Theorem divsclwd

Description: Weak division closure law. (Contributed by Scott Fenton, 12-Mar-2025)

	Ref	Expression
Hypotheses	divsclwd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	divsclwd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	divsclwd.3	⊢ ( 𝜑 → 𝐵 ≠ 0s )
	divsclwd.4	⊢ ( 𝜑 → ∃ 𝑥 ∈ No ( 𝐵 ·s 𝑥 ) = 1s )
Assertion	divsclwd	⊢ ( 𝜑 → ( 𝐴 /su 𝐵 ) ∈ No )

Proof

Step	Hyp	Ref	Expression
1		divsclwd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		divsclwd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		divsclwd.3	⊢ ( 𝜑 → 𝐵 ≠ 0s )
4		divsclwd.4	⊢ ( 𝜑 → ∃ 𝑥 ∈ No ( 𝐵 ·s 𝑥 ) = 1s )
5		divsclw	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐵 ≠ 0s ) ∧ ∃ 𝑥 ∈ No ( 𝐵 ·s 𝑥 ) = 1s ) → ( 𝐴 /su 𝐵 ) ∈ No )
6	1 2 3 4 5	syl31anc	⊢ ( 𝜑 → ( 𝐴 /su 𝐵 ) ∈ No )