Theorem recsexd

Description: A nonzero surreal has a reciprocal. (Contributed by Scott Fenton, 16-Mar-2025)

	Ref	Expression
Hypotheses	recsexd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	recsexd.2	⊢ ( 𝜑 → 𝐴 ≠ 0s )
Assertion	recsexd	⊢ ( 𝜑 → ∃ 𝑥 ∈ No ( 𝐴 ·s 𝑥 ) = 1s )

Proof

Step	Hyp	Ref	Expression
1		recsexd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		recsexd.2	⊢ ( 𝜑 → 𝐴 ≠ 0s )
3		recsex	⊢ ( ( 𝐴 ∈ No ∧ 𝐴 ≠ 0s ) → ∃ 𝑥 ∈ No ( 𝐴 ·s 𝑥 ) = 1s )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ∃ 𝑥 ∈ No ( 𝐴 ·s 𝑥 ) = 1s )