Theorem brprlng 29123

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 29122	. . . 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 6871	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
4		brprlng.l	. . . . . . . . . 10 ⊢ 𝐿 = (LineG‘𝐺)
5	3, 4	eqtr4di 2818	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
6	5	rneqd 5919	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
7	6	eleq2d 2851	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑎 ∈ ran (LineG‘𝑔) ↔ 𝑎 ∈ ran 𝐿))
8	6	eleq2d 2851	. . . . . . 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 6871	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (hlG‘𝑔) = (hlG‘𝐺))
11		brprlng.e	. . . . . . . . . . 11 ⊢ 𝐸 = (hlG‘𝐺)
12	10, 11	eqtr4di 2818	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (hlG‘𝑔) = 𝐸)
13	12	rneqd 5919	. . . . . . . . 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 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 3480	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
21	4	fvexi 6885	. . . . . . 7 ⊢ 𝐿 ∈ V
22	21	rnex 7895	. . . . . 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 8042	. . . 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 7003	. . 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 2812	. 2 ⊢ (𝜑 → ∥ = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ (𝑎 = 𝑏 ∨ (∃ℎ ∈ ran 𝐸(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)))})
29		eqeq12 2782	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 = 𝑏 ↔ 𝐴 = 𝐵))
30		sseq1 3964	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ ℎ ↔ 𝐴 ⊆ ℎ))
31		sseq1 3964	. . . . . . 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 4170	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 ∩ 𝑏) = (𝐴 ∩ 𝐵))
35	34	eqeq1d 2767	. . . . 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 5513	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 1563   ∈ wcel 2145  ∃wrex 3089  Vcvv 3457   ∩ cin 3906   ⊆ wss 3907  ∅c0 4288   class class class wbr 5105  {copab 5167  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:  prlngd  29124  prlngref  29125  prlngsym  29126