Theorem divsassd

Description: An associative law for surreal division. (Contributed by Scott Fenton, 16-Mar-2025)

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

Proof

Step	Hyp	Ref	Expression
1		divsassd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		divsassd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		divsassd.3	$$⊢\phi \to C\in \mathrm{No}$$
4		divsassd.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	divsasswd	$$⊢\phi \to \left(A{\cdot }_{s}B\right){/}_{\mathrm{su}}C=A{\cdot }_s\left(B{/}_{\mathrm{su}}C\right)$$