Theorem prlngd 29189

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 520	. 2 ⊢ (𝜑 → (𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿))
4		sseq2 3963	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝐴 ⊆ ℎ ↔ 𝐴 ⊆ 𝐻))
5		sseq2 3963	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝐵 ⊆ ℎ ↔ 𝐵 ⊆ 𝐻))
6	4, 5	anbi12d 643	. . . . 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 520	. . . . 5 ⊢ (𝜑 → (𝐴 ⊆ 𝐻 ∧ 𝐵 ⊆ 𝐻))
11	6, 7, 10	rspcedvdw 3584	. . . 4 ⊢ (𝜑 → ∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ))
12		prlngd.3	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
13	11, 12	jca 520	. . 3 ⊢ (𝜑 → (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅))
14	13	olcd 887	. 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 29188	. 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 725	1 ⊢ (𝜑 → 𝐴 ∥ 𝐵)

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  prlngplngtr  29209  prlnginn0  29210