Theorem prlngref 29167

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 2771	. . 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 29165	. 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 1568   ∈ wcel 2150  ∃wrex 3096   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294   class class class wbr 5114  ran crn 5666  ‘cfv 6540  LineGclng 28683  hlGcplng 29033  parlnGcprlng 29163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  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 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-iota 6496  df-fun 6542  df-fv 6548  df-prlng 29164

This theorem is referenced by:  perpprlng  29177  prlngex  29178  prlngplngtr  29185