Theorem pw2divsdird

Description: Distribution of surreal division over addition for powers of two. (Contributed by Scott Fenton, 7-Nov-2025)

	Ref	Expression
Hypotheses	pw2divsdird.1	$$⊢\phi \to A\in \mathrm{No}$$
	pw2divsdird.2	$$⊢\phi \to B\in \mathrm{No}$$
	pw2divsdird.3	$$⊢\phi \to N\in {\mathbb{N}}_{\mathrm{0s}}$$
Assertion	pw2divsdird	Could not format assertion : No typesetting found for |- ( ph -> ( ( A +s B ) /su ( 2s ^su N ) ) = ( ( A /su ( 2s ^su N ) ) +s ( B /su ( 2s ^su N ) ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		pw2divsdird.1	$$⊢\phi \to A\in \mathrm{No}$$
2		pw2divsdird.2	$$⊢\phi \to B\in \mathrm{No}$$
3		pw2divsdird.3	$$⊢\phi \to N\in {\mathbb{N}}_{\mathrm{0s}}$$
4		1sno	$$⊢1_s\in \mathrm{No}$$
5	4	a1i	$$⊢\phi \to 1_s\in \mathrm{No}$$
6	5 3	pw2divscld	Could not format ( ph -> ( 1s /su ( 2s ^su N ) ) e. No ) : No typesetting found for |- ( ph -> ( 1s /su ( 2s ^su N ) ) e. No ) with typecode |-
7	1 2 6	addsdird	Could not format ( ph -> ( ( A +s B ) x.s ( 1s /su ( 2s ^su N ) ) ) = ( ( A x.s ( 1s /su ( 2s ^su N ) ) ) +s ( B x.s ( 1s /su ( 2s ^su N ) ) ) ) ) : No typesetting found for |- ( ph -> ( ( A +s B ) x.s ( 1s /su ( 2s ^su N ) ) ) = ( ( A x.s ( 1s /su ( 2s ^su N ) ) ) +s ( B x.s ( 1s /su ( 2s ^su N ) ) ) ) ) with typecode |-
8	1 2	addscld	$$⊢\phi \to A{+}_{s}B\in \mathrm{No}$$
9	8 3	pw2divsrecd	Could not format ( ph -> ( ( A +s B ) /su ( 2s ^su N ) ) = ( ( A +s B ) x.s ( 1s /su ( 2s ^su N ) ) ) ) : No typesetting found for |- ( ph -> ( ( A +s B ) /su ( 2s ^su N ) ) = ( ( A +s B ) x.s ( 1s /su ( 2s ^su N ) ) ) ) with typecode |-
10	1 3	pw2divsrecd	Could not format ( ph -> ( A /su ( 2s ^su N ) ) = ( A x.s ( 1s /su ( 2s ^su N ) ) ) ) : No typesetting found for |- ( ph -> ( A /su ( 2s ^su N ) ) = ( A x.s ( 1s /su ( 2s ^su N ) ) ) ) with typecode |-
11	2 3	pw2divsrecd	Could not format ( ph -> ( B /su ( 2s ^su N ) ) = ( B x.s ( 1s /su ( 2s ^su N ) ) ) ) : No typesetting found for |- ( ph -> ( B /su ( 2s ^su N ) ) = ( B x.s ( 1s /su ( 2s ^su N ) ) ) ) with typecode |-
12	10 11	oveq12d	Could not format ( ph -> ( ( A /su ( 2s ^su N ) ) +s ( B /su ( 2s ^su N ) ) ) = ( ( A x.s ( 1s /su ( 2s ^su N ) ) ) +s ( B x.s ( 1s /su ( 2s ^su N ) ) ) ) ) : No typesetting found for |- ( ph -> ( ( A /su ( 2s ^su N ) ) +s ( B /su ( 2s ^su N ) ) ) = ( ( A x.s ( 1s /su ( 2s ^su N ) ) ) +s ( B x.s ( 1s /su ( 2s ^su N ) ) ) ) ) with typecode |-
13	7 9 12	3eqtr4d	Could not format ( ph -> ( ( A +s B ) /su ( 2s ^su N ) ) = ( ( A /su ( 2s ^su N ) ) +s ( B /su ( 2s ^su N ) ) ) ) : No typesetting found for |- ( ph -> ( ( A +s B ) /su ( 2s ^su N ) ) = ( ( A /su ( 2s ^su N ) ) +s ( B /su ( 2s ^su N ) ) ) ) with typecode |-