Theorem divsmuld

Description: Relationship between surreal division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025)

	Ref	Expression
Hypotheses	divsmuld.1	$$⊢\phi \to A\in \mathrm{No}$$
	divsmuld.2	$$⊢\phi \to B\in \mathrm{No}$$
	divsmuld.3	$$⊢\phi \to C\in \mathrm{No}$$
	divsmuld.4	$$⊢\phi \to C\ne 0_s$$
Assertion	divsmuld	$$⊢\phi \to \left(A{/}_{\mathrm{su}}C=B\leftrightarrow C{\cdot }_{s}B=A\right)$$

Proof

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