Theorem mulsfn

Description: Surreal multiplication is a function over surreals. (Contributed by Scott Fenton, 4-Feb-2025)

	Ref	Expression
Assertion	mulsfn	$$⊢{\cdot }_{s}Fn\left(\mathrm{No}\times \mathrm{No}\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-muls	$$⊢{\cdot }_s={norec2}_s\#A\#\left(\left(z\in \mathrm{V},m\in \mathrm{V}⟼⦋1^{st}\left(z\right)/x⦌⦋2^{nd}\left(z\right)/y⦌\left(\left(\left\{a|\exists p\in L\left(x\right)\exists q\in L\left(y\right)a=\left(\left(pmy\right){+}_s\left(xmq\right)\right){-}_s\left(pmq\right)\right\}\cup \left\{b|\exists r\in R\left(x\right)\exists s\in R\left(y\right)b=\left(\left(rmy\right){+}_s\left(xms\right)\right){-}_s\left(rms\right)\right\}\right){|}_s\left(\left\{c|\exists t\in L\left(x\right)\exists u\in R\left(y\right)c=\left(\left(tmy\right){+}_s\left(xmu\right)\right){-}_s\left(tmu\right)\right\}\cup \left\{d|\exists v\in R\left(x\right)\exists w\in L\left(y\right)d=\left(\left(vmy\right){+}_s\left(xmw\right)\right){-}_s\left(vmw\right)\right\}\right)\right)\right)\right)$$
2	1	norec2fn	$$⊢{\cdot }_{s}Fn\left(\mathrm{No}\times \mathrm{No}\right)$$