Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
|- ( a = b -> ( A rmY a ) = ( A rmY b ) ) |
2 |
|
oveq2 |
|- ( a = M -> ( A rmY a ) = ( A rmY M ) ) |
3 |
|
oveq2 |
|- ( a = N -> ( A rmY a ) = ( A rmY N ) ) |
4 |
|
zssre |
|- ZZ C_ RR |
5 |
|
frmy |
|- rmY : ( ( ZZ>= ` 2 ) X. ZZ ) --> ZZ |
6 |
5
|
fovcl |
|- ( ( A e. ( ZZ>= ` 2 ) /\ a e. ZZ ) -> ( A rmY a ) e. ZZ ) |
7 |
6
|
zred |
|- ( ( A e. ( ZZ>= ` 2 ) /\ a e. ZZ ) -> ( A rmY a ) e. RR ) |
8 |
|
ltrmy |
|- ( ( A e. ( ZZ>= ` 2 ) /\ a e. ZZ /\ b e. ZZ ) -> ( a < b <-> ( A rmY a ) < ( A rmY b ) ) ) |
9 |
8
|
biimpd |
|- ( ( A e. ( ZZ>= ` 2 ) /\ a e. ZZ /\ b e. ZZ ) -> ( a < b -> ( A rmY a ) < ( A rmY b ) ) ) |
10 |
9
|
3expb |
|- ( ( A e. ( ZZ>= ` 2 ) /\ ( a e. ZZ /\ b e. ZZ ) ) -> ( a < b -> ( A rmY a ) < ( A rmY b ) ) ) |
11 |
1 2 3 4 7 10
|
leord1 |
|- ( ( A e. ( ZZ>= ` 2 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M <_ N <-> ( A rmY M ) <_ ( A rmY N ) ) ) |
12 |
11
|
3impb |
|- ( ( A e. ( ZZ>= ` 2 ) /\ M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( A rmY M ) <_ ( A rmY N ) ) ) |