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 ) )