Step |
Hyp |
Ref |
Expression |
1 |
|
eluzelz |
|- ( N e. ( ZZ>= ` M ) -> N e. ZZ ) |
2 |
1
|
zcnd |
|- ( N e. ( ZZ>= ` M ) -> N e. CC ) |
3 |
|
1zzd |
|- ( N e. ( ZZ>= ` M ) -> 1 e. ZZ ) |
4 |
3
|
zcnd |
|- ( N e. ( ZZ>= ` M ) -> 1 e. CC ) |
5 |
2 4
|
npcand |
|- ( N e. ( ZZ>= ` M ) -> ( ( N - 1 ) + 1 ) = N ) |
6 |
5
|
eleq1d |
|- ( N e. ( ZZ>= ` M ) -> ( ( ( N - 1 ) + 1 ) e. ( ZZ>= ` M ) <-> N e. ( ZZ>= ` M ) ) ) |
7 |
6
|
ibir |
|- ( N e. ( ZZ>= ` M ) -> ( ( N - 1 ) + 1 ) e. ( ZZ>= ` M ) ) |
8 |
|
eluzelre |
|- ( N e. ( ZZ>= ` M ) -> N e. RR ) |
9 |
8
|
lem1d |
|- ( N e. ( ZZ>= ` M ) -> ( N - 1 ) <_ N ) |
10 |
1 3
|
zsubcld |
|- ( N e. ( ZZ>= ` M ) -> ( N - 1 ) e. ZZ ) |
11 |
|
eluz1 |
|- ( ( N - 1 ) e. ZZ -> ( N e. ( ZZ>= ` ( N - 1 ) ) <-> ( N e. ZZ /\ ( N - 1 ) <_ N ) ) ) |
12 |
10 11
|
syl |
|- ( N e. ( ZZ>= ` M ) -> ( N e. ( ZZ>= ` ( N - 1 ) ) <-> ( N e. ZZ /\ ( N - 1 ) <_ N ) ) ) |
13 |
1 9 12
|
mpbir2and |
|- ( N e. ( ZZ>= ` M ) -> N e. ( ZZ>= ` ( N - 1 ) ) ) |
14 |
|
fzsplit2 |
|- ( ( ( ( N - 1 ) + 1 ) e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` ( N - 1 ) ) ) -> ( M ... N ) = ( ( M ... ( N - 1 ) ) u. ( ( ( N - 1 ) + 1 ) ... N ) ) ) |
15 |
7 13 14
|
syl2anc |
|- ( N e. ( ZZ>= ` M ) -> ( M ... N ) = ( ( M ... ( N - 1 ) ) u. ( ( ( N - 1 ) + 1 ) ... N ) ) ) |
16 |
5
|
oveq1d |
|- ( N e. ( ZZ>= ` M ) -> ( ( ( N - 1 ) + 1 ) ... N ) = ( N ... N ) ) |
17 |
|
fzsn |
|- ( N e. ZZ -> ( N ... N ) = { N } ) |
18 |
1 17
|
syl |
|- ( N e. ( ZZ>= ` M ) -> ( N ... N ) = { N } ) |
19 |
16 18
|
eqtrd |
|- ( N e. ( ZZ>= ` M ) -> ( ( ( N - 1 ) + 1 ) ... N ) = { N } ) |
20 |
19
|
uneq2d |
|- ( N e. ( ZZ>= ` M ) -> ( ( M ... ( N - 1 ) ) u. ( ( ( N - 1 ) + 1 ) ... N ) ) = ( ( M ... ( N - 1 ) ) u. { N } ) ) |
21 |
15 20
|
eqtrd |
|- ( N e. ( ZZ>= ` M ) -> ( M ... N ) = ( ( M ... ( N - 1 ) ) u. { N } ) ) |