Theorem addsubsassd

Description: Associative-type law for surreal addition and subtraction. (Contributed by Scott Fenton, 6-Feb-2025)

	Ref	Expression
Hypotheses	addsubsassd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	addsubsassd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	addsubsassd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
Assertion	addsubsassd	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) -s 𝐶 ) = ( 𝐴 +s ( 𝐵 -s 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		addsubsassd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		addsubsassd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		addsubsassd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4	3	negscld	⊢ ( 𝜑 → ( -us ‘ 𝐶 ) ∈ No )
5	1 2 4	addsassd	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) +s ( -us ‘ 𝐶 ) ) = ( 𝐴 +s ( 𝐵 +s ( -us ‘ 𝐶 ) ) ) )
6	1 2	addscld	⊢ ( 𝜑 → ( 𝐴 +s 𝐵 ) ∈ No )
7	6 3	subsvald	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) -s 𝐶 ) = ( ( 𝐴 +s 𝐵 ) +s ( -us ‘ 𝐶 ) ) )
8	2 3	subsvald	⊢ ( 𝜑 → ( 𝐵 -s 𝐶 ) = ( 𝐵 +s ( -us ‘ 𝐶 ) ) )
9	8	oveq2d	⊢ ( 𝜑 → ( 𝐴 +s ( 𝐵 -s 𝐶 ) ) = ( 𝐴 +s ( 𝐵 +s ( -us ‘ 𝐶 ) ) ) )
10	5 7 9	3eqtr4d	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) -s 𝐶 ) = ( 𝐴 +s ( 𝐵 -s 𝐶 ) ) )