Theorem subadds

Description: Relationship between addition and subtraction for surreals. (Contributed by Scott Fenton, 3-Feb-2025)

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

Proof

Step	Hyp	Ref	Expression
1		subsval	Could not format ( ( A e. No /\ B e. No ) -> ( A -s B ) = ( A +s ( -us ` B ) ) ) : No typesetting found for |- ( ( A e. No /\ B e. No ) -> ( A -s B ) = ( A +s ( -us ` B ) ) ) with typecode |-
2	1	3adant3	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( A -s B ) = ( A +s ( -us ` B ) ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( A -s B ) = ( A +s ( -us ` B ) ) ) with typecode |-
3	2	eqeq1d	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( A -s B ) = C <-> ( A +s ( -us ` B ) ) = C ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( A -s B ) = C <-> ( A +s ( -us ` B ) ) = C ) ) with typecode |-
4		simpl	$$⊢\left(B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to B\in \mathrm{No}$$
5		simpr	$$⊢\left(B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to C\in \mathrm{No}$$
6		negscl	Could not format ( B e. No -> ( -us ` B ) e. No ) : No typesetting found for |- ( B e. No -> ( -us ` B ) e. No ) with typecode |-
7	6	adantr	Could not format ( ( B e. No /\ C e. No ) -> ( -us ` B ) e. No ) : No typesetting found for |- ( ( B e. No /\ C e. No ) -> ( -us ` B ) e. No ) with typecode |-
8	4 5 7	adds32d	Could not format ( ( B e. No /\ C e. No ) -> ( ( B +s C ) +s ( -us ` B ) ) = ( ( B +s ( -us ` B ) ) +s C ) ) : No typesetting found for |- ( ( B e. No /\ C e. No ) -> ( ( B +s C ) +s ( -us ` B ) ) = ( ( B +s ( -us ` B ) ) +s C ) ) with typecode |-
9		negsid	Could not format ( B e. No -> ( B +s ( -us ` B ) ) = 0s ) : No typesetting found for |- ( B e. No -> ( B +s ( -us ` B ) ) = 0s ) with typecode |-
10	9	adantr	Could not format ( ( B e. No /\ C e. No ) -> ( B +s ( -us ` B ) ) = 0s ) : No typesetting found for |- ( ( B e. No /\ C e. No ) -> ( B +s ( -us ` B ) ) = 0s ) with typecode |-
11	10	oveq1d	Could not format ( ( B e. No /\ C e. No ) -> ( ( B +s ( -us ` B ) ) +s C ) = ( 0s +s C ) ) : No typesetting found for |- ( ( B e. No /\ C e. No ) -> ( ( B +s ( -us ` B ) ) +s C ) = ( 0s +s C ) ) with typecode |-
12		addslid	Could not format ( C e. No -> ( 0s +s C ) = C ) : No typesetting found for |- ( C e. No -> ( 0s +s C ) = C ) with typecode |-
13	12	adantl	Could not format ( ( B e. No /\ C e. No ) -> ( 0s +s C ) = C ) : No typesetting found for |- ( ( B e. No /\ C e. No ) -> ( 0s +s C ) = C ) with typecode |-
14	8 11 13	3eqtrd	Could not format ( ( B e. No /\ C e. No ) -> ( ( B +s C ) +s ( -us ` B ) ) = C ) : No typesetting found for |- ( ( B e. No /\ C e. No ) -> ( ( B +s C ) +s ( -us ` B ) ) = C ) with typecode |-
15	14	3adant1	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( B +s C ) +s ( -us ` B ) ) = C ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( B +s C ) +s ( -us ` B ) ) = C ) with typecode |-
16	15	eqeq1d	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( ( B +s C ) +s ( -us ` B ) ) = ( A +s ( -us ` B ) ) <-> C = ( A +s ( -us ` B ) ) ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( ( B +s C ) +s ( -us ` B ) ) = ( A +s ( -us ` B ) ) <-> C = ( A +s ( -us ` B ) ) ) ) with typecode |-
17		eqcom	Could not format ( C = ( A +s ( -us ` B ) ) <-> ( A +s ( -us ` B ) ) = C ) : No typesetting found for |- ( C = ( A +s ( -us ` B ) ) <-> ( A +s ( -us ` B ) ) = C ) with typecode |-
18	16 17	bitrdi	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( ( B +s C ) +s ( -us ` B ) ) = ( A +s ( -us ` B ) ) <-> ( A +s ( -us ` B ) ) = C ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( ( B +s C ) +s ( -us ` B ) ) = ( A +s ( -us ` B ) ) <-> ( A +s ( -us ` B ) ) = C ) ) with typecode |-
19		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 |-
20	19	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 |-
21		simp1	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to A\in \mathrm{No}$$
22		simp2	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to B\in \mathrm{No}$$
23	22	negscld	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 |-
24	20 21 23	addscan2d	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( ( B +s C ) +s ( -us ` B ) ) = ( A +s ( -us ` B ) ) <-> ( B +s C ) = A ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( ( B +s C ) +s ( -us ` B ) ) = ( A +s ( -us ` B ) ) <-> ( B +s C ) = A ) ) with typecode |-
25	3 18 24	3bitr2d	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( A -s B ) = C <-> ( B +s C ) = A ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( A -s B ) = C <-> ( B +s C ) = A ) ) with typecode |-