Theorem subsdid

Description: Distribution of surreal multiplication over subtraction. (Contributed by Scott Fenton, 9-Mar-2025)

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

Proof

Step	Hyp	Ref	Expression
1		addsdid.1	$$⊢\phi \to A\in \mathrm{No}$$
2		addsdid.2	$$⊢\phi \to B\in \mathrm{No}$$
3		addsdid.3	$$⊢\phi \to C\in \mathrm{No}$$
4	2 3	subscld	Could not format ( ph -> ( B -s C ) e. No ) : No typesetting found for |- ( ph -> ( B -s C ) e. No ) with typecode |-
5	1 3 4	addsdid	Could not format ( ph -> ( A x.s ( C +s ( B -s C ) ) ) = ( ( A x.s C ) +s ( A x.s ( B -s C ) ) ) ) : No typesetting found for |- ( ph -> ( A x.s ( C +s ( B -s C ) ) ) = ( ( A x.s C ) +s ( A x.s ( B -s C ) ) ) ) with typecode |-
6		pncan3s	Could not format ( ( C e. No /\ B e. No ) -> ( C +s ( B -s C ) ) = B ) : No typesetting found for |- ( ( C e. No /\ B e. No ) -> ( C +s ( B -s C ) ) = B ) with typecode |-
7	3 2 6	syl2anc	Could not format ( ph -> ( C +s ( B -s C ) ) = B ) : No typesetting found for |- ( ph -> ( C +s ( B -s C ) ) = B ) with typecode |-
8	7	oveq2d	Could not format ( ph -> ( A x.s ( C +s ( B -s C ) ) ) = ( A x.s B ) ) : No typesetting found for |- ( ph -> ( A x.s ( C +s ( B -s C ) ) ) = ( A x.s B ) ) with typecode |-
9	5 8	eqtr3d	Could not format ( ph -> ( ( A x.s C ) +s ( A x.s ( B -s C ) ) ) = ( A x.s B ) ) : No typesetting found for |- ( ph -> ( ( A x.s C ) +s ( A x.s ( B -s C ) ) ) = ( A x.s B ) ) 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	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 |-
12	1 4	mulscld	Could not format ( ph -> ( A x.s ( B -s C ) ) e. No ) : No typesetting found for |- ( ph -> ( A x.s ( B -s C ) ) e. No ) with typecode |-
13	10 11 12	subaddsd	Could not format ( ph -> ( ( ( A x.s B ) -s ( A x.s C ) ) = ( A x.s ( B -s C ) ) <-> ( ( A x.s C ) +s ( A x.s ( B -s C ) ) ) = ( A x.s B ) ) ) : No typesetting found for |- ( ph -> ( ( ( A x.s B ) -s ( A x.s C ) ) = ( A x.s ( B -s C ) ) <-> ( ( A x.s C ) +s ( A x.s ( B -s C ) ) ) = ( A x.s B ) ) ) with typecode |-
14	9 13	mpbird	Could not format ( ph -> ( ( A x.s B ) -s ( A x.s C ) ) = ( A x.s ( B -s C ) ) ) : No typesetting found for |- ( ph -> ( ( A x.s B ) -s ( A x.s C ) ) = ( A x.s ( B -s C ) ) ) with typecode |-
15	14	eqcomd	Could not format ( ph -> ( A x.s ( B -s C ) ) = ( ( A x.s B ) -s ( A x.s C ) ) ) : No typesetting found for |- ( ph -> ( A x.s ( B -s C ) ) = ( ( A x.s B ) -s ( A x.s C ) ) ) with typecode |-