Theorem prlngref 29144

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

Hypotheses

Ref	Expression
brprlng.l	⊢ 𝐿 = (LineG‘𝐺)
brprlng.e	⊢ 𝐸 = (hlG‘𝐺)
brprlng.p	⊢ ∥ = (parlnG‘𝐺)
brprlng.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)
prlngref.1	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)

Assertion

Ref	Expression
prlngref	⊢ (𝜑 → 𝐴 ∥ 𝐴)

Proof of Theorem prlngref

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prlngref.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
2	1, 1	jca 520	. 2 ⊢ (𝜑 → (𝐴 ∈ ran 𝐿 ∧ 𝐴 ∈ ran 𝐿))
3		eqidd 2770	. . 3 ⊢ (𝜑 → 𝐴 = 𝐴)
4	3	orcd 886	. 2 ⊢ (𝜑 → (𝐴 = 𝐴 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐴 ⊆ ℎ) ∧ (𝐴 ∩ 𝐴) = ∅)))
5		brprlng.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
6		brprlng.e	. . 3 ⊢ 𝐸 = (hlG‘𝐺)
7		brprlng.p	. . 3 ⊢ ∥ = (parlnG‘𝐺)
8		brprlng.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
9	5, 6, 7, 8	brprlng 29142	. 2 ⊢ (𝜑 → (𝐴 ∥ 𝐴 ↔ ((𝐴 ∈ ran 𝐿 ∧ 𝐴 ∈ ran 𝐿) ∧ (𝐴 = 𝐴 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐴 ⊆ ℎ) ∧ (𝐴 ∩ 𝐴) = ∅)))))
10	2, 4, 9	mpbir2and 725	1 ⊢ (𝜑 → 𝐴 ∥ 𝐴)

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149  ∃wrex 3095   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294   class class class wbr 5113  ran crn 5663  ‘cfv 6537  LineGclng 28668  hlGcplng 29012  parlnGcprlng 29140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fv 6545  df-prlng 29141

This theorem is referenced by:  perpprlng  29152  prlngex  29153