Theorem adds4d

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	adds4d	|- ( ph -> ( ( A +s B ) +s ( C +s D ) ) = ( ( A +s C ) +s ( B +s D ) ) )

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	adds32d	|- ( ph -> ( ( A +s B ) +s C ) = ( ( A +s C ) +s B ) )
6	5	oveq1d	|- ( ph -> ( ( ( A +s B ) +s C ) +s D ) = ( ( ( A +s C ) +s B ) +s D ) )
7	1 2	addscld	|- ( ph -> ( A +s B ) e. No )
8	7 3 4	addsassd	|- ( ph -> ( ( ( A +s B ) +s C ) +s D ) = ( ( A +s B ) +s ( C +s D ) ) )
9	1 3	addscld	|- ( ph -> ( A +s C ) e. No )
10	9 2 4	addsassd	|- ( ph -> ( ( ( A +s C ) +s B ) +s D ) = ( ( A +s C ) +s ( B +s D ) ) )
11	6 8 10	3eqtr3d	|- ( ph -> ( ( A +s B ) +s ( C +s D ) ) = ( ( A +s C ) +s ( B +s D ) ) )