Theorem tgelrnpln 29111

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 7422	. 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 29110	. . 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 3918	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
10	7, 9	opelxpd 5702	. . . 4 ⊢ (𝜑 → ⟨𝐴, 𝑅⟩ ∈ (ran 𝐿 × 𝑃))
11	8	eldifbd 3919	. . . . 5 ⊢ (𝜑 → ¬ 𝑅 ∈ 𝐴)
12		df-br 5112	. . . . . 6 ⊢ (𝐴◡ E 𝑅 ↔ ⟨𝐴, 𝑅⟩ ∈ ◡ E )
13		brcnvg 5867	. . . . . . . 8 ⊢ ((𝐴 ∈ ran 𝐿 ∧ 𝑅 ∈ 𝑃) → (𝐴◡ E 𝑅 ↔ 𝑅 E 𝐴))
14	7, 9, 13	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (𝐴◡ E 𝑅 ↔ 𝑅 E 𝐴))
15		epelg 5564	. . . . . . . 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 3917	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝑅⟩ ∈ ((ran 𝐿 × 𝑃) ∖ ◡ E ))
21	6, 20	fnfvelrnd 7081	. 2 ⊢ (𝜑 → (𝐸‘⟨𝐴, 𝑅⟩) ∈ ran 𝐸)
22	1, 21	eqeltrid 2869	1 ⊢ (𝜑 → (𝐴𝐸𝑅) ∈ ran 𝐸)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2146   ∖ cdif 3903  ⟨cop 4597   class class class wbr 5111   E cep 5562   × cxp 5661  ^◡ccnv 5662  ran crn 5664  ‘cfv 6540  (class class class)co 7419  Basecbs 17293  LineGclng 28756  hlGcplng 29108

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-eprel 5563  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-plng 29109

This theorem is used by:  dfprlng2  29254  prlngex  29258  prlngmolem2  29260  prlngmid2  29268  quadcgrprlng  29273