Theorem prlngd 29244

Description: Deduce parallelism between two lines 𝐴 and 𝐵. (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)
prlngd.a	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
prlngd.b	⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
prlngd.h	⊢ (𝜑 → 𝐻 ∈ ran 𝐸)
prlngd.1	⊢ (𝜑 → 𝐴 ⊆ 𝐻)
prlngd.2	⊢ (𝜑 → 𝐵 ⊆ 𝐻)
prlngd.3	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)

Assertion

Ref	Expression
prlngd	⊢ (𝜑 → 𝐴 ∥ 𝐵)

Proof of Theorem prlngd

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prlngd.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
2		prlngd.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
3	1, 2	jca 521	. 2 ⊢ (𝜑 → (𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿))
4		sseq2 3964	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝐴 ⊆ ℎ ↔ 𝐴 ⊆ 𝐻))
5		sseq2 3964	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝐵 ⊆ ℎ ↔ 𝐵 ⊆ 𝐻))
6	4, 5	anbi12d 644	. . . . 5 ⊢ (ℎ = 𝐻 → ((𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ↔ (𝐴 ⊆ 𝐻 ∧ 𝐵 ⊆ 𝐻)))
7		prlngd.h	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ ran 𝐸)
8		prlngd.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐻)
9		prlngd.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐻)
10	8, 9	jca 521	. . . . 5 ⊢ (𝜑 → (𝐴 ⊆ 𝐻 ∧ 𝐵 ⊆ 𝐻))
11	6, 7, 10	rspcedvdw 3586	. . . 4 ⊢ (𝜑 → ∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ))
12		prlngd.3	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
13	11, 12	jca 521	. . 3 ⊢ (𝜑 → (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅))
14	13	olcd 888	. 2 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)))
15		brprlng.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
16		brprlng.e	. . 3 ⊢ 𝐸 = (hlG‘𝐺)
17		brprlng.p	. . 3 ⊢ ∥ = (parlnG‘𝐺)
18		brprlng.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
19	15, 16, 17, 18	brprlng 29243	. 2 ⊢ (𝜑 → (𝐴 ∥ 𝐵 ↔ ((𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿) ∧ (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)))))
20	3, 14, 19	mpbir2and 726	1 ⊢ (𝜑 → 𝐴 ∥ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146  ∃wrex 3091   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286   class class class wbr 5111  ran crn 5664  ‘cfv 6540  LineGclng 28754  hlGcplng 29106  parlnGcprlng 29241

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fv 6548  df-prlng 29242

This theorem is used by:  perpprlng  29255  prlngplngtr  29264  prlnginn0  29265