Theorem noreceuw

Description: If a surreal has a reciprocal, then it has unique division. (Contributed by Scott Fenton, 12-Mar-2025)

	Ref	Expression
Assertion	noreceuw	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ ∃ 𝑥 ∈ No ( 𝐴 ·s 𝑥 ) = 1s ) → ∃! 𝑦 ∈ No ( 𝐴 ·s 𝑦 ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		norecdiv	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ ∃ 𝑥 ∈ No ( 𝐴 ·s 𝑥 ) = 1s ) → ∃ 𝑦 ∈ No ( 𝐴 ·s 𝑦 ) = 𝐵 )
2		divsmo	⊢ ( ( 𝐴 ∈ No ∧ 𝐴 ≠ 0s ) → ∃* 𝑦 ∈ No ( 𝐴 ·s 𝑦 ) = 𝐵 )
3	2	3adant3	⊢ ( ( 𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) → ∃* 𝑦 ∈ No ( 𝐴 ·s 𝑦 ) = 𝐵 )
4	3	adantr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ ∃ 𝑥 ∈ No ( 𝐴 ·s 𝑥 ) = 1s ) → ∃* 𝑦 ∈ No ( 𝐴 ·s 𝑦 ) = 𝐵 )
5		reu5	⊢ ( ∃! 𝑦 ∈ No ( 𝐴 ·s 𝑦 ) = 𝐵 ↔ ( ∃ 𝑦 ∈ No ( 𝐴 ·s 𝑦 ) = 𝐵 ∧ ∃* 𝑦 ∈ No ( 𝐴 ·s 𝑦 ) = 𝐵 ) )
6	1 4 5	sylanbrc	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ ∃ 𝑥 ∈ No ( 𝐴 ·s 𝑥 ) = 1s ) → ∃! 𝑦 ∈ No ( 𝐴 ·s 𝑦 ) = 𝐵 )