Theorem prlngd 29063

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 519	. 2 ⊢ (𝜑 → (𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿))
4		sseq2 3962	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝐴 ⊆ ℎ ↔ 𝐴 ⊆ 𝐻))
5		sseq2 3962	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝐵 ⊆ ℎ ↔ 𝐵 ⊆ 𝐻))
6	4, 5	anbi12d 641	. . . . 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 519	. . . . 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 519	. . 3 ⊢ (𝜑 → (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅))
14	13	olcd 885	. 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 29062	. 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 723	1 ⊢ (𝜑 → 𝐴 ∥ 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 858   = wceq 1560   ∈ wcel 2142  ∃wrex 3086   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285   class class class wbr 5100  ran crn 5648  ‘cfv 6521  LineGclng 28600  hlGcplng 28977  parlnGcprlng 29060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-iota 6477  df-fun 6523  df-fv 6529  df-prlng 29061

This theorem is referenced by: (None)