Theorem tgelrnpln 29015

Description: The property of being a plane, generated by a line and a point. (Contributed by Thierry Arnoux, 17-Jun-2026.)

Hypotheses

Ref	Expression
tgplnfn.p	⊢ 𝑃 = (Base‘𝐺)
tgplnfn.l	⊢ 𝐿 = (LineG‘𝐺)
tgplnfn.i	⊢ 𝐸 = (hlG‘𝐺)
tgplnfn.1	⊢ (𝜑 → 𝐺 ∈ 𝑉)
tgelrnpln.a	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
tgelrnpln.r	⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))

Assertion

Ref	Expression
tgelrnpln	⊢ (𝜑 → (𝐴𝐸𝑅) ∈ ran 𝐸)

Proof of Theorem tgelrnpln

Step	Hyp	Ref	Expression
1		df-ov 7414	. 2 ⊢ (𝐴𝐸𝑅) = (𝐸‘⟨𝐴, 𝑅⟩)
2		tgplnfn.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
3		tgplnfn.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
4		tgplnfn.i	. . . 4 ⊢ 𝐸 = (hlG‘𝐺)
5		tgplnfn.1	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
6	2, 3, 4, 5	tgplnfn 29014	. . 3 ⊢ (𝜑 → 𝐸 Fn ((ran 𝐿 × 𝑃) ∖ ◡ E ))
7		tgelrnpln.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
8		tgelrnpln.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))
9	8	eldifad 3925	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
10	7, 9	opelxpd 5701	. . . 4 ⊢ (𝜑 → ⟨𝐴, 𝑅⟩ ∈ (ran 𝐿 × 𝑃))
11	8	eldifbd 3926	. . . . 5 ⊢ (𝜑 → ¬ 𝑅 ∈ 𝐴)
12		df-br 5114	. . . . . 6 ⊢ (𝐴◡ E 𝑅 ↔ ⟨𝐴, 𝑅⟩ ∈ ◡ E )
13		brcnvg 5866	. . . . . . . 8 ⊢ ((𝐴 ∈ ran 𝐿 ∧ 𝑅 ∈ 𝑃) → (𝐴◡ E 𝑅 ↔ 𝑅 E 𝐴))
14	7, 9, 13	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → (𝐴◡ E 𝑅 ↔ 𝑅 E 𝐴))
15		epelg 5563	. . . . . . . 8 ⊢ (𝐴 ∈ ran 𝐿 → (𝑅 E 𝐴 ↔ 𝑅 ∈ 𝐴))
16	7, 15	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑅 E 𝐴 ↔ 𝑅 ∈ 𝐴))
17	14, 16	bitrd 282	. . . . . 6 ⊢ (𝜑 → (𝐴◡ E 𝑅 ↔ 𝑅 ∈ 𝐴))
18	12, 17	bitr3id 288	. . . . 5 ⊢ (𝜑 → (⟨𝐴, 𝑅⟩ ∈ ◡ E ↔ 𝑅 ∈ 𝐴))
19	11, 18	mtbird 328	. . . 4 ⊢ (𝜑 → ¬ ⟨𝐴, 𝑅⟩ ∈ ◡ E )
20	10, 19	eldifd 3924	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝑅⟩ ∈ ((ran 𝐿 × 𝑃) ∖ ◡ E ))
21	6, 20	fnfvelrnd 7078	. 2 ⊢ (𝜑 → (𝐸‘⟨𝐴, 𝑅⟩) ∈ ran 𝐸)
22	1, 21	eqeltrid 2873	1 ⊢ (𝜑 → (𝐴𝐸𝑅) ∈ ran 𝐸)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149   ∖ cdif 3910  ⟨cop 4600   class class class wbr 5113   E cep 5561   × cxp 5660  ^◡ccnv 5661  ran crn 5663  ‘cfv 6537  (class class class)co 7411  Basecbs 17268  LineGclng 28668  hlGcplng 29012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-eprel 5562  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-plng 29013

This theorem is referenced by:  prlngex  29153  prlngmolem2  29155