Theorem slesubsubbd

Description: Equivalence for the surreal less-than or equal relationship between differences. (Contributed by Scott Fenton, 7-Mar-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	slesubsubbd	Could not format assertion : No typesetting found for |- ( ph -> ( ( A -s C ) <_s ( B -s D ) <-> ( A -s B ) <_s ( C -s D ) ) ) with typecode |-

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	2 1 4 3	sltsubsub3bd	Could not format ( ph -> ( ( B -s D ) ( C -s D ) ( ( B -s D ) ( C -s D )
6	5	notbid	Could not format ( ph -> ( -. ( B -s D ) -. ( C -s D ) ( -. ( B -s D ) -. ( C -s D )
7	1 3	subscld	Could not format ( ph -> ( A -s C ) e. No ) : No typesetting found for |- ( ph -> ( A -s C ) e. No ) with typecode |-
8	2 4	subscld	Could not format ( ph -> ( B -s D ) e. No ) : No typesetting found for |- ( ph -> ( B -s D ) e. No ) with typecode |-
9		slenlt	Could not format ( ( ( A -s C ) e. No /\ ( B -s D ) e. No ) -> ( ( A -s C ) <_s ( B -s D ) <-> -. ( B -s D ) ( ( A -s C ) <_s ( B -s D ) <-> -. ( B -s D )
10	7 8 9	syl2anc	Could not format ( ph -> ( ( A -s C ) <_s ( B -s D ) <-> -. ( B -s D ) ( ( A -s C ) <_s ( B -s D ) <-> -. ( B -s D )
11	1 2	subscld	Could not format ( ph -> ( A -s B ) e. No ) : No typesetting found for |- ( ph -> ( A -s B ) e. No ) with typecode |-
12	3 4	subscld	Could not format ( ph -> ( C -s D ) e. No ) : No typesetting found for |- ( ph -> ( C -s D ) e. No ) with typecode |-
13		slenlt	Could not format ( ( ( A -s B ) e. No /\ ( C -s D ) e. No ) -> ( ( A -s B ) <_s ( C -s D ) <-> -. ( C -s D ) ( ( A -s B ) <_s ( C -s D ) <-> -. ( C -s D )
14	11 12 13	syl2anc	Could not format ( ph -> ( ( A -s B ) <_s ( C -s D ) <-> -. ( C -s D ) ( ( A -s B ) <_s ( C -s D ) <-> -. ( C -s D )
15	6 10 14	3bitr4d	Could not format ( ph -> ( ( A -s C ) <_s ( B -s D ) <-> ( A -s B ) <_s ( C -s D ) ) ) : No typesetting found for |- ( ph -> ( ( A -s C ) <_s ( B -s D ) <-> ( A -s B ) <_s ( C -s D ) ) ) with typecode |-