Theorem prlngrcl1 29299

Description: Reverse closure for parallelism. (Contributed by Thierry Arnoux, 5-Jul-2026.)

Hypotheses

Ref	Expression
prlngin0.l	⊢ 𝐿 = (LineG‘𝐺)
prlngin0.p	⊢ ∥ = (parlnG‘𝐺)
prlngin0.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)
prlngin0.1	⊢ (𝜑 → 𝐴 ∥ 𝐵)

Assertion

Ref	Expression
prlngrcl1	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)

Proof of Theorem prlngrcl1

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prlngin0.1	. . 3 ⊢ (𝜑 → 𝐴 ∥ 𝐵)
2		prlngin0.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
3		eqid 2760	. . . 4 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
4		prlngin0.p	. . . 4 ⊢ ∥ = (parlnG‘𝐺)
5		prlngin0.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
6	2, 3, 4, 5	brprlng 29295	. . 3 ⊢ (𝜑 → (𝐴 ∥ 𝐵 ↔ ((𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿) ∧ (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran (hlG‘𝐺)(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)))))
7	1, 6	mpbid 235	. 2 ⊢ (𝜑 → ((𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿) ∧ (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran (hlG‘𝐺)(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅))))
8	7	simplld 780	1 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103  ran crn 5656  ‘cfv 6533  LineGclng 28775  hlGcplng 29130  parlnGcprlng 29293

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fv 6541  df-prlng 29294

This theorem is used by:  prlngpln3  29306  prlngeq  29314  prlngpln4  29315  prlngplngtr  29316  prlngsymquadlem  29320  prlngsymquadopp  29322  tgaltai  29324