Theorem lt2addsd

Description: Adding both sides of two surreal less-than relations. (Contributed by Scott Fenton, 15-Apr-2025)

	Ref	Expression
Hypotheses	lt2addsd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	lt2addsd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	lt2addsd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
	lt2addsd.4	⊢ ( 𝜑 → 𝐷 ∈ No )
	lt2addsd.5	⊢ ( 𝜑 → 𝐴 <s 𝐶 )
	lt2addsd.6	⊢ ( 𝜑 → 𝐵 <s 𝐷 )
Assertion	lt2addsd	⊢ ( 𝜑 → ( 𝐴 +s 𝐵 ) <s ( 𝐶 +s 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		lt2addsd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		lt2addsd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		lt2addsd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		lt2addsd.4	⊢ ( 𝜑 → 𝐷 ∈ No )
5		lt2addsd.5	⊢ ( 𝜑 → 𝐴 <s 𝐶 )
6		lt2addsd.6	⊢ ( 𝜑 → 𝐵 <s 𝐷 )
7	1 2	addscld	⊢ ( 𝜑 → ( 𝐴 +s 𝐵 ) ∈ No )
8	3 2	addscld	⊢ ( 𝜑 → ( 𝐶 +s 𝐵 ) ∈ No )
9	3 4	addscld	⊢ ( 𝜑 → ( 𝐶 +s 𝐷 ) ∈ No )
10	1 3 2	ltadds1d	⊢ ( 𝜑 → ( 𝐴 <s 𝐶 ↔ ( 𝐴 +s 𝐵 ) <s ( 𝐶 +s 𝐵 ) ) )
11	5 10	mpbid	⊢ ( 𝜑 → ( 𝐴 +s 𝐵 ) <s ( 𝐶 +s 𝐵 ) )
12	2 4 3	ltadds2d	⊢ ( 𝜑 → ( 𝐵 <s 𝐷 ↔ ( 𝐶 +s 𝐵 ) <s ( 𝐶 +s 𝐷 ) ) )
13	6 12	mpbid	⊢ ( 𝜑 → ( 𝐶 +s 𝐵 ) <s ( 𝐶 +s 𝐷 ) )
14	7 8 9 11 13	ltstrd	⊢ ( 𝜑 → ( 𝐴 +s 𝐵 ) <s ( 𝐶 +s 𝐷 ) )