Step |
Hyp |
Ref |
Expression |
1 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
2 |
|
3nn0 |
⊢ 3 ∈ ℕ0 |
3 |
|
1le3 |
⊢ 1 ≤ 3 |
4 |
|
elfz2nn0 |
⊢ ( 1 ∈ ( 0 ... 3 ) ↔ ( 1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3 ) ) |
5 |
1 2 3 4
|
mpbir3an |
⊢ 1 ∈ ( 0 ... 3 ) |
6 |
|
fzsplit |
⊢ ( 1 ∈ ( 0 ... 3 ) → ( 0 ... 3 ) = ( ( 0 ... 1 ) ∪ ( ( 1 + 1 ) ... 3 ) ) ) |
7 |
5 6
|
ax-mp |
⊢ ( 0 ... 3 ) = ( ( 0 ... 1 ) ∪ ( ( 1 + 1 ) ... 3 ) ) |
8 |
|
1e0p1 |
⊢ 1 = ( 0 + 1 ) |
9 |
8
|
oveq2i |
⊢ ( 0 ... 1 ) = ( 0 ... ( 0 + 1 ) ) |
10 |
|
0z |
⊢ 0 ∈ ℤ |
11 |
|
fzpr |
⊢ ( 0 ∈ ℤ → ( 0 ... ( 0 + 1 ) ) = { 0 , ( 0 + 1 ) } ) |
12 |
10 11
|
ax-mp |
⊢ ( 0 ... ( 0 + 1 ) ) = { 0 , ( 0 + 1 ) } |
13 |
|
0p1e1 |
⊢ ( 0 + 1 ) = 1 |
14 |
13
|
preq2i |
⊢ { 0 , ( 0 + 1 ) } = { 0 , 1 } |
15 |
9 12 14
|
3eqtri |
⊢ ( 0 ... 1 ) = { 0 , 1 } |
16 |
|
1p1e2 |
⊢ ( 1 + 1 ) = 2 |
17 |
|
df-3 |
⊢ 3 = ( 2 + 1 ) |
18 |
16 17
|
oveq12i |
⊢ ( ( 1 + 1 ) ... 3 ) = ( 2 ... ( 2 + 1 ) ) |
19 |
|
2z |
⊢ 2 ∈ ℤ |
20 |
|
fzpr |
⊢ ( 2 ∈ ℤ → ( 2 ... ( 2 + 1 ) ) = { 2 , ( 2 + 1 ) } ) |
21 |
19 20
|
ax-mp |
⊢ ( 2 ... ( 2 + 1 ) ) = { 2 , ( 2 + 1 ) } |
22 |
|
2p1e3 |
⊢ ( 2 + 1 ) = 3 |
23 |
22
|
preq2i |
⊢ { 2 , ( 2 + 1 ) } = { 2 , 3 } |
24 |
18 21 23
|
3eqtri |
⊢ ( ( 1 + 1 ) ... 3 ) = { 2 , 3 } |
25 |
15 24
|
uneq12i |
⊢ ( ( 0 ... 1 ) ∪ ( ( 1 + 1 ) ... 3 ) ) = ( { 0 , 1 } ∪ { 2 , 3 } ) |
26 |
7 25
|
eqtri |
⊢ ( 0 ... 3 ) = ( { 0 , 1 } ∪ { 2 , 3 } ) |