Step |
Hyp |
Ref |
Expression |
1 |
|
eluzelz |
|- ( M e. ( ZZ>= ` 2 ) -> M e. ZZ ) |
2 |
|
eluzelz |
|- ( N e. ( ZZ>= ` 2 ) -> N e. ZZ ) |
3 |
|
zmulcl |
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ ) |
4 |
1 2 3
|
syl2an |
|- ( ( M e. ( ZZ>= ` 2 ) /\ N e. ( ZZ>= ` 2 ) ) -> ( M x. N ) e. ZZ ) |
5 |
|
eluz2b1 |
|- ( M e. ( ZZ>= ` 2 ) <-> ( M e. ZZ /\ 1 < M ) ) |
6 |
|
zre |
|- ( M e. ZZ -> M e. RR ) |
7 |
6
|
anim1i |
|- ( ( M e. ZZ /\ 1 < M ) -> ( M e. RR /\ 1 < M ) ) |
8 |
5 7
|
sylbi |
|- ( M e. ( ZZ>= ` 2 ) -> ( M e. RR /\ 1 < M ) ) |
9 |
|
eluz2b1 |
|- ( N e. ( ZZ>= ` 2 ) <-> ( N e. ZZ /\ 1 < N ) ) |
10 |
|
zre |
|- ( N e. ZZ -> N e. RR ) |
11 |
10
|
anim1i |
|- ( ( N e. ZZ /\ 1 < N ) -> ( N e. RR /\ 1 < N ) ) |
12 |
9 11
|
sylbi |
|- ( N e. ( ZZ>= ` 2 ) -> ( N e. RR /\ 1 < N ) ) |
13 |
|
mulgt1 |
|- ( ( ( M e. RR /\ N e. RR ) /\ ( 1 < M /\ 1 < N ) ) -> 1 < ( M x. N ) ) |
14 |
13
|
an4s |
|- ( ( ( M e. RR /\ 1 < M ) /\ ( N e. RR /\ 1 < N ) ) -> 1 < ( M x. N ) ) |
15 |
8 12 14
|
syl2an |
|- ( ( M e. ( ZZ>= ` 2 ) /\ N e. ( ZZ>= ` 2 ) ) -> 1 < ( M x. N ) ) |
16 |
|
eluz2b1 |
|- ( ( M x. N ) e. ( ZZ>= ` 2 ) <-> ( ( M x. N ) e. ZZ /\ 1 < ( M x. N ) ) ) |
17 |
4 15 16
|
sylanbrc |
|- ( ( M e. ( ZZ>= ` 2 ) /\ N e. ( ZZ>= ` 2 ) ) -> ( M x. N ) e. ( ZZ>= ` 2 ) ) |