Theorem prlngsym 29298

Description: Parallelism is symmetric. Theorem 12.5 of [Schwabhauser] p. 122. (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)
prlngsym.1	⊢ (𝜑 → 𝐴 ∥ 𝐵)

Assertion

Ref	Expression
prlngsym	⊢ (𝜑 → 𝐵 ∥ 𝐴)

Proof of Theorem prlngsym

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prlngsym.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∥ 𝐵)
2		brprlng.l	. . . . . . 7 ⊢ 𝐿 = (LineG‘𝐺)
3		brprlng.e	. . . . . . 7 ⊢ 𝐸 = (hlG‘𝐺)
4		brprlng.p	. . . . . . 7 ⊢ ∥ = (parlnG‘𝐺)
5		brprlng.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
6	2, 3, 4, 5	brprlng 29295	. . . . . 6 ⊢ (𝜑 → (𝐴 ∥ 𝐵 ↔ ((𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿) ∧ (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)))))
7	1, 6	mpbid 235	. . . . 5 ⊢ (𝜑 → ((𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿) ∧ (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅))))
8	7	simpld 500	. . . 4 ⊢ (𝜑 → (𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿))
9	8	simprd 501	. . 3 ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
10	8	simpld 500	. . 3 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
11		eqcom 2767	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
12	11	bilani 510	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 = 𝐴)
13		ancom 466	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ↔ (𝐵 ⊆ ℎ ∧ 𝐴 ⊆ ℎ))
14	13	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ↔ (𝐵 ⊆ ℎ ∧ 𝐴 ⊆ ℎ)))
15	14	rexbidv 3186	. . . . . 6 ⊢ (𝜑 → (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ↔ ∃ℎ ∈ ran 𝐸(𝐵 ⊆ ℎ ∧ 𝐴 ⊆ ℎ)))
16		incom 4155	. . . . . . . 8 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
17	16	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴))
18	17	eqeq1d 2762	. . . . . 6 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅))
19	15, 18	anbi12d 644	. . . . 5 ⊢ (𝜑 → ((∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅) ↔ (∃ℎ ∈ ran 𝐸(𝐵 ⊆ ℎ ∧ 𝐴 ⊆ ℎ) ∧ (𝐵 ∩ 𝐴) = ∅)))
20	19	biimpa 482	. . . 4 ⊢ ((𝜑 ∧ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)) → (∃ℎ ∈ ran 𝐸(𝐵 ⊆ ℎ ∧ 𝐴 ⊆ ℎ) ∧ (𝐵 ∩ 𝐴) = ∅))
21	7	simprd 501	. . . 4 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)))
22	12, 20, 21	orim12da 980	. . 3 ⊢ (𝜑 → (𝐵 = 𝐴 ∨ (∃ℎ ∈ ran 𝐸(𝐵 ⊆ ℎ ∧ 𝐴 ⊆ ℎ) ∧ (𝐵 ∩ 𝐴) = ∅)))
23	9, 10, 22	jca31 524	. 2 ⊢ (𝜑 → ((𝐵 ∈ ran 𝐿 ∧ 𝐴 ∈ ran 𝐿) ∧ (𝐵 = 𝐴 ∨ (∃ℎ ∈ ran 𝐸(𝐵 ⊆ ℎ ∧ 𝐴 ⊆ ℎ) ∧ (𝐵 ∩ 𝐴) = ∅))))
24	2, 3, 4, 5	brprlng 29295	. 2 ⊢ (𝜑 → (𝐵 ∥ 𝐴 ↔ ((𝐵 ∈ ran 𝐿 ∧ 𝐴 ∈ ran 𝐿) ∧ (𝐵 = 𝐴 ∨ (∃ℎ ∈ ran 𝐸(𝐵 ⊆ ℎ ∧ 𝐴 ⊆ ℎ) ∧ (𝐵 ∩ 𝐴) = ∅)))))
25	23, 24	mpbird 260	1 ⊢ (𝜑 → 𝐵 ∥ 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103  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:  prlngplngtr  29316  quadcgrprlng  29323