Theorem prlngrcl2 29308

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
prlngrcl2	⊢ (𝜑 → 𝐵 ∈ ran 𝐿)

Proof of Theorem prlngrcl2

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 2762	. . . 4 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
4		prlngin0.p	. . . 4 ⊢ ∥ = (parlnG‘𝐺)
5		prlngin0.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
6	2, 3, 4, 5	brprlng 29303	. . 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	simplrd 782	1 ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∃wrex 3088   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282   class class class wbr 5107  ran crn 5660  ‘cfv 6537  LineGclng 28783  hlGcplng 29138  parlnGcprlng 29301

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fv 6545  df-prlng 29302

This theorem is used by:  prlngpln3  29314  prlngeq  29322  prlngplngtr  29324  prlnginn0  29325  prlngsymquadlem  29328  prlngsymquadopp  29330  quadcgrprlng  29331  tgaltai  29332