Theorem subsubs4d

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

	Ref	Expression
Hypotheses	subsubs4d.1	$$⊢\phi \to A\in \mathrm{No}$$
	subsubs4d.2	$$⊢\phi \to B\in \mathrm{No}$$
	subsubs4d.3	$$⊢\phi \to C\in \mathrm{No}$$
Assertion	subsubs4d	Could not format assertion : No typesetting found for |- ( ph -> ( ( A -s B ) -s C ) = ( A -s ( B +s C ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		subsubs4d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		subsubs4d.2	$$⊢\phi \to B\in \mathrm{No}$$
3		subsubs4d.3	$$⊢\phi \to C\in \mathrm{No}$$
4	2	negscld	Could not format ( ph -> ( -us ` B ) e. No ) : No typesetting found for |- ( ph -> ( -us ` B ) e. No ) with typecode |-
5	3	negscld	Could not format ( ph -> ( -us ` C ) e. No ) : No typesetting found for |- ( ph -> ( -us ` C ) e. No ) with typecode |-
6	1 4 5	addsassd	Could not format ( ph -> ( ( A +s ( -us ` B ) ) +s ( -us ` C ) ) = ( A +s ( ( -us ` B ) +s ( -us ` C ) ) ) ) : No typesetting found for |- ( ph -> ( ( A +s ( -us ` B ) ) +s ( -us ` C ) ) = ( A +s ( ( -us ` B ) +s ( -us ` C ) ) ) ) with typecode |-
7	1 2	subsvald	Could not format ( ph -> ( A -s B ) = ( A +s ( -us ` B ) ) ) : No typesetting found for |- ( ph -> ( A -s B ) = ( A +s ( -us ` B ) ) ) with typecode |-
8	7	oveq1d	Could not format ( ph -> ( ( A -s B ) -s C ) = ( ( A +s ( -us ` B ) ) -s C ) ) : No typesetting found for |- ( ph -> ( ( A -s B ) -s C ) = ( ( A +s ( -us ` B ) ) -s C ) ) with typecode |-
9	1 4	addscld	Could not format ( ph -> ( A +s ( -us ` B ) ) e. No ) : No typesetting found for |- ( ph -> ( A +s ( -us ` B ) ) e. No ) with typecode |-
10	9 3	subsvald	Could not format ( ph -> ( ( A +s ( -us ` B ) ) -s C ) = ( ( A +s ( -us ` B ) ) +s ( -us ` C ) ) ) : No typesetting found for |- ( ph -> ( ( A +s ( -us ` B ) ) -s C ) = ( ( A +s ( -us ` B ) ) +s ( -us ` C ) ) ) with typecode |-
11	8 10	eqtrd	Could not format ( ph -> ( ( A -s B ) -s C ) = ( ( A +s ( -us ` B ) ) +s ( -us ` C ) ) ) : No typesetting found for |- ( ph -> ( ( A -s B ) -s C ) = ( ( A +s ( -us ` B ) ) +s ( -us ` C ) ) ) with typecode |-
12	2 3	addscld	Could not format ( ph -> ( B +s C ) e. No ) : No typesetting found for |- ( ph -> ( B +s C ) e. No ) with typecode |-
13	1 12	subsvald	Could not format ( ph -> ( A -s ( B +s C ) ) = ( A +s ( -us ` ( B +s C ) ) ) ) : No typesetting found for |- ( ph -> ( A -s ( B +s C ) ) = ( A +s ( -us ` ( B +s C ) ) ) ) with typecode |-
14		negsdi	Could not format ( ( B e. No /\ C e. No ) -> ( -us ` ( B +s C ) ) = ( ( -us ` B ) +s ( -us ` C ) ) ) : No typesetting found for |- ( ( B e. No /\ C e. No ) -> ( -us ` ( B +s C ) ) = ( ( -us ` B ) +s ( -us ` C ) ) ) with typecode |-
15	2 3 14	syl2anc	Could not format ( ph -> ( -us ` ( B +s C ) ) = ( ( -us ` B ) +s ( -us ` C ) ) ) : No typesetting found for |- ( ph -> ( -us ` ( B +s C ) ) = ( ( -us ` B ) +s ( -us ` C ) ) ) with typecode |-
16	15	oveq2d	Could not format ( ph -> ( A +s ( -us ` ( B +s C ) ) ) = ( A +s ( ( -us ` B ) +s ( -us ` C ) ) ) ) : No typesetting found for |- ( ph -> ( A +s ( -us ` ( B +s C ) ) ) = ( A +s ( ( -us ` B ) +s ( -us ` C ) ) ) ) with typecode |-
17	13 16	eqtrd	Could not format ( ph -> ( A -s ( B +s C ) ) = ( A +s ( ( -us ` B ) +s ( -us ` C ) ) ) ) : No typesetting found for |- ( ph -> ( A -s ( B +s C ) ) = ( A +s ( ( -us ` B ) +s ( -us ` C ) ) ) ) with typecode |-
18	6 11 17	3eqtr4d	Could not format ( ph -> ( ( A -s B ) -s C ) = ( A -s ( B +s C ) ) ) : No typesetting found for |- ( ph -> ( ( A -s B ) -s C ) = ( A -s ( B +s C ) ) ) with typecode |-