Theorem leadds1d

Description: Addition to both sides of surreal less-than or equal. Theorem 5 of Conway p. 18. (Contributed by Scott Fenton, 21-Jan-2025)

	Ref	Expression
Hypotheses	addscand.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	addscand.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	addscand.3	⊢ ( 𝜑 → 𝐶 ∈ No )
Assertion	leadds1d	⊢ ( 𝜑 → ( 𝐴 ≤s 𝐵 ↔ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		addscand.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		addscand.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		addscand.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		leadds1	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐵 ↔ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 ≤s 𝐵 ↔ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )