Theorem lnoppnhpg

Description: If two points lie on the opposite side of a line D , they are not on the same half-plane. Theorem 9.9 of Schwabhauser p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020)

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

Proof

Step	Hyp	Ref	Expression
1		ishpg.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		ishpg.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
3		ishpg.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
4		ishpg.o	⊢ 𝑂 = { ⟨ 𝑎 , 𝑏 ⟩ ∣ ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐷 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐷 ) ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) }
5		ishpg.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
6		ishpg.d	⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 )
7		hpgbr.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
8		hpgbr.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
9		lnoppnhpg.1	⊢ ( 𝜑 → 𝐴 𝑂 𝐵 )
10		eqid	⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 )
11	1 10 2 4 3 6 5 8	oppnid	⊢ ( 𝜑 → ¬ 𝐵 𝑂 𝐵 )
12	1 2 3 4 5 6 7 8 8 9	lnopp2hpgb	⊢ ( 𝜑 → ( 𝐵 𝑂 𝐵 ↔ 𝐴 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐵 ) )
13	11 12	mtbid	⊢ ( 𝜑 → ¬ 𝐴 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐵 )