Theorem elplnglnid

Description: The line A itself is a subset of a plane defined by the line A and a point R . (Contributed by Thierry Arnoux, 17-Jun-2026)

	Ref	Expression
Hypotheses	plngval.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	plngval.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	plngval.1	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	plngval.e	⊢ 𝐸 = ( hlG ‘ 𝐺 )
	plngval.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	elplng.a	⊢ ( 𝜑 → 𝐴 ∈ ran 𝐿 )
	elplng.r	⊢ ( 𝜑 → 𝑅 ∈ ( 𝑃 ∖ 𝐴 ) )
Assertion	elplnglnid	⊢ ( 𝜑 → 𝐴 ⊆ ( 𝐴 𝐸 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		plngval.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		plngval.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
3		plngval.1	⊢ 𝐿 = ( LineG ‘ 𝐺 )
4		plngval.e	⊢ 𝐸 = ( hlG ‘ 𝐺 )
5		plngval.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
6		elplng.a	⊢ ( 𝜑 → 𝐴 ∈ ran 𝐿 )
7		elplng.r	⊢ ( 𝜑 → 𝑅 ∈ ( 𝑃 ∖ 𝐴 ) )
8		simpr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝑧 ∈ 𝐴 )
9	8	3mix1d	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ( 𝑧 ∈ 𝐴 ∨ 𝑧 ( ( hpG ‘ 𝐺 ) ‘ 𝐴 ) 𝑅 ∨ 𝑧 { ⟨ 𝑎 , 𝑏 ⟩ ∣ ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐴 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐴 ) ) ∧ ∃ 𝑡 ∈ 𝐴 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) } 𝑅 ) )
10	5	adantr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝐺 ∈ TarskiG )
11	6	adantr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝐴 ∈ ran 𝐿 )
12	7	adantr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝑅 ∈ ( 𝑃 ∖ 𝐴 ) )
13		eqid	⊢ { ⟨ 𝑎 , 𝑏 ⟩ ∣ ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐴 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐴 ) ) ∧ ∃ 𝑡 ∈ 𝐴 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) } = { ⟨ 𝑎 , 𝑏 ⟩ ∣ ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐴 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐴 ) ) ∧ ∃ 𝑡 ∈ 𝐴 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) }
14	1 3 2 10 11 8	tglnpt	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝑧 ∈ 𝑃 )
15	1 2 3 4 10 11 12 13 14	elplng	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ( 𝑧 ∈ ( 𝐴 𝐸 𝑅 ) ↔ ( 𝑧 ∈ 𝐴 ∨ 𝑧 ( ( hpG ‘ 𝐺 ) ‘ 𝐴 ) 𝑅 ∨ 𝑧 { ⟨ 𝑎 , 𝑏 ⟩ ∣ ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐴 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐴 ) ) ∧ ∃ 𝑡 ∈ 𝐴 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) } 𝑅 ) ) )
16	9 15	mpbird	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝑧 ∈ ( 𝐴 𝐸 𝑅 ) )
17	16	ex	⊢ ( 𝜑 → ( 𝑧 ∈ 𝐴 → 𝑧 ∈ ( 𝐴 𝐸 𝑅 ) ) )
18	17	ssrdv	⊢ ( 𝜑 → 𝐴 ⊆ ( 𝐴 𝐸 𝑅 ) )