Theorem zs12subscl

Description: The dyadics are closed under subtraction. (Contributed by Scott Fenton, 12-Dec-2025)

	Ref	Expression
Assertion	zs12subscl	Could not format assertion : No typesetting found for |- ( ( A e. ZZ_s[1/2] /\ B e. ZZ_s[1/2] ) -> ( A -s B ) e. ZZ_s[1/2] ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		zs12no	Could not format ( A e. ZZ_s[1/2] -> A e. No ) : No typesetting found for |- ( A e. ZZ_s[1/2] -> A e. No ) with typecode |-
2		zs12no	Could not format ( B e. ZZ_s[1/2] -> B e. No ) : No typesetting found for |- ( B e. ZZ_s[1/2] -> B e. No ) with typecode |-
3		subsval	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to A{-}_{s}B=A{+}_s{+}_s\left(B\right)$$
4	1 2 3	syl2an	Could not format ( ( A e. ZZ_s[1/2] /\ B e. ZZ_s[1/2] ) -> ( A -s B ) = ( A +s ( -us ` B ) ) ) : No typesetting found for |- ( ( A e. ZZ_s[1/2] /\ B e. ZZ_s[1/2] ) -> ( A -s B ) = ( A +s ( -us ` B ) ) ) with typecode |-
5		zs12negscl	Could not format ( B e. ZZ_s[1/2] -> ( -us ` B ) e. ZZ_s[1/2] ) : No typesetting found for |- ( B e. ZZ_s[1/2] -> ( -us ` B ) e. ZZ_s[1/2] ) with typecode |-
6		zs12addscl	Could not format ( ( A e. ZZ_s[1/2] /\ ( -us ` B ) e. ZZ_s[1/2] ) -> ( A +s ( -us ` B ) ) e. ZZ_s[1/2] ) : No typesetting found for |- ( ( A e. ZZ_s[1/2] /\ ( -us ` B ) e. ZZ_s[1/2] ) -> ( A +s ( -us ` B ) ) e. ZZ_s[1/2] ) with typecode |-
7	5 6	sylan2	Could not format ( ( A e. ZZ_s[1/2] /\ B e. ZZ_s[1/2] ) -> ( A +s ( -us ` B ) ) e. ZZ_s[1/2] ) : No typesetting found for |- ( ( A e. ZZ_s[1/2] /\ B e. ZZ_s[1/2] ) -> ( A +s ( -us ` B ) ) e. ZZ_s[1/2] ) with typecode |-
8	4 7	eqeltrd	Could not format ( ( A e. ZZ_s[1/2] /\ B e. ZZ_s[1/2] ) -> ( A -s B ) e. ZZ_s[1/2] ) : No typesetting found for |- ( ( A e. ZZ_s[1/2] /\ B e. ZZ_s[1/2] ) -> ( A -s B ) e. ZZ_s[1/2] ) with typecode |-