Metamath Proof Explorer


Theorem lermxnn0

Description: The X-sequence is monotonic on NN0 . (Contributed by Stefan O'Rear, 4-Oct-2014)

Ref Expression
Assertion lermxnn0 A 2 M 0 N 0 M N A X rm M A X rm N

Proof

Step Hyp Ref Expression
1 oveq2 a = b A X rm a = A X rm b
2 oveq2 a = M A X rm a = A X rm M
3 oveq2 a = N A X rm a = A X rm N
4 nn0ssre 0
5 nn0z a 0 a
6 frmx X rm : 2 × 0
7 6 fovcl A 2 a A X rm a 0
8 5 7 sylan2 A 2 a 0 A X rm a 0
9 8 nn0red A 2 a 0 A X rm a
10 ltrmxnn0 A 2 a 0 b 0 a < b A X rm a < A X rm b
11 10 biimpd A 2 a 0 b 0 a < b A X rm a < A X rm b
12 11 3expb A 2 a 0 b 0 a < b A X rm a < A X rm b
13 1 2 3 4 9 12 leord1 A 2 M 0 N 0 M N A X rm M A X rm N
14 13 3impb A 2 M 0 N 0 M N A X rm M A X rm N