Theorem sltsub2

Description: Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 4-Feb-2025)

	Ref	Expression
Assertion	sltsub2	Could not format assertion : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( A ( C -s B )

Proof

Step	Hyp	Ref	Expression
1		negscl	Could not format ( B e. No -> ( -us ` B ) e. No ) : No typesetting found for |- ( B e. No -> ( -us ` B ) e. No ) with typecode |-
2	1	3ad2ant2	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( -us ` B ) e. No ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( -us ` B ) e. No ) with typecode |-
3		negscl	Could not format ( A e. No -> ( -us ` A ) e. No ) : No typesetting found for |- ( A e. No -> ( -us ` A ) e. No ) with typecode |-
4	3	3ad2ant1	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( -us ` A ) e. No ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( -us ` A ) e. No ) with typecode |-
5		simp3	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to C\in \mathrm{No}$$
6	2 4 5	sltadd2d	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( -us ` B ) ( C +s ( -us ` B ) ) ( ( -us ` B ) ( C +s ( -us ` B ) )
7		sltneg	Could not format ( ( A e. No /\ B e. No ) -> ( A ( -us ` B ) ( A ( -us ` B )
8	7	3adant3	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( A ( -us ` B ) ( A ( -us ` B )
9		simp2	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to B\in \mathrm{No}$$
10	5 9	subsvald	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( C -s B ) = ( C +s ( -us ` B ) ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( C -s B ) = ( C +s ( -us ` B ) ) ) with typecode |-
11		simp1	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to A\in \mathrm{No}$$
12	5 11	subsvald	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( C -s A ) = ( C +s ( -us ` A ) ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( C -s A ) = ( C +s ( -us ` A ) ) ) with typecode |-
13	10 12	breq12d	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( C -s B ) ( C +s ( -us ` B ) ) ( ( C -s B ) ( C +s ( -us ` B ) )
14	6 8 13	3bitr4d	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( A ( C -s B ) ( A ( C -s B )