Metamath Proof Explorer


Theorem slesubaddd

Description: Surreal less-than or equal relationship between subtraction and addition. (Contributed by Scott Fenton, 26-May-2025)

Ref Expression
Hypotheses sltsubadd.1 φ A No
sltsubadd.2 φ B No
sltsubadd.3 φ C No
Assertion slesubaddd φ A - s B s C A s C + s B

Proof

Step Hyp Ref Expression
1 sltsubadd.1 φ A No
2 sltsubadd.2 φ B No
3 sltsubadd.3 φ C No
4 3 2 1 sltaddsubd φ C + s B < s A C < s A - s B
5 4 notbid φ ¬ C + s B < s A ¬ C < s A - s B
6 3 2 addscld φ C + s B No
7 slenlt A No C + s B No A s C + s B ¬ C + s B < s A
8 1 6 7 syl2anc φ A s C + s B ¬ C + s B < s A
9 1 2 subscld φ A - s B No
10 slenlt A - s B No C No A - s B s C ¬ C < s A - s B
11 9 3 10 syl2anc φ A - s B s C ¬ C < s A - s B
12 5 8 11 3bitr4rd φ A - s B s C A s C + s B