Theorem addsubsassd

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

	Ref	Expression
Hypotheses	addsubsassd.1	|- ( ph -> A e. No )
	addsubsassd.2	|- ( ph -> B e. No )
	addsubsassd.3	|- ( ph -> C e. No )
Assertion	addsubsassd	|- ( ph -> ( ( A +s B ) -s C ) = ( A +s ( B -s C ) ) )

Proof

Step	Hyp	Ref	Expression
1		addsubsassd.1	|- ( ph -> A e. No )
2		addsubsassd.2	|- ( ph -> B e. No )
3		addsubsassd.3	|- ( ph -> C e. No )
4	3	negscld	|- ( ph -> ( -us ` C ) e. No )
5	1 2 4	addsassd	|- ( ph -> ( ( A +s B ) +s ( -us ` C ) ) = ( A +s ( B +s ( -us ` C ) ) ) )
6	1 2	addscld	|- ( ph -> ( A +s B ) e. No )
7	6 3	subsvald	|- ( ph -> ( ( A +s B ) -s C ) = ( ( A +s B ) +s ( -us ` C ) ) )
8	2 3	subsvald	|- ( ph -> ( B -s C ) = ( B +s ( -us ` C ) ) )
9	8	oveq2d	|- ( ph -> ( A +s ( B -s C ) ) = ( A +s ( B +s ( -us ` C ) ) ) )
10	5 7 9	3eqtr4d	|- ( ph -> ( ( A +s B ) -s C ) = ( A +s ( B -s C ) ) )