Theorem sleadd2

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

	Ref	Expression
Assertion	sleadd2	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐵 ↔ ( 𝐶 +s 𝐴 ) ≤s ( 𝐶 +s 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sleadd1	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐵 ↔ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )
2		addscom	⊢ ( ( 𝐴 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 +s 𝐶 ) = ( 𝐶 +s 𝐴 ) )
3	2	3adant2	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 +s 𝐶 ) = ( 𝐶 +s 𝐴 ) )
4		addscom	⊢ ( ( 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 +s 𝐶 ) = ( 𝐶 +s 𝐵 ) )
5	4	3adant1	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 +s 𝐶 ) = ( 𝐶 +s 𝐵 ) )
6	3 5	breq12d	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ↔ ( 𝐶 +s 𝐴 ) ≤s ( 𝐶 +s 𝐵 ) ) )
7	1 6	bitrd	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐵 ↔ ( 𝐶 +s 𝐴 ) ≤s ( 𝐶 +s 𝐵 ) ) )