Theorem brprlng 29188

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 29187	. . . 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 6881	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
4		brprlng.l	. . . . . . . . . 10 ⊢ 𝐿 = (LineG‘𝐺)
5	3, 4	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
6	5	rneqd 5928	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
7	6	eleq2d 2849	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑎 ∈ ran (LineG‘𝑔) ↔ 𝑎 ∈ ran 𝐿))
8	6	eleq2d 2849	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑏 ∈ ran (LineG‘𝑔) ↔ 𝑏 ∈ ran 𝐿))
9	7, 8	anbi12d 643	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ↔ (𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿)))
10		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (hlG‘𝑔) = (hlG‘𝐺))
11		brprlng.e	. . . . . . . . . . 11 ⊢ 𝐸 = (hlG‘𝐺)
12	10, 11	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (hlG‘𝑔) = 𝐸)
13	12	rneqd 5928	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ran (hlG‘𝑔) = ran 𝐸)
14	13	rexeqdv 3324	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (∃ℎ ∈ ran (hlG‘𝑔)(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ↔ ∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ)))
15	14	anbi1d 642	. . . . . . 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 928	. . . . . 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 643	. . . . 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 5177	. . . 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 3478	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
21	4	fvexi 6895	. . . . . . 7 ⊢ 𝐿 ∈ V
22	21	rnex 7903	. . . . . 6 ⊢ ran 𝐿 ∈ V
23	22	a1i 11	. . . . 5 ⊢ (𝜑 → ran 𝐿 ∈ V)
24		simprll 790	. . . . 5 ⊢ ((𝜑 ∧ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ (𝑎 = 𝑏 ∨ (∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)))) → 𝑎 ∈ ran 𝐿)
25		simprlr 791	. . . . 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 8050	. . . 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 7013	. . 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 2810	. 2 ⊢ (𝜑 → ∥ = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ (𝑎 = 𝑏 ∨ (∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)))})
29		eqeq12 2780	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 = 𝑏 ↔ 𝐴 = 𝐵))
30		sseq1 3962	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ ℎ ↔ 𝐴 ⊆ ℎ))
31		sseq1 3962	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ⊆ ℎ ↔ 𝐵 ⊆ ℎ))
32	30, 31	bi2anan9 649	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ↔ (𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ)))
33	32	rexbidv 3189	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ↔ ∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ)))
34		ineq12 4168	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 ∩ 𝑏) = (𝐴 ∩ 𝐵))
35	34	eqeq1d 2765	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎 ∩ 𝑏) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅))
36	33, 35	anbi12d 643	. . . 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 931	. . 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 486	. 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 5522	1 ⊢ (𝜑 → (𝐴 ∥ 𝐵 ↔ ((𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿) ∧ (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  Vcvv 3455   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286   class class class wbr 5109  {copab 5173  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:  prlngd  29189  prlngref  29190  prlngsym  29191  prlngrcl1  29192  prlngrcl2  29193  prlngin0  29194  prlngpln  29195  prlnghpg  29196  dfprlng2  29197