Step |
Hyp |
Ref |
Expression |
1 |
|
elnn0 |
⊢ ( 𝐴 ∈ ℕ0 ↔ ( 𝐴 ∈ ℕ ∨ 𝐴 = 0 ) ) |
2 |
|
nnge1 |
⊢ ( 𝐴 ∈ ℕ → 1 ≤ 𝐴 ) |
3 |
2
|
adantr |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) ) → 1 ≤ 𝐴 ) |
4 |
|
simprr |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) ) → 𝐴 ≤ 𝐵 ) |
5 |
|
nnz |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℤ ) |
6 |
5
|
adantr |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) ) → 𝐴 ∈ ℤ ) |
7 |
|
1zzd |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) ) → 1 ∈ ℤ ) |
8 |
|
nn0z |
⊢ ( 𝐵 ∈ ℕ0 → 𝐵 ∈ ℤ ) |
9 |
8
|
ad2antrl |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) ) → 𝐵 ∈ ℤ ) |
10 |
|
elfz |
⊢ ( ( 𝐴 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 ∈ ( 1 ... 𝐵 ) ↔ ( 1 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ) |
11 |
6 7 9 10
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) ) → ( 𝐴 ∈ ( 1 ... 𝐵 ) ↔ ( 1 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ) |
12 |
3 4 11
|
mpbir2and |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) ) → 𝐴 ∈ ( 1 ... 𝐵 ) ) |
13 |
|
fzsplit |
⊢ ( 𝐴 ∈ ( 1 ... 𝐵 ) → ( 1 ... 𝐵 ) = ( ( 1 ... 𝐴 ) ∪ ( ( 𝐴 + 1 ) ... 𝐵 ) ) ) |
14 |
12 13
|
syl |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) ) → ( 1 ... 𝐵 ) = ( ( 1 ... 𝐴 ) ∪ ( ( 𝐴 + 1 ) ... 𝐵 ) ) ) |
15 |
|
uncom |
⊢ ( ( 1 ... 𝐴 ) ∪ ( ( 𝐴 + 1 ) ... 𝐵 ) ) = ( ( ( 𝐴 + 1 ) ... 𝐵 ) ∪ ( 1 ... 𝐴 ) ) |
16 |
|
oveq1 |
⊢ ( 𝐴 = 0 → ( 𝐴 + 1 ) = ( 0 + 1 ) ) |
17 |
16
|
adantr |
⊢ ( ( 𝐴 = 0 ∧ ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) ) → ( 𝐴 + 1 ) = ( 0 + 1 ) ) |
18 |
|
0p1e1 |
⊢ ( 0 + 1 ) = 1 |
19 |
17 18
|
eqtrdi |
⊢ ( ( 𝐴 = 0 ∧ ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) ) → ( 𝐴 + 1 ) = 1 ) |
20 |
19
|
oveq1d |
⊢ ( ( 𝐴 = 0 ∧ ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) ) → ( ( 𝐴 + 1 ) ... 𝐵 ) = ( 1 ... 𝐵 ) ) |
21 |
|
oveq2 |
⊢ ( 𝐴 = 0 → ( 1 ... 𝐴 ) = ( 1 ... 0 ) ) |
22 |
21
|
adantr |
⊢ ( ( 𝐴 = 0 ∧ ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) ) → ( 1 ... 𝐴 ) = ( 1 ... 0 ) ) |
23 |
|
fz10 |
⊢ ( 1 ... 0 ) = ∅ |
24 |
22 23
|
eqtrdi |
⊢ ( ( 𝐴 = 0 ∧ ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) ) → ( 1 ... 𝐴 ) = ∅ ) |
25 |
20 24
|
uneq12d |
⊢ ( ( 𝐴 = 0 ∧ ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) ) → ( ( ( 𝐴 + 1 ) ... 𝐵 ) ∪ ( 1 ... 𝐴 ) ) = ( ( 1 ... 𝐵 ) ∪ ∅ ) ) |
26 |
|
un0 |
⊢ ( ( 1 ... 𝐵 ) ∪ ∅ ) = ( 1 ... 𝐵 ) |
27 |
25 26
|
eqtrdi |
⊢ ( ( 𝐴 = 0 ∧ ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) ) → ( ( ( 𝐴 + 1 ) ... 𝐵 ) ∪ ( 1 ... 𝐴 ) ) = ( 1 ... 𝐵 ) ) |
28 |
15 27
|
syl5req |
⊢ ( ( 𝐴 = 0 ∧ ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) ) → ( 1 ... 𝐵 ) = ( ( 1 ... 𝐴 ) ∪ ( ( 𝐴 + 1 ) ... 𝐵 ) ) ) |
29 |
14 28
|
jaoian |
⊢ ( ( ( 𝐴 ∈ ℕ ∨ 𝐴 = 0 ) ∧ ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) ) → ( 1 ... 𝐵 ) = ( ( 1 ... 𝐴 ) ∪ ( ( 𝐴 + 1 ) ... 𝐵 ) ) ) |
30 |
29
|
ex |
⊢ ( ( 𝐴 ∈ ℕ ∨ 𝐴 = 0 ) → ( ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) → ( 1 ... 𝐵 ) = ( ( 1 ... 𝐴 ) ∪ ( ( 𝐴 + 1 ) ... 𝐵 ) ) ) ) |
31 |
1 30
|
sylbi |
⊢ ( 𝐴 ∈ ℕ0 → ( ( 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) → ( 1 ... 𝐵 ) = ( ( 1 ... 𝐴 ) ∪ ( ( 𝐴 + 1 ) ... 𝐵 ) ) ) ) |
32 |
31
|
3impib |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵 ) → ( 1 ... 𝐵 ) = ( ( 1 ... 𝐴 ) ∪ ( ( 𝐴 + 1 ) ... 𝐵 ) ) ) |