Theorem tgelrnpln 29311

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 7454	. 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 29310	. . 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 4054	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
10	7, 9	opelxpd 5697	. . . 4 ⊢ (𝜑 → ⟨𝐴, 𝑅⟩ ∈ (ran 𝐿 × 𝑃))
11	8	eldifbd 4055	. . . . 5 ⊢ (𝜑 → ¬ 𝑅 ∈ 𝐴)
12		df-br 5109	. . . . . 6 ⊢ (𝐴◡ E 𝑅 ↔ ⟨𝐴, 𝑅⟩ ∈ ◡ E )
13		brcnvg 5869	. . . . . . . 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 5559	. . . . . . . 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 4053	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝑅⟩ ∈ ((ran 𝐿 × 𝑃) ∖ ◡ E ))
21	6, 20	fnfvelrnd 7114	. 2 ⊢ (𝜑 → (𝐸‘⟨𝐴, 𝑅⟩) ∈ ran 𝐸)
22	1, 21	eqeltrid 2868	1 ⊢ (𝜑 → (𝐴𝐸𝑅) ∈ ran 𝐸)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147   ∖ cdif 4048  ⟨cop 4840   class class class wbr 5108   E cep 5557   × cxp 5656  ^◡ccnv 5657  ran crn 5659  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  LineGclng 28953  hlGcplng 29308

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-eprel 5558  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-plng 29309

This theorem is used by:  dfprlng2  29482  prlngex  29486  prlngmolem2  29488  prlngmid2  29496  quadcgrprlng  29501