lnperpexs

Description: Existence of a perpendicular to a line L at a given point A . Theorem 10.15 of Schwabhauser p. 92. (Contributed by Thierry Arnoux, 2-Aug-2020)

	Ref	Expression
Hypotheses	lnperpexs.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	lnperpexs.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	lnperpexs.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	lnperpexs.h	⊢ ( 𝜑 → 𝐺 Dim_TarskiG≥ 2 )
	lnperpexs.d	⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 )
	lnperpexs.a	⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
	lnperpexs.q	⊢ ( 𝜑 → 𝑄 ∈ 𝑃 )
	lnperpexs.1	⊢ ( 𝜑 → ¬ 𝑄 ∈ 𝐷 )
Assertion	lnperpexs	⊢ ( 𝜑 → ∃ 𝑝 ∈ 𝑃 ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝑝 𝐿 𝐴 ) ∧ 𝑝 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝑄 ) )

Proof

Step	Hyp	Ref	Expression
1		lnperpexs.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		lnperpexs.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
3		lnperpexs.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
4		lnperpexs.h	⊢ ( 𝜑 → 𝐺 Dim_TarskiG≥ 2 )
5		lnperpexs.d	⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 )
6		lnperpexs.a	⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
7		lnperpexs.q	⊢ ( 𝜑 → 𝑄 ∈ 𝑃 )
8		lnperpexs.1	⊢ ( 𝜑 → ¬ 𝑄 ∈ 𝐷 )
9		eqid	⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 )
10		eqid	⊢ ( Itv ‘ 𝐺 ) = ( Itv ‘ 𝐺 )
11		eleq1w	⊢ ( 𝑎 = 𝑐 → ( 𝑎 ∈ ( 𝑃 ∖ 𝐷 ) ↔ 𝑐 ∈ ( 𝑃 ∖ 𝐷 ) ) )
12		eleq1w	⊢ ( 𝑏 = 𝑑 → ( 𝑏 ∈ ( 𝑃 ∖ 𝐷 ) ↔ 𝑑 ∈ ( 𝑃 ∖ 𝐷 ) ) )
13	11 12	bi2anan9	⊢ ( ( 𝑎 = 𝑐 ∧ 𝑏 = 𝑑 ) → ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐷 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐷 ) ) ↔ ( 𝑐 ∈ ( 𝑃 ∖ 𝐷 ) ∧ 𝑑 ∈ ( 𝑃 ∖ 𝐷 ) ) ) )
14		oveq12	⊢ ( ( 𝑎 = 𝑐 ∧ 𝑏 = 𝑑 ) → ( 𝑎 ( Itv ‘ 𝐺 ) 𝑏 ) = ( 𝑐 ( Itv ‘ 𝐺 ) 𝑑 ) )
15	14	eleq2d	⊢ ( ( 𝑎 = 𝑐 ∧ 𝑏 = 𝑑 ) → ( 𝑠 ∈ ( 𝑎 ( Itv ‘ 𝐺 ) 𝑏 ) ↔ 𝑠 ∈ ( 𝑐 ( Itv ‘ 𝐺 ) 𝑑 ) ) )
16	15	rexbidv	⊢ ( ( 𝑎 = 𝑐 ∧ 𝑏 = 𝑑 ) → ( ∃ 𝑠 ∈ 𝐷 𝑠 ∈ ( 𝑎 ( Itv ‘ 𝐺 ) 𝑏 ) ↔ ∃ 𝑠 ∈ 𝐷 𝑠 ∈ ( 𝑐 ( Itv ‘ 𝐺 ) 𝑑 ) ) )
17		eleq1w	⊢ ( 𝑠 = 𝑡 → ( 𝑠 ∈ ( 𝑐 ( Itv ‘ 𝐺 ) 𝑑 ) ↔ 𝑡 ∈ ( 𝑐 ( Itv ‘ 𝐺 ) 𝑑 ) ) )
18	17	cbvrexvw	⊢ ( ∃ 𝑠 ∈ 𝐷 𝑠 ∈ ( 𝑐 ( Itv ‘ 𝐺 ) 𝑑 ) ↔ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝑐 ( Itv ‘ 𝐺 ) 𝑑 ) )
19	16 18	bitrdi	⊢ ( ( 𝑎 = 𝑐 ∧ 𝑏 = 𝑑 ) → ( ∃ 𝑠 ∈ 𝐷 𝑠 ∈ ( 𝑎 ( Itv ‘ 𝐺 ) 𝑏 ) ↔ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝑐 ( Itv ‘ 𝐺 ) 𝑑 ) ) )
20	13 19	anbi12d	⊢ ( ( 𝑎 = 𝑐 ∧ 𝑏 = 𝑑 ) → ( ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐷 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐷 ) ) ∧ ∃ 𝑠 ∈ 𝐷 𝑠 ∈ ( 𝑎 ( Itv ‘ 𝐺 ) 𝑏 ) ) ↔ ( ( 𝑐 ∈ ( 𝑃 ∖ 𝐷 ) ∧ 𝑑 ∈ ( 𝑃 ∖ 𝐷 ) ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝑐 ( Itv ‘ 𝐺 ) 𝑑 ) ) ) )
21	20	cbvopabv	⊢ { ⟨ 𝑎 , 𝑏 ⟩ ∣ ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐷 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐷 ) ) ∧ ∃ 𝑠 ∈ 𝐷 𝑠 ∈ ( 𝑎 ( Itv ‘ 𝐺 ) 𝑏 ) ) } = { ⟨ 𝑐 , 𝑑 ⟩ ∣ ( ( 𝑐 ∈ ( 𝑃 ∖ 𝐷 ) ∧ 𝑑 ∈ ( 𝑃 ∖ 𝐷 ) ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝑐 ( Itv ‘ 𝐺 ) 𝑑 ) ) }
22	1 9 10 2 3 4 5 21 6 7 8	lnperpex	⊢ ( 𝜑 → ∃ 𝑝 ∈ 𝑃 ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝑝 𝐿 𝐴 ) ∧ 𝑝 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝑄 ) )

Metamath Proof Explorer

Theorem lnperpexs

Proof