . . 7
⊢ (𝑥 = 𝑛 → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛))) |
15 | 12, 14 | eqeq12d 2625 |
. . . . . 6
⊢ (𝑥 = 𝑛 → ((seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) ↔ (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)))) |
16 | 11, 15 | imbi12d 333 |
. . . . 5
⊢ (𝑥 = 𝑛 → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥))) ↔ (𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛))))) |
17 | 16 | imbi2d 329 |
. . . 4
⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)))) ↔ (𝜑 → (𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)))))) |
18 | | eleq1 2676 |
. . . . . 6
⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) |
19 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑥 = (𝑛 + 1) → (seq𝐾( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐹)‘(𝑛 + 1))) |
20 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑥 = (𝑛 + 1) → (seq(𝑀 + 1)( + , 𝐹)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))) |
21 | 20 | oveq2d 6565 |
. . . . . . 7
⊢ (𝑥 = (𝑛 + 1) → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)))) |
22 | 19, 21 | eqeq12d 2625 |
. . . . . 6
⊢ (𝑥 = (𝑛 + 1) → ((seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) ↔ (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))))) |
23 | 18, 22 | imbi12d 333 |
. . . . 5
⊢ (𝑥 = (𝑛 + 1) → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥))) ↔ ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)))))) |
24 | 23 | imbi2d 329 |
. . . 4
⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)))) ↔ (𝜑 → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))))))) |
25 | | eleq1 2676 |
. . . . . 6
⊢ (𝑥 = 𝑁 → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ 𝑁 ∈ ((𝑀 + 1)...𝑁))) |
26 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑥 = 𝑁 → (seq𝐾( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐹)‘𝑁)) |
27 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑥 = 𝑁 → (seq(𝑀 + 1)( + , 𝐹)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹)‘𝑁)) |
28 | 27 | oveq2d 6565 |
. . . . . . 7
⊢ (𝑥 = 𝑁 → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁))) |
29 | 26, 28 | eqeq12d 2625 |
. . . . . 6
⊢ (𝑥 = 𝑁 → ((seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) ↔ (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))) |
30 | 25, 29 | imbi12d 333 |
. . . . 5
⊢ (𝑥 = 𝑁 → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥))) ↔ (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁))))) |
31 | 30 | imbi2d 329 |
. . . 4
⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)))) ↔ (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))))) |
32 | | seqsplit.4 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐾)) |
33 | | seqp1 12678 |
. . . . . . . 8
⊢ (𝑀 ∈
(ℤ≥‘𝐾) → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (𝐹‘(𝑀 + 1)))) |
34 | 32, 33 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (𝐹‘(𝑀 + 1)))) |
35 | | eluzel2 11568 |
. . . . . . . . 9
⊢ (𝑁 ∈
(ℤ≥‘(𝑀 + 1)) → (𝑀 + 1) ∈ ℤ) |
36 | | seq1 12676 |
. . . . . . . . 9
⊢ ((𝑀 + 1) ∈ ℤ →
(seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)) = (𝐹‘(𝑀 + 1))) |
37 | 1, 35, 36 | 3syl 18 |
. . . . . . . 8
⊢ (𝜑 → (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)) = (𝐹‘(𝑀 + 1))) |
38 | 37 | oveq2d 6565 |
. . . . . . 7
⊢ (𝜑 → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1))) = ((seq𝐾( + , 𝐹)‘𝑀) + (𝐹‘(𝑀 + 1)))) |
39 | 34, 38 | eqtr4d 2647 |
. . . . . 6
⊢ (𝜑 → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)))) |
40 | 39 | a1d 25 |
. . . . 5
⊢ (𝜑 → ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1))))) |
41 | 40 | a1i 11 |
. . . 4
⊢ ((𝑀 + 1) ∈ ℤ →
(𝜑 → ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)))))) |
42 | | peano2fzr 12225 |
. . . . . . . . . 10
⊢ ((𝑛 ∈
(ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → 𝑛 ∈ ((𝑀 + 1)...𝑁)) |
43 | 42 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ ((𝑀 + 1)...𝑁)) |
44 | 43 | expr 641 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → 𝑛 ∈ ((𝑀 + 1)...𝑁))) |
45 | 44 | imim1d 80 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛))))) |
46 | | oveq1 6556 |
. . . . . . . . . 10
⊢
((seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) → ((seq𝐾( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) = (((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) + (𝐹‘(𝑛 + 1)))) |
47 | | simprl 790 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (ℤ≥‘(𝑀 + 1))) |
48 | | peano2uz 11617 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 ∈
(ℤ≥‘𝐾) → (𝑀 + 1) ∈
(ℤ≥‘𝐾)) |
49 | 32, 48 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑀 + 1) ∈
(ℤ≥‘𝐾)) |
50 | 49 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝑀 + 1) ∈
(ℤ≥‘𝐾)) |
51 | | uztrn 11580 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈
(ℤ≥‘(𝑀 + 1)) ∧ (𝑀 + 1) ∈
(ℤ≥‘𝐾)) → 𝑛 ∈ (ℤ≥‘𝐾)) |
52 | 47, 50, 51 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (ℤ≥‘𝐾)) |
53 | | seqp1 12678 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈
(ℤ≥‘𝐾) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) |
54 | 52, 53 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) |
55 | | seqp1 12678 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈
(ℤ≥‘(𝑀 + 1)) → (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)) = ((seq(𝑀 + 1)( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) |
56 | 47, 55 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)) = ((seq(𝑀 + 1)( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) |
57 | 56 | oveq2d 6565 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))) = ((seq𝐾( + , 𝐹)‘𝑀) + ((seq(𝑀 + 1)( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))) |
58 | | simpl 472 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝜑) |
59 | | eluzelz 11573 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑀 ∈
(ℤ≥‘𝐾) → 𝑀 ∈ ℤ) |
60 | 32, 59 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑀 ∈ ℤ) |
61 | | peano2uzr 11619 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈
(ℤ≥‘(𝑀 + 1))) → 𝑁 ∈ (ℤ≥‘𝑀)) |
62 | 60, 1, 61 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
63 | | fzss2 12252 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝐾...𝑀) ⊆ (𝐾...𝑁)) |
64 | 62, 63 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐾...𝑀) ⊆ (𝐾...𝑁)) |
65 | 64 | sselda 3568 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑀)) → 𝑥 ∈ (𝐾...𝑁)) |
66 | | seqsplit.5 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) |
67 | 65, 66 | syldan 486 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
68 | | seqsplit.1 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
69 | 32, 67, 68 | seqcl 12683 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝑀) ∈ 𝑆) |
70 | 69 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq𝐾( + , 𝐹)‘𝑀) ∈ 𝑆) |
71 | | elfzuz3 12210 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ((𝑀 + 1)...𝑁) → 𝑁 ∈ (ℤ≥‘𝑛)) |
72 | | fzss2 12252 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈
(ℤ≥‘𝑛) → ((𝑀 + 1)...𝑛) ⊆ ((𝑀 + 1)...𝑁)) |
73 | 43, 71, 72 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((𝑀 + 1)...𝑛) ⊆ ((𝑀 + 1)...𝑁)) |
74 | | fzss1 12251 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑀 + 1) ∈
(ℤ≥‘𝐾) → ((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁)) |
75 | 32, 48, 74 | 3syl 18 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁)) |
76 | 75 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁)) |
77 | 73, 76 | sstrd 3578 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((𝑀 + 1)...𝑛) ⊆ (𝐾...𝑁)) |
78 | 77 | sselda 3568 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ 𝑥 ∈ ((𝑀 + 1)...𝑛)) → 𝑥 ∈ (𝐾...𝑁)) |
79 | 66 | adantlr 747 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) |
80 | 78, 79 | syldan 486 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ 𝑥 ∈ ((𝑀 + 1)...𝑛)) → (𝐹‘𝑥) ∈ 𝑆) |
81 | 68 | adantlr 747 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
82 | 47, 80, 81 | seqcl 12683 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq(𝑀 + 1)( + , 𝐹)‘𝑛) ∈ 𝑆) |
83 | | simpr 476 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈
(ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) |
84 | | ssel2 3563 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → (𝑛 + 1) ∈ (𝐾...𝑁)) |
85 | 75, 83, 84 | syl2an 493 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝑛 + 1) ∈ (𝐾...𝑁)) |
86 | 66 | ralrimiva 2949 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∀𝑥 ∈ (𝐾...𝑁)(𝐹‘𝑥) ∈ 𝑆) |
87 | 86 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ∀𝑥 ∈ (𝐾...𝑁)(𝐹‘𝑥) ∈ 𝑆) |
88 | | fveq2 6103 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1))) |
89 | 88 | eleq1d 2672 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆)) |
90 | 89 | rspcv 3278 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 + 1) ∈ (𝐾...𝑁) → (∀𝑥 ∈ (𝐾...𝑁)(𝐹‘𝑥) ∈ 𝑆 → (𝐹‘(𝑛 + 1)) ∈ 𝑆)) |
91 | 85, 87, 90 | sylc 63 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ 𝑆) |
92 | | seqsplit.2 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
93 | 92 | caovassg 6730 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((seq𝐾( + , 𝐹)‘𝑀) ∈ 𝑆 ∧ (seq(𝑀 + 1)( + , 𝐹)‘𝑛) ∈ 𝑆 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑆)) → (((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) + (𝐹‘(𝑛 + 1))) = ((seq𝐾( + , 𝐹)‘𝑀) + ((seq(𝑀 + 1)( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))) |
94 | 58, 70, 82, 91, 93 | syl13anc 1320 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) + (𝐹‘(𝑛 + 1))) = ((seq𝐾( + , 𝐹)‘𝑀) + ((seq(𝑀 + 1)( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))) |
95 | 57, 94 | eqtr4d 2647 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))) = (((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) + (𝐹‘(𝑛 + 1)))) |
96 | 54, 95 | eqeq12d 2625 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))) ↔ ((seq𝐾( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) = (((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) + (𝐹‘(𝑛 + 1))))) |
97 | 46, 96 | syl5ibr 235 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))))) |
98 | 97 | expr 641 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → ((seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)))))) |
99 | 98 | a2d 29 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑀 + 1))) → (((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)))))) |
100 | 45, 99 | syld 46 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)))))) |
101 | 100 | expcom 450 |
. . . . 5
⊢ (𝑛 ∈
(ℤ≥‘(𝑀 + 1)) → (𝜑 → ((𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))))))) |
102 | 101 | a2d 29 |
. . . 4
⊢ (𝑛 ∈
(ℤ≥‘(𝑀 + 1)) → ((𝜑 → (𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)))) → (𝜑 → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))))))) |
103 | 10, 17, 24, 31, 41, 102 | uzind4 11622 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘(𝑀 + 1)) → (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁))))) |
104 | 1, 103 | mpcom 37 |
. 2
⊢ (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))) |
105 | 3, 104 | mpd 15 |
1
⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁))) |