Theorem plngcp

Description: The plane defined by a line A and a point R can also be defined using a different point R on the same plane: changes the point used to define the plane. Theorem 9.21 of Schwabhauser p. 74. (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 )
	plngcp.a	⊢ ( 𝜑 → 𝐴 ∈ ran 𝐿 )
	plngcp.r	⊢ ( 𝜑 → 𝑅 ∈ ( 𝑃 ∖ 𝐴 ) )
	plngcp.s	⊢ ( 𝜑 → 𝑆 ∈ ( ( 𝐴 𝐸 𝑅 ) ∖ 𝐴 ) )
Assertion	plngcp	⊢ ( 𝜑 → ( 𝐴 𝐸 𝑅 ) = ( 𝐴 𝐸 𝑆 ) )

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		plngcp.a	⊢ ( 𝜑 → 𝐴 ∈ ran 𝐿 )
7		plngcp.r	⊢ ( 𝜑 → 𝑅 ∈ ( 𝑃 ∖ 𝐴 ) )
8		plngcp.s	⊢ ( 𝜑 → 𝑆 ∈ ( ( 𝐴 𝐸 𝑅 ) ∖ 𝐴 ) )
9		eleq1w	⊢ ( 𝑎 = 𝑐 → ( 𝑎 ∈ ( 𝑃 ∖ 𝐴 ) ↔ 𝑐 ∈ ( 𝑃 ∖ 𝐴 ) ) )
10		eleq1w	⊢ ( 𝑏 = 𝑑 → ( 𝑏 ∈ ( 𝑃 ∖ 𝐴 ) ↔ 𝑑 ∈ ( 𝑃 ∖ 𝐴 ) ) )
11	9 10	bi2anan9	⊢ ( ( 𝑎 = 𝑐 ∧ 𝑏 = 𝑑 ) → ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐴 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐴 ) ) ↔ ( 𝑐 ∈ ( 𝑃 ∖ 𝐴 ) ∧ 𝑑 ∈ ( 𝑃 ∖ 𝐴 ) ) ) )
12		eleq1w	⊢ ( 𝑡 = 𝑠 → ( 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ↔ 𝑠 ∈ ( 𝑎 𝐼 𝑏 ) ) )
13	12	cbvrexvw	⊢ ( ∃ 𝑡 ∈ 𝐴 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ↔ ∃ 𝑠 ∈ 𝐴 𝑠 ∈ ( 𝑎 𝐼 𝑏 ) )
14		oveq12	⊢ ( ( 𝑎 = 𝑐 ∧ 𝑏 = 𝑑 ) → ( 𝑎 𝐼 𝑏 ) = ( 𝑐 𝐼 𝑑 ) )
15	14	eleq2d	⊢ ( ( 𝑎 = 𝑐 ∧ 𝑏 = 𝑑 ) → ( 𝑠 ∈ ( 𝑎 𝐼 𝑏 ) ↔ 𝑠 ∈ ( 𝑐 𝐼 𝑑 ) ) )
16	15	rexbidv	⊢ ( ( 𝑎 = 𝑐 ∧ 𝑏 = 𝑑 ) → ( ∃ 𝑠 ∈ 𝐴 𝑠 ∈ ( 𝑎 𝐼 𝑏 ) ↔ ∃ 𝑠 ∈ 𝐴 𝑠 ∈ ( 𝑐 𝐼 𝑑 ) ) )
17	13 16	bitrid	⊢ ( ( 𝑎 = 𝑐 ∧ 𝑏 = 𝑑 ) → ( ∃ 𝑡 ∈ 𝐴 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ↔ ∃ 𝑠 ∈ 𝐴 𝑠 ∈ ( 𝑐 𝐼 𝑑 ) ) )
18	11 17	anbi12d	⊢ ( ( 𝑎 = 𝑐 ∧ 𝑏 = 𝑑 ) → ( ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐴 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐴 ) ) ∧ ∃ 𝑡 ∈ 𝐴 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) ↔ ( ( 𝑐 ∈ ( 𝑃 ∖ 𝐴 ) ∧ 𝑑 ∈ ( 𝑃 ∖ 𝐴 ) ) ∧ ∃ 𝑠 ∈ 𝐴 𝑠 ∈ ( 𝑐 𝐼 𝑑 ) ) ) )
19	18	cbvopabv	⊢ { ⟨ 𝑎 , 𝑏 ⟩ ∣ ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐴 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐴 ) ) ∧ ∃ 𝑡 ∈ 𝐴 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) } = { ⟨ 𝑐 , 𝑑 ⟩ ∣ ( ( 𝑐 ∈ ( 𝑃 ∖ 𝐴 ) ∧ 𝑑 ∈ ( 𝑃 ∖ 𝐴 ) ) ∧ ∃ 𝑠 ∈ 𝐴 𝑠 ∈ ( 𝑐 𝐼 𝑑 ) ) }
20	1 2 3 4 5 6 7 8 19	plngcplem	⊢ ( 𝜑 → ( 𝐴 𝐸 𝑅 ) = ( 𝐴 𝐸 𝑆 ) )