Theorem addsubsd

Description: Law for surreal addition and subtraction. (Contributed by Scott Fenton, 4-Mar-2025)

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

Proof

Step	Hyp	Ref	Expression
1		addsubsassd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		addsubsassd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		addsubsassd.3	$$⊢\phi \to C\in \mathrm{No}$$
4	3	negscld	Could not format ( ph -> ( -us ` C ) e. No ) : No typesetting found for |- ( ph -> ( -us ` C ) e. No ) with typecode |-
5	1 2 4	adds32d	Could not format ( ph -> ( ( A +s B ) +s ( -us ` C ) ) = ( ( A +s ( -us ` C ) ) +s B ) ) : No typesetting found for |- ( ph -> ( ( A +s B ) +s ( -us ` C ) ) = ( ( A +s ( -us ` C ) ) +s B ) ) with typecode |-
6	1 2	addscld	Could not format ( ph -> ( A +s B ) e. No ) : No typesetting found for |- ( ph -> ( A +s B ) e. No ) with typecode |-
7	6 3	subsvald	Could not format ( ph -> ( ( A +s B ) -s C ) = ( ( A +s B ) +s ( -us ` C ) ) ) : No typesetting found for |- ( ph -> ( ( A +s B ) -s C ) = ( ( A +s B ) +s ( -us ` C ) ) ) with typecode |-
8	1 3	subsvald	Could not format ( ph -> ( A -s C ) = ( A +s ( -us ` C ) ) ) : No typesetting found for |- ( ph -> ( A -s C ) = ( A +s ( -us ` C ) ) ) with typecode |-
9	8	oveq1d	Could not format ( ph -> ( ( A -s C ) +s B ) = ( ( A +s ( -us ` C ) ) +s B ) ) : No typesetting found for |- ( ph -> ( ( A -s C ) +s B ) = ( ( A +s ( -us ` C ) ) +s B ) ) with typecode |-
10	5 7 9	3eqtr4d	Could not format ( ph -> ( ( A +s B ) -s C ) = ( ( A -s C ) +s B ) ) : No typesetting found for |- ( ph -> ( ( A +s B ) -s C ) = ( ( A -s C ) +s B ) ) with typecode |-