Theorem prlngsym 29065

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 29062	. . . . . 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 234	. . . . 5 ⊢ (𝜑 → ((𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿) ∧ (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅))))
8	7	simpld 498	. . . 4 ⊢ (𝜑 → (𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿))
9	8	simprd 499	. . 3 ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
10	8	simpld 498	. . 3 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
11		eqcom 2769	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
12	11	bilani 508	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 = 𝐴)
13		ancom 464	. . . . . . . 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 4161	. . . . . . . 8 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
17	16	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴))
18	17	eqeq1d 2764	. . . . . 6 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅))
19	15, 18	anbi12d 641	. . . . 5 ⊢ (𝜑 → ((∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅) ↔ (∃ℎ ∈ ran 𝐸(𝐵 ⊆ ℎ ∧ 𝐴 ⊆ ℎ) ∧ (𝐵 ∩ 𝐴) = ∅)))
20	19	biimpa 480	. . . 4 ⊢ ((𝜑 ∧ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)) → (∃ℎ ∈ ran 𝐸(𝐵 ⊆ ℎ ∧ 𝐴 ⊆ ℎ) ∧ (𝐵 ∩ 𝐴) = ∅))
21	7	simprd 499	. . . 4 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)))
22	12, 20, 21	orim12da 978	. . 3 ⊢ (𝜑 → (𝐵 = 𝐴 ∨ (∃ℎ ∈ ran 𝐸(𝐵 ⊆ ℎ ∧ 𝐴 ⊆ ℎ) ∧ (𝐵 ∩ 𝐴) = ∅)))
23	9, 10, 22	jca31 522	. 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 29062	. 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 259	1 ⊢ (𝜑 → 𝐵 ∥ 𝐴)

Syntax hints:   → wi 4   ↔ wb 208   ∧ 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)