Step |
Hyp |
Ref |
Expression |
1 |
|
2p1e3 |
⊢ ( 2 + 1 ) = 3 |
2 |
|
0z |
⊢ 0 ∈ ℤ |
3 |
|
3z |
⊢ 3 ∈ ℤ |
4 |
|
0re |
⊢ 0 ∈ ℝ |
5 |
|
3re |
⊢ 3 ∈ ℝ |
6 |
|
3pos |
⊢ 0 < 3 |
7 |
4 5 6
|
ltleii |
⊢ 0 ≤ 3 |
8 |
|
eluz2 |
⊢ ( 3 ∈ ( ℤ≥ ‘ 0 ) ↔ ( 0 ∈ ℤ ∧ 3 ∈ ℤ ∧ 0 ≤ 3 ) ) |
9 |
2 3 7 8
|
mpbir3an |
⊢ 3 ∈ ( ℤ≥ ‘ 0 ) |
10 |
1 9
|
eqeltri |
⊢ ( 2 + 1 ) ∈ ( ℤ≥ ‘ 0 ) |
11 |
|
2z |
⊢ 2 ∈ ℤ |
12 |
|
5nn0 |
⊢ 5 ∈ ℕ0 |
13 |
12
|
nn0zi |
⊢ 5 ∈ ℤ |
14 |
|
2re |
⊢ 2 ∈ ℝ |
15 |
|
5re |
⊢ 5 ∈ ℝ |
16 |
|
2lt5 |
⊢ 2 < 5 |
17 |
14 15 16
|
ltleii |
⊢ 2 ≤ 5 |
18 |
|
eluz2 |
⊢ ( 5 ∈ ( ℤ≥ ‘ 2 ) ↔ ( 2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 2 ≤ 5 ) ) |
19 |
11 13 17 18
|
mpbir3an |
⊢ 5 ∈ ( ℤ≥ ‘ 2 ) |
20 |
|
fzsplit2 |
⊢ ( ( ( 2 + 1 ) ∈ ( ℤ≥ ‘ 0 ) ∧ 5 ∈ ( ℤ≥ ‘ 2 ) ) → ( 0 ... 5 ) = ( ( 0 ... 2 ) ∪ ( ( 2 + 1 ) ... 5 ) ) ) |
21 |
10 19 20
|
mp2an |
⊢ ( 0 ... 5 ) = ( ( 0 ... 2 ) ∪ ( ( 2 + 1 ) ... 5 ) ) |
22 |
|
fz0tp |
⊢ ( 0 ... 2 ) = { 0 , 1 , 2 } |
23 |
1
|
oveq1i |
⊢ ( ( 2 + 1 ) ... 5 ) = ( 3 ... 5 ) |
24 |
|
3p2e5 |
⊢ ( 3 + 2 ) = 5 |
25 |
24
|
eqcomi |
⊢ 5 = ( 3 + 2 ) |
26 |
25
|
oveq2i |
⊢ ( 3 ... 5 ) = ( 3 ... ( 3 + 2 ) ) |
27 |
|
fztp |
⊢ ( 3 ∈ ℤ → ( 3 ... ( 3 + 2 ) ) = { 3 , ( 3 + 1 ) , ( 3 + 2 ) } ) |
28 |
3 27
|
ax-mp |
⊢ ( 3 ... ( 3 + 2 ) ) = { 3 , ( 3 + 1 ) , ( 3 + 2 ) } |
29 |
|
eqid |
⊢ 3 = 3 |
30 |
|
id |
⊢ ( 3 = 3 → 3 = 3 ) |
31 |
|
3p1e4 |
⊢ ( 3 + 1 ) = 4 |
32 |
31
|
a1i |
⊢ ( 3 = 3 → ( 3 + 1 ) = 4 ) |
33 |
24
|
a1i |
⊢ ( 3 = 3 → ( 3 + 2 ) = 5 ) |
34 |
30 32 33
|
tpeq123d |
⊢ ( 3 = 3 → { 3 , ( 3 + 1 ) , ( 3 + 2 ) } = { 3 , 4 , 5 } ) |
35 |
29 34
|
ax-mp |
⊢ { 3 , ( 3 + 1 ) , ( 3 + 2 ) } = { 3 , 4 , 5 } |
36 |
26 28 35
|
3eqtri |
⊢ ( 3 ... 5 ) = { 3 , 4 , 5 } |
37 |
23 36
|
eqtri |
⊢ ( ( 2 + 1 ) ... 5 ) = { 3 , 4 , 5 } |
38 |
22 37
|
uneq12i |
⊢ ( ( 0 ... 2 ) ∪ ( ( 2 + 1 ) ... 5 ) ) = ( { 0 , 1 , 2 } ∪ { 3 , 4 , 5 } ) |
39 |
21 38
|
eqtri |
⊢ ( 0 ... 5 ) = ( { 0 , 1 , 2 } ∪ { 3 , 4 , 5 } ) |