Theorem subsubs4d

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

	Ref	Expression
Hypotheses	subsubs4d.1	|- ( ph -> A e. No )
	subsubs4d.2	|- ( ph -> B e. No )
	subsubs4d.3	|- ( ph -> C e. No )
Assertion	subsubs4d	|- ( ph -> ( ( A -s B ) -s C ) = ( A -s ( B +s C ) ) )

Proof

Step	Hyp	Ref	Expression
1		subsubs4d.1	|- ( ph -> A e. No )
2		subsubs4d.2	|- ( ph -> B e. No )
3		subsubs4d.3	|- ( ph -> C e. No )
4	2	negscld	|- ( ph -> ( -us ` B ) e. No )
5	3	negscld	|- ( ph -> ( -us ` C ) e. No )
6	1 4 5	addsassd	|- ( ph -> ( ( A +s ( -us ` B ) ) +s ( -us ` C ) ) = ( A +s ( ( -us ` B ) +s ( -us ` C ) ) ) )
7	1 2	subsvald	|- ( ph -> ( A -s B ) = ( A +s ( -us ` B ) ) )
8	7	oveq1d	|- ( ph -> ( ( A -s B ) -s C ) = ( ( A +s ( -us ` B ) ) -s C ) )
9	1 4	addscld	|- ( ph -> ( A +s ( -us ` B ) ) e. No )
10	9 3	subsvald	|- ( ph -> ( ( A +s ( -us ` B ) ) -s C ) = ( ( A +s ( -us ` B ) ) +s ( -us ` C ) ) )
11	8 10	eqtrd	|- ( ph -> ( ( A -s B ) -s C ) = ( ( A +s ( -us ` B ) ) +s ( -us ` C ) ) )
12	2 3	addscld	|- ( ph -> ( B +s C ) e. No )
13	1 12	subsvald	|- ( ph -> ( A -s ( B +s C ) ) = ( A +s ( -us ` ( B +s C ) ) ) )
14		negsdi	|- ( ( B e. No /\ C e. No ) -> ( -us ` ( B +s C ) ) = ( ( -us ` B ) +s ( -us ` C ) ) )
15	2 3 14	syl2anc	|- ( ph -> ( -us ` ( B +s C ) ) = ( ( -us ` B ) +s ( -us ` C ) ) )
16	15	oveq2d	|- ( ph -> ( A +s ( -us ` ( B +s C ) ) ) = ( A +s ( ( -us ` B ) +s ( -us ` C ) ) ) )
17	13 16	eqtrd	|- ( ph -> ( A -s ( B +s C ) ) = ( A +s ( ( -us ` B ) +s ( -us ` C ) ) ) )
18	6 11 17	3eqtr4d	|- ( ph -> ( ( A -s B ) -s C ) = ( A -s ( B +s C ) ) )