Theorem divmuldivsd

Description: Multiplication of two surreal ratios. (Contributed by Scott Fenton, 16-Apr-2025)

	Ref	Expression
Hypotheses	divmuldivsd.1	$$⊢\phi \to A\in \mathrm{No}$$
	divmuldivsd.2	$$⊢\phi \to B\in \mathrm{No}$$
	divmuldivsd.3	$$⊢\phi \to C\in \mathrm{No}$$
	divmuldivsd.4	$$⊢\phi \to D\in \mathrm{No}$$
	divmuldivsd.5	No typesetting found for |- ( ph -> B =/= 0s ) with typecode |-
	divmuldivsd.6	No typesetting found for |- ( ph -> D =/= 0s ) with typecode |-
Assertion	divmuldivsd	Could not format assertion : No typesetting found for |- ( ph -> ( ( A /su B ) x.s ( C /su D ) ) = ( ( A x.s C ) /su ( B x.s D ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		divmuldivsd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		divmuldivsd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		divmuldivsd.3	$$⊢\phi \to C\in \mathrm{No}$$
4		divmuldivsd.4	$$⊢\phi \to D\in \mathrm{No}$$
5		divmuldivsd.5	Could not format ( ph -> B =/= 0s ) : No typesetting found for |- ( ph -> B =/= 0s ) with typecode |-
6		divmuldivsd.6	Could not format ( ph -> D =/= 0s ) : No typesetting found for |- ( ph -> D =/= 0s ) with typecode |-
7	1 2 5	divscld	Could not format ( ph -> ( A /su B ) e. No ) : No typesetting found for |- ( ph -> ( A /su B ) e. No ) with typecode |-
8	3 4 6	divscld	Could not format ( ph -> ( C /su D ) e. No ) : No typesetting found for |- ( ph -> ( C /su D ) e. No ) with typecode |-
9	2 4 7 8	muls4d	Could not format ( ph -> ( ( B x.s D ) x.s ( ( A /su B ) x.s ( C /su D ) ) ) = ( ( B x.s ( A /su B ) ) x.s ( D x.s ( C /su D ) ) ) ) : No typesetting found for |- ( ph -> ( ( B x.s D ) x.s ( ( A /su B ) x.s ( C /su D ) ) ) = ( ( B x.s ( A /su B ) ) x.s ( D x.s ( C /su D ) ) ) ) with typecode |-
10	1 2 5	divscan2d	Could not format ( ph -> ( B x.s ( A /su B ) ) = A ) : No typesetting found for |- ( ph -> ( B x.s ( A /su B ) ) = A ) with typecode |-
11	3 4 6	divscan2d	Could not format ( ph -> ( D x.s ( C /su D ) ) = C ) : No typesetting found for |- ( ph -> ( D x.s ( C /su D ) ) = C ) with typecode |-
12	10 11	oveq12d	Could not format ( ph -> ( ( B x.s ( A /su B ) ) x.s ( D x.s ( C /su D ) ) ) = ( A x.s C ) ) : No typesetting found for |- ( ph -> ( ( B x.s ( A /su B ) ) x.s ( D x.s ( C /su D ) ) ) = ( A x.s C ) ) with typecode |-
13	9 12	eqtrd	Could not format ( ph -> ( ( B x.s D ) x.s ( ( A /su B ) x.s ( C /su D ) ) ) = ( A x.s C ) ) : No typesetting found for |- ( ph -> ( ( B x.s D ) x.s ( ( A /su B ) x.s ( C /su D ) ) ) = ( A x.s C ) ) with typecode |-
14	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 |-
15	7 8	mulscld	Could not format ( ph -> ( ( A /su B ) x.s ( C /su D ) ) e. No ) : No typesetting found for |- ( ph -> ( ( A /su B ) x.s ( C /su D ) ) e. No ) with typecode |-
16	2 4	mulscld	Could not format ( ph -> ( B x.s D ) e. No ) : No typesetting found for |- ( ph -> ( B x.s D ) e. No ) with typecode |-
17	2 4	mulsne0bd	Could not format ( ph -> ( ( B x.s D ) =/= 0s <-> ( B =/= 0s /\ D =/= 0s ) ) ) : No typesetting found for |- ( ph -> ( ( B x.s D ) =/= 0s <-> ( B =/= 0s /\ D =/= 0s ) ) ) with typecode |-
18	5 6 17	mpbir2and	Could not format ( ph -> ( B x.s D ) =/= 0s ) : No typesetting found for |- ( ph -> ( B x.s D ) =/= 0s ) with typecode |-
19	14 15 16 18	divsmuld	Could not format ( ph -> ( ( ( A x.s C ) /su ( B x.s D ) ) = ( ( A /su B ) x.s ( C /su D ) ) <-> ( ( B x.s D ) x.s ( ( A /su B ) x.s ( C /su D ) ) ) = ( A x.s C ) ) ) : No typesetting found for |- ( ph -> ( ( ( A x.s C ) /su ( B x.s D ) ) = ( ( A /su B ) x.s ( C /su D ) ) <-> ( ( B x.s D ) x.s ( ( A /su B ) x.s ( C /su D ) ) ) = ( A x.s C ) ) ) with typecode |-
20	13 19	mpbird	Could not format ( ph -> ( ( A x.s C ) /su ( B x.s D ) ) = ( ( A /su B ) x.s ( C /su D ) ) ) : No typesetting found for |- ( ph -> ( ( A x.s C ) /su ( B x.s D ) ) = ( ( A /su B ) x.s ( C /su D ) ) ) with typecode |-
21	20	eqcomd	Could not format ( ph -> ( ( A /su B ) x.s ( C /su D ) ) = ( ( A x.s C ) /su ( B x.s D ) ) ) : No typesetting found for |- ( ph -> ( ( A /su B ) x.s ( C /su D ) ) = ( ( A x.s C ) /su ( B x.s D ) ) ) with typecode |-