Theorem ltsubs1

Description: Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 4-Feb-2025)

	Ref	Expression
Assertion	ltsubs1	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 <s 𝐵 ↔ ( 𝐴 -s 𝐶 ) <s ( 𝐵 -s 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		negscl	⊢ ( 𝐶 ∈ No → ( -us ‘ 𝐶 ) ∈ No )
2		ltadds1	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( -us ‘ 𝐶 ) ∈ No ) → ( 𝐴 <s 𝐵 ↔ ( 𝐴 +s ( -us ‘ 𝐶 ) ) <s ( 𝐵 +s ( -us ‘ 𝐶 ) ) ) )
3	1 2	syl3an3	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 <s 𝐵 ↔ ( 𝐴 +s ( -us ‘ 𝐶 ) ) <s ( 𝐵 +s ( -us ‘ 𝐶 ) ) ) )
4		subsval	⊢ ( ( 𝐴 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 -s 𝐶 ) = ( 𝐴 +s ( -us ‘ 𝐶 ) ) )
5	4	3adant2	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 -s 𝐶 ) = ( 𝐴 +s ( -us ‘ 𝐶 ) ) )
6		subsval	⊢ ( ( 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 -s 𝐶 ) = ( 𝐵 +s ( -us ‘ 𝐶 ) ) )
7	6	3adant1	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 -s 𝐶 ) = ( 𝐵 +s ( -us ‘ 𝐶 ) ) )
8	5 7	breq12d	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 -s 𝐶 ) <s ( 𝐵 -s 𝐶 ) ↔ ( 𝐴 +s ( -us ‘ 𝐶 ) ) <s ( 𝐵 +s ( -us ‘ 𝐶 ) ) ) )
9	3 8	bitr4d	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 <s 𝐵 ↔ ( 𝐴 -s 𝐶 ) <s ( 𝐵 -s 𝐶 ) ) )