Theorem sltadd2

Description: Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 21-Jan-2025)

	Ref	Expression
Assertion	sltadd2	Could not format assertion : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( A ( C +s A )

Proof

Step	Hyp	Ref	Expression
1		sleadd2	Could not format ( ( B e. No /\ A e. No /\ C e. No ) -> ( B <_s A <-> ( C +s B ) <_s ( C +s A ) ) ) : No typesetting found for |- ( ( B e. No /\ A e. No /\ C e. No ) -> ( B <_s A <-> ( C +s B ) <_s ( C +s A ) ) ) with typecode |-
2	1	3com12	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( B <_s A <-> ( C +s B ) <_s ( C +s A ) ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( B <_s A <-> ( C +s B ) <_s ( C +s A ) ) ) with typecode |-
3	2	notbid	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( -. B <_s A <-> -. ( C +s B ) <_s ( C +s A ) ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( -. B <_s A <-> -. ( C +s B ) <_s ( C +s A ) ) ) with typecode |-
4		sltnle	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(A{<}_{s}B\leftrightarrow \neg B{\le }_{s}A\right)$$
5	4	3adant3	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to \left(A{<}_{s}B\leftrightarrow \neg B{\le }_{s}A\right)$$
6		simp3	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to C\in \mathrm{No}$$
7		simp1	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to A\in \mathrm{No}$$
8	6 7	addscld	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( C +s A ) e. No ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( C +s A ) e. No ) with typecode |-
9		simp2	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to B\in \mathrm{No}$$
10	6 9	addscld	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( C +s B ) e. No ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( C +s B ) e. No ) with typecode |-
11		sltnle	Could not format ( ( ( C +s A ) e. No /\ ( C +s B ) e. No ) -> ( ( C +s A ) -. ( C +s B ) <_s ( C +s A ) ) ) : No typesetting found for |- ( ( ( C +s A ) e. No /\ ( C +s B ) e. No ) -> ( ( C +s A ) -. ( C +s B ) <_s ( C +s A ) ) ) with typecode |-
12	8 10 11	syl2anc	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( C +s A ) -. ( C +s B ) <_s ( C +s A ) ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( C +s A ) -. ( C +s B ) <_s ( C +s A ) ) ) with typecode |-
13	3 5 12	3bitr4d	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( A ( C +s A ) ( A ( C +s A )