Theorem mulscan2dlem

Description: Lemma for mulscan2d . Cancellation of multiplication when the right term is positive. (Contributed by Scott Fenton, 10-Mar-2025)

	Ref	Expression
Hypotheses	mulscan2d.1	$$⊢\phi \to A\in \mathrm{No}$$
	mulscan2d.2	$$⊢\phi \to B\in \mathrm{No}$$
	mulscan2d.3	$$⊢\phi \to C\in \mathrm{No}$$
	mulscan2dlem.1	No typesetting found for |- ( ph -> 0s
Assertion	mulscan2dlem	Could not format assertion : No typesetting found for |- ( ph -> ( ( A x.s C ) = ( B x.s C ) <-> A = B ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		mulscan2d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		mulscan2d.2	$$⊢\phi \to B\in \mathrm{No}$$
3		mulscan2d.3	$$⊢\phi \to C\in \mathrm{No}$$
4		mulscan2dlem.1	Could not format ( ph -> 0s 0s
5	1 2 3 4	slemul1d	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 |-
6	2 1 3 4	slemul1d	Could not format ( ph -> ( B <_s A <-> ( B x.s C ) <_s ( A x.s C ) ) ) : No typesetting found for |- ( ph -> ( B <_s A <-> ( B x.s C ) <_s ( A x.s C ) ) ) with typecode |-
7	5 6	anbi12d	Could not format ( ph -> ( ( A <_s B /\ B <_s A ) <-> ( ( A x.s C ) <_s ( B x.s C ) /\ ( B x.s C ) <_s ( A x.s C ) ) ) ) : No typesetting found for |- ( ph -> ( ( A <_s B /\ B <_s A ) <-> ( ( A x.s C ) <_s ( B x.s C ) /\ ( B x.s C ) <_s ( A x.s C ) ) ) ) with typecode |-
8		sletri3	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(A=B\leftrightarrow \left(A{\le }_{s}B\wedge B{\le }_{s}A\right)\right)$$
9	1 2 8	syl2anc	$$⊢\phi \to \left(A=B\leftrightarrow \left(A{\le }_{s}B\wedge B{\le }_{s}A\right)\right)$$
10	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 |-
11	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 |-
12		sletri3	Could not format ( ( ( A x.s C ) e. No /\ ( B x.s C ) e. No ) -> ( ( A x.s C ) = ( B x.s C ) <-> ( ( A x.s C ) <_s ( B x.s C ) /\ ( B x.s C ) <_s ( A x.s C ) ) ) ) : No typesetting found for |- ( ( ( A x.s C ) e. No /\ ( B x.s C ) e. No ) -> ( ( A x.s C ) = ( B x.s C ) <-> ( ( A x.s C ) <_s ( B x.s C ) /\ ( B x.s C ) <_s ( A x.s C ) ) ) ) with typecode |-
13	10 11 12	syl2anc	Could not format ( ph -> ( ( A x.s C ) = ( B x.s C ) <-> ( ( A x.s C ) <_s ( B x.s C ) /\ ( B x.s C ) <_s ( A x.s C ) ) ) ) : No typesetting found for |- ( ph -> ( ( A x.s C ) = ( B x.s C ) <-> ( ( A x.s C ) <_s ( B x.s C ) /\ ( B x.s C ) <_s ( A x.s C ) ) ) ) with typecode |-
14	7 9 13	3bitr4rd	Could not format ( ph -> ( ( A x.s C ) = ( B x.s C ) <-> A = B ) ) : No typesetting found for |- ( ph -> ( ( A x.s C ) = ( B x.s C ) <-> A = B ) ) with typecode |-