Step |
Hyp |
Ref |
Expression |
1 |
|
uzp1 |
|- ( N e. ( ZZ>= ` ( M - 1 ) ) -> ( N = ( M - 1 ) \/ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) ) |
2 |
|
zcn |
|- ( M e. ZZ -> M e. CC ) |
3 |
|
ax-1cn |
|- 1 e. CC |
4 |
|
npcan |
|- ( ( M e. CC /\ 1 e. CC ) -> ( ( M - 1 ) + 1 ) = M ) |
5 |
2 3 4
|
sylancl |
|- ( M e. ZZ -> ( ( M - 1 ) + 1 ) = M ) |
6 |
5
|
oveq2d |
|- ( M e. ZZ -> ( M ... ( ( M - 1 ) + 1 ) ) = ( M ... M ) ) |
7 |
|
uncom |
|- ( (/) u. { M } ) = ( { M } u. (/) ) |
8 |
|
un0 |
|- ( { M } u. (/) ) = { M } |
9 |
7 8
|
eqtri |
|- ( (/) u. { M } ) = { M } |
10 |
|
zre |
|- ( M e. ZZ -> M e. RR ) |
11 |
10
|
ltm1d |
|- ( M e. ZZ -> ( M - 1 ) < M ) |
12 |
|
peano2zm |
|- ( M e. ZZ -> ( M - 1 ) e. ZZ ) |
13 |
|
fzn |
|- ( ( M e. ZZ /\ ( M - 1 ) e. ZZ ) -> ( ( M - 1 ) < M <-> ( M ... ( M - 1 ) ) = (/) ) ) |
14 |
12 13
|
mpdan |
|- ( M e. ZZ -> ( ( M - 1 ) < M <-> ( M ... ( M - 1 ) ) = (/) ) ) |
15 |
11 14
|
mpbid |
|- ( M e. ZZ -> ( M ... ( M - 1 ) ) = (/) ) |
16 |
5
|
sneqd |
|- ( M e. ZZ -> { ( ( M - 1 ) + 1 ) } = { M } ) |
17 |
15 16
|
uneq12d |
|- ( M e. ZZ -> ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) = ( (/) u. { M } ) ) |
18 |
|
fzsn |
|- ( M e. ZZ -> ( M ... M ) = { M } ) |
19 |
9 17 18
|
3eqtr4a |
|- ( M e. ZZ -> ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) = ( M ... M ) ) |
20 |
6 19
|
eqtr4d |
|- ( M e. ZZ -> ( M ... ( ( M - 1 ) + 1 ) ) = ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) ) |
21 |
|
oveq1 |
|- ( N = ( M - 1 ) -> ( N + 1 ) = ( ( M - 1 ) + 1 ) ) |
22 |
21
|
oveq2d |
|- ( N = ( M - 1 ) -> ( M ... ( N + 1 ) ) = ( M ... ( ( M - 1 ) + 1 ) ) ) |
23 |
|
oveq2 |
|- ( N = ( M - 1 ) -> ( M ... N ) = ( M ... ( M - 1 ) ) ) |
24 |
21
|
sneqd |
|- ( N = ( M - 1 ) -> { ( N + 1 ) } = { ( ( M - 1 ) + 1 ) } ) |
25 |
23 24
|
uneq12d |
|- ( N = ( M - 1 ) -> ( ( M ... N ) u. { ( N + 1 ) } ) = ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) ) |
26 |
22 25
|
eqeq12d |
|- ( N = ( M - 1 ) -> ( ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) <-> ( M ... ( ( M - 1 ) + 1 ) ) = ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) ) ) |
27 |
20 26
|
syl5ibrcom |
|- ( M e. ZZ -> ( N = ( M - 1 ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) ) ) |
28 |
27
|
imp |
|- ( ( M e. ZZ /\ N = ( M - 1 ) ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) ) |
29 |
5
|
fveq2d |
|- ( M e. ZZ -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) ) |
30 |
29
|
eleq2d |
|- ( M e. ZZ -> ( N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) <-> N e. ( ZZ>= ` M ) ) ) |
31 |
30
|
biimpa |
|- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) -> N e. ( ZZ>= ` M ) ) |
32 |
|
fzsuc |
|- ( N e. ( ZZ>= ` M ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) ) |
33 |
31 32
|
syl |
|- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) ) |
34 |
28 33
|
jaodan |
|- ( ( M e. ZZ /\ ( N = ( M - 1 ) \/ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) ) |
35 |
1 34
|
sylan2 |
|- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M - 1 ) ) ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) ) |