Theorem hpgid

Description: The half-plane relation is reflexive. Theorem 9.11 of Schwabhauser p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020)

	Ref	Expression
Hypotheses	hpgid.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	hpgid.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	hpgid.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	hpgid.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	hpgid.d	⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 )
	hpgid.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	hpgid.o	⊢ 𝑂 = { ⟨ 𝑎 , 𝑏 ⟩ ∣ ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐷 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐷 ) ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) }
	hpgid.1	⊢ ( 𝜑 → ¬ 𝐴 ∈ 𝐷 )
Assertion	hpgid	⊢ ( 𝜑 → 𝐴 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		hpgid.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		hpgid.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
3		hpgid.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
4		hpgid.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		hpgid.d	⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 )
6		hpgid.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
7		hpgid.o	⊢ 𝑂 = { ⟨ 𝑎 , 𝑏 ⟩ ∣ ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐷 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐷 ) ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) }
8		hpgid.1	⊢ ( 𝜑 → ¬ 𝐴 ∈ 𝐷 )
9		simprr	⊢ ( ( 𝜑 ∧ ( 𝑐 ∈ 𝑃 ∧ 𝐴 𝑂 𝑐 ) ) → 𝐴 𝑂 𝑐 )
10	9 9	jca	⊢ ( ( 𝜑 ∧ ( 𝑐 ∈ 𝑃 ∧ 𝐴 𝑂 𝑐 ) ) → ( 𝐴 𝑂 𝑐 ∧ 𝐴 𝑂 𝑐 ) )
11	1 2 3 4 5 6 7 8	hpgerlem	⊢ ( 𝜑 → ∃ 𝑐 ∈ 𝑃 𝐴 𝑂 𝑐 )
12	10 11	reximddv	⊢ ( 𝜑 → ∃ 𝑐 ∈ 𝑃 ( 𝐴 𝑂 𝑐 ∧ 𝐴 𝑂 𝑐 ) )
13	1 2 3 7 4 5 6 6	hpgbr	⊢ ( 𝜑 → ( 𝐴 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐴 ↔ ∃ 𝑐 ∈ 𝑃 ( 𝐴 𝑂 𝑐 ∧ 𝐴 𝑂 𝑐 ) ) )
14	12 13	mpbird	⊢ ( 𝜑 → 𝐴 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐴 )