Database
SURREAL NUMBERS
Surreal arithmetic
Negation and Subtraction
subsge0d
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Multiplication
Metamath Proof Explorer
Ascii
Unicode
Theorem
subsge0d
Description:
Non-negative subtraction.
(Contributed by
Scott Fenton
, 26-May-2025)
Ref
Expression
Hypotheses
subsge0d.1
⊢
φ
→
A
∈
No
subsge0d.2
⊢
φ
→
B
∈
No
Assertion
subsge0d
⊢
φ
→
0
s
≤
s
A
-
s
B
↔
B
≤
s
A
Proof
Step
Hyp
Ref
Expression
1
subsge0d.1
⊢
φ
→
A
∈
No
2
subsge0d.2
⊢
φ
→
B
∈
No
3
0sno
⊢
0
s
∈
No
4
3
a1i
⊢
φ
→
0
s
∈
No
5
1
2
subscld
⊢
φ
→
A
-
s
B
∈
No
6
4
5
2
sleadd1d
⊢
φ
→
0
s
≤
s
A
-
s
B
↔
0
s
+
s
B
≤
s
A
-
s
B
+
s
B
7
addslid
⊢
B
∈
No
→
0
s
+
s
B
=
B
8
2
7
syl
⊢
φ
→
0
s
+
s
B
=
B
9
npcans
⊢
A
∈
No
∧
B
∈
No
→
A
-
s
B
+
s
B
=
A
10
1
2
9
syl2anc
⊢
φ
→
A
-
s
B
+
s
B
=
A
11
8
10
breq12d
⊢
φ
→
0
s
+
s
B
≤
s
A
-
s
B
+
s
B
↔
B
≤
s
A
12
6
11
bitrd
⊢
φ
→
0
s
≤
s
A
-
s
B
↔
B
≤
s
A