Theorem addscan2

Description: Cancellation law for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025)

	Ref	Expression
Assertion	addscan2	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 +s 𝐶 ) = ( 𝐵 +s 𝐶 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		sleadd1	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐵 ↔ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )
2		sleadd1	⊢ ( ( 𝐵 ∈ No ∧ 𝐴 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 ≤s 𝐴 ↔ ( 𝐵 +s 𝐶 ) ≤s ( 𝐴 +s 𝐶 ) ) )
3	2	3com12	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 ≤s 𝐴 ↔ ( 𝐵 +s 𝐶 ) ≤s ( 𝐴 +s 𝐶 ) ) )
4	1 3	anbi12d	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐴 ) ↔ ( ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ∧ ( 𝐵 +s 𝐶 ) ≤s ( 𝐴 +s 𝐶 ) ) ) )
5		sletri3	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 = 𝐵 ↔ ( 𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐴 ) ) )
6	5	3adant3	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 = 𝐵 ↔ ( 𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐴 ) ) )
7		addscl	⊢ ( ( 𝐴 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 +s 𝐶 ) ∈ No )
8	7	3adant2	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 +s 𝐶 ) ∈ No )
9		addscl	⊢ ( ( 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 +s 𝐶 ) ∈ No )
10	9	3adant1	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 +s 𝐶 ) ∈ No )
11		sletri3	⊢ ( ( ( 𝐴 +s 𝐶 ) ∈ No ∧ ( 𝐵 +s 𝐶 ) ∈ No ) → ( ( 𝐴 +s 𝐶 ) = ( 𝐵 +s 𝐶 ) ↔ ( ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ∧ ( 𝐵 +s 𝐶 ) ≤s ( 𝐴 +s 𝐶 ) ) ) )
12	8 10 11	syl2anc	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 +s 𝐶 ) = ( 𝐵 +s 𝐶 ) ↔ ( ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ∧ ( 𝐵 +s 𝐶 ) ≤s ( 𝐴 +s 𝐶 ) ) ) )
13	4 6 12	3bitr4rd	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 +s 𝐶 ) = ( 𝐵 +s 𝐶 ) ↔ 𝐴 = 𝐵 ) )