Theorem sleadd1im

Description: Surreal less-than or equal cancels under addition. (Contributed by Scott Fenton, 21-Jan-2025)

	Ref	Expression
Assertion	sleadd1im	Could not format assertion : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( A +s C ) <_s ( B +s C ) -> A <_s B ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		sltadd1im	Could not format ( ( B e. No /\ A e. No /\ C e. No ) -> ( B ( B +s C ) ( B ( B +s C )
2	1	3com12	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( B ( B +s C ) ( B ( B +s C )
3		sltnle	$$⊢\left(B\in \mathrm{No}\wedge A\in \mathrm{No}\right)\to \left(B{<}_{s}A\leftrightarrow \neg A{\le }_{s}B\right)$$
4	3	ancoms	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(B{<}_{s}A\leftrightarrow \neg A{\le }_{s}B\right)$$
5	4	3adant3	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to \left(B{<}_{s}A\leftrightarrow \neg A{\le }_{s}B\right)$$
6		addscl	Could not format ( ( B e. No /\ C e. No ) -> ( B +s C ) e. No ) : No typesetting found for |- ( ( B e. No /\ C e. No ) -> ( B +s C ) e. No ) with typecode |-
7	6	3adant1	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( B +s C ) e. No ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( B +s C ) e. No ) with typecode |-
8		addscl	Could not format ( ( A e. No /\ C e. No ) -> ( A +s C ) e. No ) : No typesetting found for |- ( ( A e. No /\ C e. No ) -> ( A +s C ) e. No ) with typecode |-
9		sltnle	Could not format ( ( ( B +s C ) e. No /\ ( A +s C ) e. No ) -> ( ( B +s C ) -. ( A +s C ) <_s ( B +s C ) ) ) : No typesetting found for |- ( ( ( B +s C ) e. No /\ ( A +s C ) e. No ) -> ( ( B +s C ) -. ( A +s C ) <_s ( B +s C ) ) ) with typecode |-
10	7 8 9	3imp3i2an	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( B +s C ) -. ( A +s C ) <_s ( B +s C ) ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( B +s C ) -. ( A +s C ) <_s ( B +s C ) ) ) with typecode |-
11	2 5 10	3imtr3d	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( -. A <_s B -> -. ( A +s C ) <_s ( B +s C ) ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( -. A <_s B -> -. ( A +s C ) <_s ( B +s C ) ) ) with typecode |-
12	11	con4d	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( A +s C ) <_s ( B +s C ) -> A <_s B ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( A +s C ) <_s ( B +s C ) -> A <_s B ) ) with typecode |-