Metamath Proof Explorer


Theorem lermy

Description: Y is monotonic (non-strict). (Contributed by Stefan O'Rear, 3-Oct-2014)

Ref Expression
Assertion lermy
|- ( ( A e. ( ZZ>= ` 2 ) /\ M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( A rmY M ) <_ ( A rmY N ) ) )

Proof

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 ) ) )