Theorem prlngd 29410

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 3957	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝐴 ⊆ ℎ ↔ 𝐴 ⊆ 𝐻))
5		sseq2 3957	. . . . . 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 3580	. . . 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 29409	. 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 2145  ∃wrex 3087   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103  ran crn 5652  ‘cfv 6537  LineGclng 28889  hlGcplng 29244  parlnGcprlng 29407

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 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  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 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6493  df-fun 6539  df-fv 6545  df-prlng 29408

This theorem is used by:  perpprlng  29421  prlngplngtr  29430  prlnginn0  29431