Theorem addscan1d

Description: Cancellation law for surreal addition. (Contributed by Scott Fenton, 5-Feb-2025)

	Ref	Expression
Hypotheses	addscand.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	addscand.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	addscand.3	⊢ ( 𝜑 → 𝐶 ∈ No )
Assertion	addscan1d	⊢ ( 𝜑 → ( ( 𝐶 +s 𝐴 ) = ( 𝐶 +s 𝐵 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		addscand.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		addscand.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		addscand.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		addscan1	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐶 +s 𝐴 ) = ( 𝐶 +s 𝐵 ) ↔ 𝐴 = 𝐵 ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( ( 𝐶 +s 𝐴 ) = ( 𝐶 +s 𝐵 ) ↔ 𝐴 = 𝐵 ) )