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	⊢ ( ( 𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘ 𝑋 ) = ( bday ‘ 𝑌 ) ) → ( ( L ‘ 𝑋 ) ∪ ( R ‘ 𝑋 ) ) = ( ( L ‘ 𝑌 ) ∪ ( R ‘ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		fveq2	⊢ ( ( bday ‘ 𝑋 ) = ( bday ‘ 𝑌 ) → ( O ‘ ( bday ‘ 𝑋 ) ) = ( O ‘ ( bday ‘ 𝑌 ) ) )
2	1	3ad2ant3	⊢ ( ( 𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘ 𝑋 ) = ( bday ‘ 𝑌 ) ) → ( O ‘ ( bday ‘ 𝑋 ) ) = ( O ‘ ( bday ‘ 𝑌 ) ) )
3		lrold	⊢ ( ( L ‘ 𝑋 ) ∪ ( R ‘ 𝑋 ) ) = ( O ‘ ( bday ‘ 𝑋 ) )
4		lrold	⊢ ( ( L ‘ 𝑌 ) ∪ ( R ‘ 𝑌 ) ) = ( O ‘ ( bday ‘ 𝑌 ) )
5	2 3 4	3eqtr4g	⊢ ( ( 𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘ 𝑋 ) = ( bday ‘ 𝑌 ) ) → ( ( L ‘ 𝑋 ) ∪ ( R ‘ 𝑋 ) ) = ( ( L ‘ 𝑌 ) ∪ ( R ‘ 𝑌 ) ) )