Theorem tgelrnpln 29006

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 7403	. 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 29005	. . 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 3919	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
10	7, 9	opelxpd 5691	. . . 4 ⊢ (𝜑 → ⟨𝐴, 𝑅⟩ ∈ (ran 𝐿 × 𝑃))
11	8	eldifbd 3920	. . . . 5 ⊢ (𝜑 → ¬ 𝑅 ∈ 𝐴)
12		df-br 5106	. . . . . 6 ⊢ (𝐴◡ E 𝑅 ↔ ⟨𝐴, 𝑅⟩ ∈ ◡ E )
13		brcnvg 5856	. . . . . . . 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 5553	. . . . . . . 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 3918	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝑅⟩ ∈ ((ran 𝐿 × 𝑃) ∖ ◡ E ))
21	6, 20	fnfvelrnd 7067	. 2 ⊢ (𝜑 → (𝐸‘⟨𝐴, 𝑅⟩) ∈ ran 𝐸)
22	1, 21	eqeltrid 2869	1 ⊢ (𝜑 → (𝐴𝐸𝑅) ∈ ran 𝐸)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1563   ∈ wcel 2145   ∖ cdif 3904  ⟨cop 4591   class class class wbr 5105   E cep 5551   × cxp 5650  ^◡ccnv 5651  ran crn 5653  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  LineGclng 28661  hlGcplng 29003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-eprel 5552  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-plng 29004

This theorem is referenced by: (None)