Theorem subsubs4d

Description: Law for double surreal subtraction. (Contributed by Scott Fenton, 9-Mar-2025)

	Ref	Expression
Hypotheses	subsubs4d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	subsubs4d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	subsubs4d.3	⊢ ( 𝜑 → 𝐶 ∈ No )
Assertion	subsubs4d	⊢ ( 𝜑 → ( ( 𝐴 -s 𝐵 ) -s 𝐶 ) = ( 𝐴 -s ( 𝐵 +s 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		subsubs4d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		subsubs4d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		subsubs4d.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4	2	negscld	⊢ ( 𝜑 → ( -us ‘ 𝐵 ) ∈ No )
5	3	negscld	⊢ ( 𝜑 → ( -us ‘ 𝐶 ) ∈ No )
6	1 4 5	addsassd	⊢ ( 𝜑 → ( ( 𝐴 +s ( -us ‘ 𝐵 ) ) +s ( -us ‘ 𝐶 ) ) = ( 𝐴 +s ( ( -us ‘ 𝐵 ) +s ( -us ‘ 𝐶 ) ) ) )
7	1 2	subsvald	⊢ ( 𝜑 → ( 𝐴 -s 𝐵 ) = ( 𝐴 +s ( -us ‘ 𝐵 ) ) )
8	7	oveq1d	⊢ ( 𝜑 → ( ( 𝐴 -s 𝐵 ) -s 𝐶 ) = ( ( 𝐴 +s ( -us ‘ 𝐵 ) ) -s 𝐶 ) )
9	1 4	addscld	⊢ ( 𝜑 → ( 𝐴 +s ( -us ‘ 𝐵 ) ) ∈ No )
10	9 3	subsvald	⊢ ( 𝜑 → ( ( 𝐴 +s ( -us ‘ 𝐵 ) ) -s 𝐶 ) = ( ( 𝐴 +s ( -us ‘ 𝐵 ) ) +s ( -us ‘ 𝐶 ) ) )
11	8 10	eqtrd	⊢ ( 𝜑 → ( ( 𝐴 -s 𝐵 ) -s 𝐶 ) = ( ( 𝐴 +s ( -us ‘ 𝐵 ) ) +s ( -us ‘ 𝐶 ) ) )
12	2 3	addscld	⊢ ( 𝜑 → ( 𝐵 +s 𝐶 ) ∈ No )
13	1 12	subsvald	⊢ ( 𝜑 → ( 𝐴 -s ( 𝐵 +s 𝐶 ) ) = ( 𝐴 +s ( -us ‘ ( 𝐵 +s 𝐶 ) ) ) )
14		negsdi	⊢ ( ( 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( -us ‘ ( 𝐵 +s 𝐶 ) ) = ( ( -us ‘ 𝐵 ) +s ( -us ‘ 𝐶 ) ) )
15	2 3 14	syl2anc	⊢ ( 𝜑 → ( -us ‘ ( 𝐵 +s 𝐶 ) ) = ( ( -us ‘ 𝐵 ) +s ( -us ‘ 𝐶 ) ) )
16	15	oveq2d	⊢ ( 𝜑 → ( 𝐴 +s ( -us ‘ ( 𝐵 +s 𝐶 ) ) ) = ( 𝐴 +s ( ( -us ‘ 𝐵 ) +s ( -us ‘ 𝐶 ) ) ) )
17	13 16	eqtrd	⊢ ( 𝜑 → ( 𝐴 -s ( 𝐵 +s 𝐶 ) ) = ( 𝐴 +s ( ( -us ‘ 𝐵 ) +s ( -us ‘ 𝐶 ) ) ) )
18	6 11 17	3eqtr4d	⊢ ( 𝜑 → ( ( 𝐴 -s 𝐵 ) -s 𝐶 ) = ( 𝐴 -s ( 𝐵 +s 𝐶 ) ) )