Step |
Hyp |
Ref |
Expression |
1 |
|
1p1e2 |
|- ( 1 + 1 ) = 2 |
2 |
|
2p1e3 |
|- ( 2 + 1 ) = 3 |
3 |
|
eluzle |
|- ( M e. ( ZZ>= ` 3 ) -> 3 <_ M ) |
4 |
2 3
|
eqbrtrid |
|- ( M e. ( ZZ>= ` 3 ) -> ( 2 + 1 ) <_ M ) |
5 |
|
2z |
|- 2 e. ZZ |
6 |
|
eluzelz |
|- ( M e. ( ZZ>= ` 3 ) -> M e. ZZ ) |
7 |
|
zltp1le |
|- ( ( 2 e. ZZ /\ M e. ZZ ) -> ( 2 < M <-> ( 2 + 1 ) <_ M ) ) |
8 |
5 6 7
|
sylancr |
|- ( M e. ( ZZ>= ` 3 ) -> ( 2 < M <-> ( 2 + 1 ) <_ M ) ) |
9 |
4 8
|
mpbird |
|- ( M e. ( ZZ>= ` 3 ) -> 2 < M ) |
10 |
1 9
|
eqbrtrid |
|- ( M e. ( ZZ>= ` 3 ) -> ( 1 + 1 ) < M ) |
11 |
|
1red |
|- ( M e. ( ZZ>= ` 3 ) -> 1 e. RR ) |
12 |
|
eluzelre |
|- ( M e. ( ZZ>= ` 3 ) -> M e. RR ) |
13 |
11 11 12
|
ltaddsub2d |
|- ( M e. ( ZZ>= ` 3 ) -> ( ( 1 + 1 ) < M <-> 1 < ( M - 1 ) ) ) |
14 |
10 13
|
mpbid |
|- ( M e. ( ZZ>= ` 3 ) -> 1 < ( M - 1 ) ) |
15 |
|
eluzge3nn |
|- ( M e. ( ZZ>= ` 3 ) -> M e. NN ) |
16 |
|
m1modnnsub1 |
|- ( M e. NN -> ( -u 1 mod M ) = ( M - 1 ) ) |
17 |
15 16
|
syl |
|- ( M e. ( ZZ>= ` 3 ) -> ( -u 1 mod M ) = ( M - 1 ) ) |
18 |
14 17
|
breqtrrd |
|- ( M e. ( ZZ>= ` 3 ) -> 1 < ( -u 1 mod M ) ) |