Theorem prlngref

Description: Parallelism is reflexive. Theorem 12.4 of Schwabhauser p. 122. (Contributed by Thierry Arnoux, 18-Jun-2026)

	Ref	Expression
Hypotheses	brprlng.l	|- L = ( LineG ` G )
	brprlng.e	|- E = ( PlnG ` G )
	brprlng.p	|- .|| = ( parlnG ` G )
	brprlng.g	|- ( ph -> G e. V )
	prlngref.1	|- ( ph -> A e. ran L )
Assertion	prlngref	|- ( ph -> A .|| A )

Proof

Step	Hyp	Ref	Expression
1		brprlng.l	|- L = ( LineG ` G )
2		brprlng.e	|- E = ( PlnG ` G )
3		brprlng.p	|- .|| = ( parlnG ` G )
4		brprlng.g	|- ( ph -> G e. V )
5		prlngref.1	|- ( ph -> A e. ran L )
6	5 5	jca	|- ( ph -> ( A e. ran L /\ A e. ran L ) )
7		eqidd	|- ( ph -> A = A )
8	7	orcd	|- ( ph -> ( A = A \/ ( E. h e. ran E ( A C_ h /\ A C_ h ) /\ ( A i^i A ) = (/) ) ) )
9	1 2 3 4	brprlng	|- ( ph -> ( A .|| A <-> ( ( A e. ran L /\ A e. ran L ) /\ ( A = A \/ ( E. h e. ran E ( A C_ h /\ A C_ h ) /\ ( A i^i A ) = (/) ) ) ) ) )
10	6 8 9	mpbir2and	|- ( ph -> A .|| A )