Theorem tgelrnpln 29266

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 7430	. 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 29265	. . 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 4053	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
10	7, 9	opelxpd 5693	. . . 4 ⊢ (𝜑 → ⟨𝐴, 𝑅⟩ ∈ (ran 𝐿 × 𝑃))
11	8	eldifbd 4054	. . . . 5 ⊢ (𝜑 → ¬ 𝑅 ∈ 𝐴)
12		df-br 5106	. . . . . 6 ⊢ (𝐴◡ E 𝑅 ↔ ⟨𝐴, 𝑅⟩ ∈ ◡ E )
13		brcnvg 5861	. . . . . . . 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 5555	. . . . . . . 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 4052	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝑅⟩ ∈ ((ran 𝐿 × 𝑃) ∖ ◡ E ))
21	6, 20	fnfvelrnd 7090	. 2 ⊢ (𝜑 → (𝐸‘⟨𝐴, 𝑅⟩) ∈ ran 𝐸)
22	1, 21	eqeltrid 2867	1 ⊢ (𝜑 → (𝐴𝐸𝑅) ∈ ran 𝐸)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2146   ∖ cdif 4047  ⟨cop 4837   class class class wbr 5105   E cep 5553   × cxp 5652  ^◡ccnv 5653  ran crn 5655  ‘cfv 6546  (class class class)co 7427  Basecbs 17399  LineGclng 28908  hlGcplng 29263

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-eprel 5554  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-1st 8008  df-2nd 8009  df-plng 29264

This theorem is used by:  dfprlng2  29437  prlngex  29441  prlngmolem2  29443  prlngmid2  29451  quadcgrprlng  29456