Theorem sltsubsubbd

Description: Equivalence for the surreal less-than relationship between differences. (Contributed by Scott Fenton, 6-Feb-2025)

	Ref	Expression
Hypotheses	sltsubsubbd.1	$$⊢\phi \to A\in \mathrm{No}$$
	sltsubsubbd.2	$$⊢\phi \to B\in \mathrm{No}$$
	sltsubsubbd.3	$$⊢\phi \to C\in \mathrm{No}$$
	sltsubsubbd.4	$$⊢\phi \to D\in \mathrm{No}$$
Assertion	sltsubsubbd	Could not format assertion : No typesetting found for |- ( ph -> ( ( A -s C ) ( A -s B )

Proof

Step	Hyp	Ref	Expression
1		sltsubsubbd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		sltsubsubbd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		sltsubsubbd.3	$$⊢\phi \to C\in \mathrm{No}$$
4		sltsubsubbd.4	$$⊢\phi \to D\in \mathrm{No}$$
5		npcans	Could not format ( ( A e. No /\ C e. No ) -> ( ( A -s C ) +s C ) = A ) : No typesetting found for |- ( ( A e. No /\ C e. No ) -> ( ( A -s C ) +s C ) = A ) with typecode |-
6	1 3 5	syl2anc	Could not format ( ph -> ( ( A -s C ) +s C ) = A ) : No typesetting found for |- ( ph -> ( ( A -s C ) +s C ) = A ) with typecode |-
7		npcans	Could not format ( ( A e. No /\ B e. No ) -> ( ( A -s B ) +s B ) = A ) : No typesetting found for |- ( ( A e. No /\ B e. No ) -> ( ( A -s B ) +s B ) = A ) with typecode |-
8	1 2 7	syl2anc	Could not format ( ph -> ( ( A -s B ) +s B ) = A ) : No typesetting found for |- ( ph -> ( ( A -s B ) +s B ) = A ) with typecode |-
9	6 8	eqtr4d	Could not format ( ph -> ( ( A -s C ) +s C ) = ( ( A -s B ) +s B ) ) : No typesetting found for |- ( ph -> ( ( A -s C ) +s C ) = ( ( A -s B ) +s B ) ) with typecode |-
10	2 3	addscomd	Could not format ( ph -> ( B +s C ) = ( C +s B ) ) : No typesetting found for |- ( ph -> ( B +s C ) = ( C +s B ) ) with typecode |-
11	10	oveq1d	Could not format ( ph -> ( ( B +s C ) +s ( -us ` D ) ) = ( ( C +s B ) +s ( -us ` D ) ) ) : No typesetting found for |- ( ph -> ( ( B +s C ) +s ( -us ` D ) ) = ( ( C +s B ) +s ( -us ` D ) ) ) with typecode |-
12	2 4	subsvald	Could not format ( ph -> ( B -s D ) = ( B +s ( -us ` D ) ) ) : No typesetting found for |- ( ph -> ( B -s D ) = ( B +s ( -us ` D ) ) ) with typecode |-
13	12	oveq1d	Could not format ( ph -> ( ( B -s D ) +s C ) = ( ( B +s ( -us ` D ) ) +s C ) ) : No typesetting found for |- ( ph -> ( ( B -s D ) +s C ) = ( ( B +s ( -us ` D ) ) +s C ) ) with typecode |-
14	4	negscld	Could not format ( ph -> ( -us ` D ) e. No ) : No typesetting found for |- ( ph -> ( -us ` D ) e. No ) with typecode |-
15	2 14 3	adds32d	Could not format ( ph -> ( ( B +s ( -us ` D ) ) +s C ) = ( ( B +s C ) +s ( -us ` D ) ) ) : No typesetting found for |- ( ph -> ( ( B +s ( -us ` D ) ) +s C ) = ( ( B +s C ) +s ( -us ` D ) ) ) with typecode |-
16	13 15	eqtrd	Could not format ( ph -> ( ( B -s D ) +s C ) = ( ( B +s C ) +s ( -us ` D ) ) ) : No typesetting found for |- ( ph -> ( ( B -s D ) +s C ) = ( ( B +s C ) +s ( -us ` D ) ) ) with typecode |-
17	3 4	subsvald	Could not format ( ph -> ( C -s D ) = ( C +s ( -us ` D ) ) ) : No typesetting found for |- ( ph -> ( C -s D ) = ( C +s ( -us ` D ) ) ) with typecode |-
18	17	oveq1d	Could not format ( ph -> ( ( C -s D ) +s B ) = ( ( C +s ( -us ` D ) ) +s B ) ) : No typesetting found for |- ( ph -> ( ( C -s D ) +s B ) = ( ( C +s ( -us ` D ) ) +s B ) ) with typecode |-
19	3 14 2	adds32d	Could not format ( ph -> ( ( C +s ( -us ` D ) ) +s B ) = ( ( C +s B ) +s ( -us ` D ) ) ) : No typesetting found for |- ( ph -> ( ( C +s ( -us ` D ) ) +s B ) = ( ( C +s B ) +s ( -us ` D ) ) ) with typecode |-
20	18 19	eqtrd	Could not format ( ph -> ( ( C -s D ) +s B ) = ( ( C +s B ) +s ( -us ` D ) ) ) : No typesetting found for |- ( ph -> ( ( C -s D ) +s B ) = ( ( C +s B ) +s ( -us ` D ) ) ) with typecode |-
21	11 16 20	3eqtr4d	Could not format ( ph -> ( ( B -s D ) +s C ) = ( ( C -s D ) +s B ) ) : No typesetting found for |- ( ph -> ( ( B -s D ) +s C ) = ( ( C -s D ) +s B ) ) with typecode |-
22	9 21	breq12d	Could not format ( ph -> ( ( ( A -s C ) +s C ) ( ( A -s B ) +s B ) ( ( ( A -s C ) +s C ) ( ( A -s B ) +s B )
23	1 3	subscld	Could not format ( ph -> ( A -s C ) e. No ) : No typesetting found for |- ( ph -> ( A -s C ) e. No ) with typecode |-
24	2 4	subscld	Could not format ( ph -> ( B -s D ) e. No ) : No typesetting found for |- ( ph -> ( B -s D ) e. No ) with typecode |-
25	23 24 3	sltadd1d	Could not format ( ph -> ( ( A -s C ) ( ( A -s C ) +s C ) ( ( A -s C ) ( ( A -s C ) +s C )
26	1 2	subscld	Could not format ( ph -> ( A -s B ) e. No ) : No typesetting found for |- ( ph -> ( A -s B ) e. No ) with typecode |-
27	3 4	subscld	Could not format ( ph -> ( C -s D ) e. No ) : No typesetting found for |- ( ph -> ( C -s D ) e. No ) with typecode |-
28	26 27 2	sltadd1d	Could not format ( ph -> ( ( A -s B ) ( ( A -s B ) +s B ) ( ( A -s B ) ( ( A -s B ) +s B )
29	22 25 28	3bitr4d	Could not format ( ph -> ( ( A -s C ) ( A -s B ) ( ( A -s C ) ( A -s B )