Theorem prlngref 29190

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 2764	. . 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 29188	. 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 1570   ∈ wcel 2143  ∃wrex 3089   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286   class class class wbr 5109  ran crn 5662  ‘cfv 6536  LineGclng 28703  hlGcplng 29055  parlnGcprlng 29186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fv 6544  df-prlng 29187

This theorem is referenced by:  perpprlng  29200  prlngex  29201  prlngplngtr  29209  prlngsymquadopp  29215  quadcgrprlng  29216