Metamath Proof Explorer

Theorem addscan1

Description: Cancellation law for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025)

Ref Expression
Assertion addscan1 ⊢ A ∈ No ∧ B ∈ No ∧ C ∈ No → C + s A = C + s B ↔ A = B

Proof

Step Hyp Ref Expression
1 addscom ⊢ A ∈ No ∧ C ∈ No → A + s C = C + s A
2 1 3adant2 ⊢ A ∈ No ∧ B ∈ No ∧ C ∈ No → A + s C = C + s A
3 addscom ⊢ B ∈ No ∧ C ∈ No → B + s C = C + s B
4 3 3adant1 ⊢ A ∈ No ∧ B ∈ No ∧ C ∈ No → B + s C = C + s B
5 2 4 eqeq12d ⊢ A ∈ No ∧ B ∈ No ∧ C ∈ No → A + s C = B + s C ↔ C + s A = C + s B
6 addscan2 ⊢ A ∈ No ∧ B ∈ No ∧ C ∈ No → A + s C = B + s C ↔ A = B
7 5 6 bitr3d ⊢ A ∈ No ∧ B ∈ No ∧ C ∈ No → C + s A = C + s B ↔ A = B