Database
SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
Mathbox for Stefan O'Rear
X and Y sequences 2: Order properties
lermxnn0
Next ⟩
rmxnn
Metamath Proof Explorer
Ascii
Unicode
Theorem
lermxnn0
Description:
The X-sequence is monotonic on
NN0
.
(Contributed by
Stefan O'Rear
, 4-Oct-2014)
Ref
Expression
Assertion
lermxnn0
⊢
A
∈
ℤ
≥
2
∧
M
∈
ℕ
0
∧
N
∈
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
⊢
ℕ
0
⊆
ℝ
5
nn0z
⊢
a
∈
ℕ
0
→
a
∈
ℤ
6
frmx
⊢
X
rm
:
ℤ
≥
2
×
ℤ
⟶
ℕ
0
7
6
fovcl
⊢
A
∈
ℤ
≥
2
∧
a
∈
ℤ
→
A
X
rm
a
∈
ℕ
0
8
5
7
sylan2
⊢
A
∈
ℤ
≥
2
∧
a
∈
ℕ
0
→
A
X
rm
a
∈
ℕ
0
9
8
nn0red
⊢
A
∈
ℤ
≥
2
∧
a
∈
ℕ
0
→
A
X
rm
a
∈
ℝ
10
ltrmxnn0
⊢
A
∈
ℤ
≥
2
∧
a
∈
ℕ
0
∧
b
∈
ℕ
0
→
a
<
b
↔
A
X
rm
a
<
A
X
rm
b
11
10
biimpd
⊢
A
∈
ℤ
≥
2
∧
a
∈
ℕ
0
∧
b
∈
ℕ
0
→
a
<
b
→
A
X
rm
a
<
A
X
rm
b
12
11
3expb
⊢
A
∈
ℤ
≥
2
∧
a
∈
ℕ
0
∧
b
∈
ℕ
0
→
a
<
b
→
A
X
rm
a
<
A
X
rm
b
13
1
2
3
4
9
12
leord1
⊢
A
∈
ℤ
≥
2
∧
M
∈
ℕ
0
∧
N
∈
ℕ
0
→
M
≤
N
↔
A
X
rm
M
≤
A
X
rm
N
14
13
3impb
⊢
A
∈
ℤ
≥
2
∧
M
∈
ℕ
0
∧
N
∈
ℕ
0
→
M
≤
N
↔
A
X
rm
M
≤
A
X
rm
N