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	Could not format assertion : No typesetting found for |- ( ( X e. No /\ Y e. No /\ ( bday ` X ) = ( bday ` Y ) ) -> ( ( _Left ` X ) u. ( _Right ` X ) ) = ( ( _Left ` Y ) u. ( _Right ` Y ) ) ) with typecode |-

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	Could not format ( ( _Left ` X ) u. ( _Right ` X ) ) = ( _Old ` ( bday ` X ) ) : No typesetting found for |- ( ( _Left ` X ) u. ( _Right ` X ) ) = ( _Old ` ( bday ` X ) ) with typecode |-
4		lrold	Could not format ( ( _Left ` Y ) u. ( _Right ` Y ) ) = ( _Old ` ( bday ` Y ) ) : No typesetting found for |- ( ( _Left ` Y ) u. ( _Right ` Y ) ) = ( _Old ` ( bday ` Y ) ) with typecode |-
5	2 3 4	3eqtr4g	Could not format ( ( X e. No /\ Y e. No /\ ( bday ` X ) = ( bday ` Y ) ) -> ( ( _Left ` X ) u. ( _Right ` X ) ) = ( ( _Left ` Y ) u. ( _Right ` Y ) ) ) : No typesetting found for |- ( ( X e. No /\ Y e. No /\ ( bday ` X ) = ( bday ` Y ) ) -> ( ( _Left ` X ) u. ( _Right ` X ) ) = ( ( _Left ` Y ) u. ( _Right ` Y ) ) ) with typecode |-