Theorem brprlng 29295

Description: Property of two lines 𝐴 and 𝐵 to be parallel. (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)

Assertion

Ref	Expression
brprlng	⊢ (𝜑 → (𝐴 ∥ 𝐵 ↔ ((𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿) ∧ (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)))))

Distinct variable groups:   𝐴,ℎ   𝐵,ℎ   ℎ,𝐸   ℎ,𝐺

Allowed substitution hints:   𝜑(ℎ)   ∥ (ℎ)   𝐿(ℎ)   𝑉(ℎ)

Proof of Theorem brprlng

Dummy variables 𝑎 𝑏 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brprlng.p	. . 3 ⊢ ∥ = (parlnG‘𝐺)
2		df-prlng 29294	. . . 4 ⊢ parlnG = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ (𝑎 = 𝑏 ∨ (∃ℎ ∈ ran (hlG‘𝑔)(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)))})
3		fveq2 6878	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
4		brprlng.l	. . . . . . . . . 10 ⊢ 𝐿 = (LineG‘𝐺)
5	3, 4	eqtr4di 2813	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
6	5	rneqd 5922	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
7	6	eleq2d 2846	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑎 ∈ ran (LineG‘𝑔) ↔ 𝑎 ∈ ran 𝐿))
8	6	eleq2d 2846	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑏 ∈ ran (LineG‘𝑔) ↔ 𝑏 ∈ ran 𝐿))
9	7, 8	anbi12d 644	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ↔ (𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿)))
10		fveq2 6878	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (hlG‘𝑔) = (hlG‘𝐺))
11		brprlng.e	. . . . . . . . . . 11 ⊢ 𝐸 = (hlG‘𝐺)
12	10, 11	eqtr4di 2813	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (hlG‘𝑔) = 𝐸)
13	12	rneqd 5922	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ran (hlG‘𝑔) = ran 𝐸)
14	13	rexeqdv 3320	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (∃ℎ ∈ ran (hlG‘𝑔)(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ↔ ∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ)))
15	14	anbi1d 643	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((∃ℎ ∈ ran (hlG‘𝑔)(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅) ↔ (∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)))
16	15	orbi2d 929	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑎 = 𝑏 ∨ (∃ℎ ∈ ran (hlG‘𝑔)(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)) ↔ (𝑎 = 𝑏 ∨ (∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅))))
17	9, 16	anbi12d 644	. . . . 5 ⊢ (𝑔 = 𝐺 → (((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ (𝑎 = 𝑏 ∨ (∃ℎ ∈ ran (hlG‘𝑔)(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅))) ↔ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ (𝑎 = 𝑏 ∨ (∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)))))
18	17	opabbidv 5171	. . . 4 ⊢ (𝑔 = 𝐺 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ (𝑎 = 𝑏 ∨ (∃ℎ ∈ ran (hlG‘𝑔)(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ (𝑎 = 𝑏 ∨ (∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)))})
19		brprlng.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
20	19	elexd 3473	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
21	4	fvexi 6892	. . . . . . 7 ⊢ 𝐿 ∈ V
22	21	rnex 7907	. . . . . 6 ⊢ ran 𝐿 ∈ V
23	22	a1i 11	. . . . 5 ⊢ (𝜑 → ran 𝐿 ∈ V)
24		simprll 791	. . . . 5 ⊢ ((𝜑 ∧ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ (𝑎 = 𝑏 ∨ (∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)))) → 𝑎 ∈ ran 𝐿)
25		simprlr 792	. . . . 5 ⊢ ((𝜑 ∧ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ (𝑎 = 𝑏 ∨ (∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)))) → 𝑏 ∈ ran 𝐿)
26	23, 23, 24, 25	opabex2 8054	. . . 4 ⊢ (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ (𝑎 = 𝑏 ∨ (∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)))} ∈ V)
27	2, 18, 20, 26	fvmptd3 7010	. . 3 ⊢ (𝜑 → (parlnG‘𝐺) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ (𝑎 = 𝑏 ∨ (∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)))})
28	1, 27	eqtrid 2807	. 2 ⊢ (𝜑 → ∥ = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ (𝑎 = 𝑏 ∨ (∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)))})
29		eqeq12 2777	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 = 𝑏 ↔ 𝐴 = 𝐵))
30		sseq1 3956	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ ℎ ↔ 𝐴 ⊆ ℎ))
31		sseq1 3956	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ⊆ ℎ ↔ 𝐵 ⊆ ℎ))
32	30, 31	bi2anan9 650	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ↔ (𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ)))
33	32	rexbidv 3186	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ↔ ∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ)))
34		ineq12 4161	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 ∩ 𝑏) = (𝐴 ∩ 𝐵))
35	34	eqeq1d 2762	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎 ∩ 𝑏) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅))
36	33, 35	anbi12d 644	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅) ↔ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)))
37	29, 36	orbi12d 932	. . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎 = 𝑏 ∨ (∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)) ↔ (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅))))
38	37	adantl 487	. 2 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → ((𝑎 = 𝑏 ∨ (∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)) ↔ (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅))))
39	28, 38	brab2d 5516	1 ⊢ (𝜑 → (𝐴 ∥ 𝐵 ↔ ((𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿) ∧ (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  Vcvv 3450   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103  {copab 5167  ran crn 5656  ‘cfv 6533  LineGclng 28775  hlGcplng 29130  parlnGcprlng 29293

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 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  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 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fv 6541  df-prlng 29294

This theorem is used by:  prlngd  29296  prlngref  29297  prlngsym  29298  prlngrcl1  29299  prlngrcl2  29300  prlngin0  29301  prlngpln  29302  prlnghpg  29303  dfprlng2  29304