Theorem adds42d

Description: Rearrangement of four terms in a surreal sum. (Contributed by Scott Fenton, 5-Feb-2025)

	Ref	Expression
Hypotheses	adds4d.1	|- ( ph -> A e. No )
	adds4d.2	|- ( ph -> B e. No )
	adds4d.3	|- ( ph -> C e. No )
	adds4d.4	|- ( ph -> D e. No )
Assertion	adds42d	|- ( ph -> ( ( A +s B ) +s ( C +s D ) ) = ( ( A +s C ) +s ( D +s B ) ) )

Proof

Step	Hyp	Ref	Expression
1		adds4d.1	|- ( ph -> A e. No )
2		adds4d.2	|- ( ph -> B e. No )
3		adds4d.3	|- ( ph -> C e. No )
4		adds4d.4	|- ( ph -> D e. No )
5	1 2 3 4	adds4d	|- ( ph -> ( ( A +s B ) +s ( C +s D ) ) = ( ( A +s C ) +s ( B +s D ) ) )
6	2 4	addscomd	|- ( ph -> ( B +s D ) = ( D +s B ) )
7	6	oveq2d	|- ( ph -> ( ( A +s C ) +s ( B +s D ) ) = ( ( A +s C ) +s ( D +s B ) ) )
8	5 7	eqtrd	|- ( ph -> ( ( A +s B ) +s ( C +s D ) ) = ( ( A +s C ) +s ( D +s B ) ) )