Theorem ltadds2

Description: Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 21-Jan-2025)

	Ref	Expression
Assertion	ltadds2	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 <s 𝐵 ↔ ( 𝐶 +s 𝐴 ) <s ( 𝐶 +s 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		leadds2	⊢ ( ( 𝐵 ∈ No ∧ 𝐴 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 ≤s 𝐴 ↔ ( 𝐶 +s 𝐵 ) ≤s ( 𝐶 +s 𝐴 ) ) )
2	1	3com12	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 ≤s 𝐴 ↔ ( 𝐶 +s 𝐵 ) ≤s ( 𝐶 +s 𝐴 ) ) )
3	2	notbid	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ¬ 𝐵 ≤s 𝐴 ↔ ¬ ( 𝐶 +s 𝐵 ) ≤s ( 𝐶 +s 𝐴 ) ) )
4		ltnles	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴 ) )
5	4	3adant3	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴 ) )
6		simp3	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐶 ∈ No )
7		simp1	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐴 ∈ No )
8	6 7	addscld	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐶 +s 𝐴 ) ∈ No )
9		simp2	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐵 ∈ No )
10	6 9	addscld	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐶 +s 𝐵 ) ∈ No )
11		ltnles	⊢ ( ( ( 𝐶 +s 𝐴 ) ∈ No ∧ ( 𝐶 +s 𝐵 ) ∈ No ) → ( ( 𝐶 +s 𝐴 ) <s ( 𝐶 +s 𝐵 ) ↔ ¬ ( 𝐶 +s 𝐵 ) ≤s ( 𝐶 +s 𝐴 ) ) )
12	8 10 11	syl2anc	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐶 +s 𝐴 ) <s ( 𝐶 +s 𝐵 ) ↔ ¬ ( 𝐶 +s 𝐵 ) ≤s ( 𝐶 +s 𝐴 ) ) )
13	3 5 12	3bitr4d	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 <s 𝐵 ↔ ( 𝐶 +s 𝐴 ) <s ( 𝐶 +s 𝐵 ) ) )