Theorem slemul1d

Description: Multiplication of both sides of surreal less-than or equal by a positive number. (Contributed by Scott Fenton, 10-Mar-2025)

	Ref	Expression
Hypotheses	sltmul12d.1	$$⊢\phi \to A\in \mathrm{No}$$
	sltmul12d.2	$$⊢\phi \to B\in \mathrm{No}$$
	sltmul12d.3	$$⊢\phi \to C\in \mathrm{No}$$
	sltmul12d.4	No typesetting found for |- ( ph -> 0s
Assertion	slemul1d	Could not format assertion : No typesetting found for |- ( ph -> ( A <_s B <-> ( A x.s C ) <_s ( B x.s C ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		sltmul12d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		sltmul12d.2	$$⊢\phi \to B\in \mathrm{No}$$
3		sltmul12d.3	$$⊢\phi \to C\in \mathrm{No}$$
4		sltmul12d.4	Could not format ( ph -> 0s 0s
5	2 1 3 4	sltmul1d	Could not format ( ph -> ( B ( B x.s C ) ( B ( B x.s C )
6	5	notbid	Could not format ( ph -> ( -. B -. ( B x.s C ) ( -. B -. ( B x.s C )
7		slenlt	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(A{\le }_{s}B\leftrightarrow \neg B{<}_{s}A\right)$$
8	1 2 7	syl2anc	$$⊢\phi \to \left(A{\le }_{s}B\leftrightarrow \neg B{<}_{s}A\right)$$
9	1 3	mulscld	Could not format ( ph -> ( A x.s C ) e. No ) : No typesetting found for |- ( ph -> ( A x.s C ) e. No ) with typecode |-
10	2 3	mulscld	Could not format ( ph -> ( B x.s C ) e. No ) : No typesetting found for |- ( ph -> ( B x.s C ) e. No ) with typecode |-
11		slenlt	Could not format ( ( ( A x.s C ) e. No /\ ( B x.s C ) e. No ) -> ( ( A x.s C ) <_s ( B x.s C ) <-> -. ( B x.s C ) ( ( A x.s C ) <_s ( B x.s C ) <-> -. ( B x.s C )
12	9 10 11	syl2anc	Could not format ( ph -> ( ( A x.s C ) <_s ( B x.s C ) <-> -. ( B x.s C ) ( ( A x.s C ) <_s ( B x.s C ) <-> -. ( B x.s C )
13	6 8 12	3bitr4d	Could not format ( ph -> ( A <_s B <-> ( A x.s C ) <_s ( B x.s C ) ) ) : No typesetting found for |- ( ph -> ( A <_s B <-> ( A x.s C ) <_s ( B x.s C ) ) ) with typecode |-