Metamath Proof Explorer


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