Theorem lruneq

Description: If two surreals share a birthday, then the union of their left and right sets are equal. (Contributed by Scott Fenton, 17-Sep-2024)

	Ref	Expression
Assertion	lruneq	$$⊢\left(X\in \mathrm{No}\wedge Y\in \mathrm{No}\wedge \mathrm{bday}\left(X\right)=\mathrm{bday}\left(Y\right)\right)\to L\left(X\right)\cup R\left(X\right)=L\left(Y\right)\cup R\left(Y\right)$$

Proof

Step	Hyp	Ref	Expression
1		fveq2	$$⊢\mathrm{bday}\left(X\right)=\mathrm{bday}\left(Y\right)\to \mathrm{Old}\left(\mathrm{bday}\left(X\right)\right)=\mathrm{Old}\left(\mathrm{bday}\left(Y\right)\right)$$
2	1	3ad2ant3	$$⊢\left(X\in \mathrm{No}\wedge Y\in \mathrm{No}\wedge \mathrm{bday}\left(X\right)=\mathrm{bday}\left(Y\right)\right)\to \mathrm{Old}\left(\mathrm{bday}\left(X\right)\right)=\mathrm{Old}\left(\mathrm{bday}\left(Y\right)\right)$$
3		lrold	$$⊢L\left(X\right)\cup R\left(X\right)=\mathrm{Old}\left(\mathrm{bday}\left(X\right)\right)$$
4		lrold	$$⊢L\left(Y\right)\cup R\left(Y\right)=\mathrm{Old}\left(\mathrm{bday}\left(Y\right)\right)$$
5	2 3 4	3eqtr4g	$$⊢\left(X\in \mathrm{No}\wedge Y\in \mathrm{No}\wedge \mathrm{bday}\left(X\right)=\mathrm{bday}\left(Y\right)\right)\to L\left(X\right)\cup R\left(X\right)=L\left(Y\right)\cup R\left(Y\right)$$