Step |
Hyp |
Ref |
Expression |
1 |
|
0elfz |
|- ( N e. NN0 -> 0 e. ( 0 ... N ) ) |
2 |
|
fzsplit |
|- ( 0 e. ( 0 ... N ) -> ( 0 ... N ) = ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... N ) ) ) |
3 |
|
0p1e1 |
|- ( 0 + 1 ) = 1 |
4 |
3
|
oveq1i |
|- ( ( 0 + 1 ) ... N ) = ( 1 ... N ) |
5 |
4
|
uneq2i |
|- ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... N ) ) = ( ( 0 ... 0 ) u. ( 1 ... N ) ) |
6 |
2 5
|
eqtrdi |
|- ( 0 e. ( 0 ... N ) -> ( 0 ... N ) = ( ( 0 ... 0 ) u. ( 1 ... N ) ) ) |
7 |
1 6
|
syl |
|- ( N e. NN0 -> ( 0 ... N ) = ( ( 0 ... 0 ) u. ( 1 ... N ) ) ) |
8 |
|
0z |
|- 0 e. ZZ |
9 |
|
fzsn |
|- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } ) |
10 |
8 9
|
mp1i |
|- ( N e. NN0 -> ( 0 ... 0 ) = { 0 } ) |
11 |
10
|
uneq1d |
|- ( N e. NN0 -> ( ( 0 ... 0 ) u. ( 1 ... N ) ) = ( { 0 } u. ( 1 ... N ) ) ) |
12 |
7 11
|
eqtrd |
|- ( N e. NN0 -> ( 0 ... N ) = ( { 0 } u. ( 1 ... N ) ) ) |