Theorem divsasswd

Description: An associative law for surreal division. Weak version. (Contributed by Scott Fenton, 14-Mar-2025)

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

Proof

Step	Hyp	Ref	Expression
1		divsasswd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		divsasswd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		divsasswd.3	$$⊢\phi \to C\in \mathrm{No}$$
4		divsasswd.4	Could not format ( ph -> C =/= 0s ) : No typesetting found for |- ( ph -> C =/= 0s ) with typecode |-
5		divsasswd.5	Could not format ( ph -> E. x e. No ( C x.s x ) = 1s ) : No typesetting found for |- ( ph -> E. x e. No ( C x.s x ) = 1s ) with typecode |-
6	2 3 4 5	divscan2wd	Could not format ( ph -> ( C x.s ( B /su C ) ) = B ) : No typesetting found for |- ( ph -> ( C x.s ( B /su C ) ) = B ) with typecode |-
7	6	oveq2d	Could not format ( ph -> ( A x.s ( C x.s ( B /su C ) ) ) = ( A x.s B ) ) : No typesetting found for |- ( ph -> ( A x.s ( C x.s ( B /su C ) ) ) = ( A x.s B ) ) with typecode |-
8	2 3 4 5	divsclwd	Could not format ( ph -> ( B /su C ) e. No ) : No typesetting found for |- ( ph -> ( B /su C ) e. No ) with typecode |-
9	3 1 8	muls12d	Could not format ( ph -> ( C x.s ( A x.s ( B /su C ) ) ) = ( A x.s ( C x.s ( B /su C ) ) ) ) : No typesetting found for |- ( ph -> ( C x.s ( A x.s ( B /su C ) ) ) = ( A x.s ( C x.s ( B /su C ) ) ) ) with typecode |-
10	1 2	mulscld	Could not format ( ph -> ( A x.s B ) e. No ) : No typesetting found for |- ( ph -> ( A x.s B ) e. No ) with typecode |-
11	10 3 4 5	divscan2wd	Could not format ( ph -> ( C x.s ( ( A x.s B ) /su C ) ) = ( A x.s B ) ) : No typesetting found for |- ( ph -> ( C x.s ( ( A x.s B ) /su C ) ) = ( A x.s B ) ) with typecode |-
12	7 9 11	3eqtr4rd	Could not format ( ph -> ( C x.s ( ( A x.s B ) /su C ) ) = ( C x.s ( A x.s ( B /su C ) ) ) ) : No typesetting found for |- ( ph -> ( C x.s ( ( A x.s B ) /su C ) ) = ( C x.s ( A x.s ( B /su C ) ) ) ) with typecode |-
13	10 3 4 5	divsclwd	Could not format ( ph -> ( ( A x.s B ) /su C ) e. No ) : No typesetting found for |- ( ph -> ( ( A x.s B ) /su C ) e. No ) with typecode |-
14	1 8	mulscld	Could not format ( ph -> ( A x.s ( B /su C ) ) e. No ) : No typesetting found for |- ( ph -> ( A x.s ( B /su C ) ) e. No ) with typecode |-
15	13 14 3 4	mulscan1d	Could not format ( ph -> ( ( C x.s ( ( A x.s B ) /su C ) ) = ( C x.s ( A x.s ( B /su C ) ) ) <-> ( ( A x.s B ) /su C ) = ( A x.s ( B /su C ) ) ) ) : No typesetting found for |- ( ph -> ( ( C x.s ( ( A x.s B ) /su C ) ) = ( C x.s ( A x.s ( B /su C ) ) ) <-> ( ( A x.s B ) /su C ) = ( A x.s ( B /su C ) ) ) ) with typecode |-
16	12 15	mpbid	Could not format ( ph -> ( ( A x.s B ) /su C ) = ( A x.s ( B /su C ) ) ) : No typesetting found for |- ( ph -> ( ( A x.s B ) /su C ) = ( A x.s ( B /su C ) ) ) with typecode |-