Theorem prlngd 29124

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 3965	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝐴 ⊆ ℎ ↔ 𝐴 ⊆ 𝐻))
5		sseq2 3965	. . . . . 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 3587	. . . 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 29123	. 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 1563   ∈ wcel 2145  ∃wrex 3089   ∩ cin 3906   ⊆ wss 3907  ∅c0 4288   class class class wbr 5105  ran crn 5653  ‘cfv 6525  LineGclng 28661  hlGcplng 29003  parlnGcprlng 29121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-iota 6481  df-fun 6527  df-fv 6533  df-prlng 29122

This theorem is referenced by: (None)