Theorem pw2ltsdiv1d

Description: Surreal less-than relationship for division by a power of two. (Contributed by Scott Fenton, 18-Jan-2026)

	Ref	Expression
Hypotheses	pw2ltdivmulsd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	pw2ltdivmulsd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	pw2ltdivmulsd.3	⊢ ( 𝜑 → 𝑁 ∈ ℕ0s )
Assertion	pw2ltsdiv1d	⊢ ( 𝜑 → ( 𝐴 <s 𝐵 ↔ ( 𝐴 /su ( 2s ↑s 𝑁 ) ) <s ( 𝐵 /su ( 2s ↑s 𝑁 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		pw2ltdivmulsd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		pw2ltdivmulsd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		pw2ltdivmulsd.3	⊢ ( 𝜑 → 𝑁 ∈ ℕ0s )
4	2 3	pw2divscld	⊢ ( 𝜑 → ( 𝐵 /su ( 2s ↑s 𝑁 ) ) ∈ No )
5	1 4 3	pw2ltdivmulsd	⊢ ( 𝜑 → ( ( 𝐴 /su ( 2s ↑s 𝑁 ) ) <s ( 𝐵 /su ( 2s ↑s 𝑁 ) ) ↔ 𝐴 <s ( ( 2s ↑s 𝑁 ) ·s ( 𝐵 /su ( 2s ↑s 𝑁 ) ) ) ) )
6	2 3	pw2divscan2d	⊢ ( 𝜑 → ( ( 2s ↑s 𝑁 ) ·s ( 𝐵 /su ( 2s ↑s 𝑁 ) ) ) = 𝐵 )
7	6	breq2d	⊢ ( 𝜑 → ( 𝐴 <s ( ( 2s ↑s 𝑁 ) ·s ( 𝐵 /su ( 2s ↑s 𝑁 ) ) ) ↔ 𝐴 <s 𝐵 ) )
8	5 7	bitr2d	⊢ ( 𝜑 → ( 𝐴 <s 𝐵 ↔ ( 𝐴 /su ( 2s ↑s 𝑁 ) ) <s ( 𝐵 /su ( 2s ↑s 𝑁 ) ) ) )