Theorem divsclwd

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

	Ref	Expression
Hypotheses	divsclwd.1	$$⊢\phi \to A\in \mathrm{No}$$
	divsclwd.2	$$⊢\phi \to B\in \mathrm{No}$$
	divsclwd.3	$$⊢\phi \to B\ne 0_s$$
	divsclwd.4	$$⊢\phi \to \exists x\in \mathrm{No}B{\cdot }_{s}x=1_s$$
Assertion	divsclwd	$$⊢\phi \to A{/}_{\mathrm{su}}B\in \mathrm{No}$$

Proof

Step	Hyp	Ref	Expression
1		divsclwd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		divsclwd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		divsclwd.3	$$⊢\phi \to B\ne 0_s$$
4		divsclwd.4	$$⊢\phi \to \exists x\in \mathrm{No}B{\cdot }_{s}x=1_s$$
5		divsclw	$$⊢\left(\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge B\ne 0_s\right)\wedge \exists x\in \mathrm{No}B{\cdot }_{s}x=1_s\right)\to A{/}_{\mathrm{su}}B\in \mathrm{No}$$
6	1 2 3 4 5	syl31anc	$$⊢\phi \to A{/}_{\mathrm{su}}B\in \mathrm{No}$$