Theorem hpgne1

Description: Points on the open half plane cannot lie on its border. (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 )
	hpgne1.1	⊢ ( 𝜑 → 𝐴 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐵 )
Assertion	hpgne1	⊢ ( 𝜑 → ¬ 𝐴 ∈ 𝐷 )

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		hpgne1.1	⊢ ( 𝜑 → 𝐴 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐵 )
10		eqid	⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 )
11	6	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝑃 ) ∧ ( 𝐴 𝑂 𝑐 ∧ 𝐵 𝑂 𝑐 ) ) → 𝐷 ∈ ran 𝐿 )
12	5	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝑃 ) ∧ ( 𝐴 𝑂 𝑐 ∧ 𝐵 𝑂 𝑐 ) ) → 𝐺 ∈ TarskiG )
13	7	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝑃 ) ∧ ( 𝐴 𝑂 𝑐 ∧ 𝐵 𝑂 𝑐 ) ) → 𝐴 ∈ 𝑃 )
14		simplr	⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝑃 ) ∧ ( 𝐴 𝑂 𝑐 ∧ 𝐵 𝑂 𝑐 ) ) → 𝑐 ∈ 𝑃 )
15		simprl	⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝑃 ) ∧ ( 𝐴 𝑂 𝑐 ∧ 𝐵 𝑂 𝑐 ) ) → 𝐴 𝑂 𝑐 )
16	1 10 2 4 3 11 12 13 14 15	oppne1	⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝑃 ) ∧ ( 𝐴 𝑂 𝑐 ∧ 𝐵 𝑂 𝑐 ) ) → ¬ 𝐴 ∈ 𝐷 )
17	1 2 3 4 5 6 7 8	hpgbr	⊢ ( 𝜑 → ( 𝐴 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐵 ↔ ∃ 𝑐 ∈ 𝑃 ( 𝐴 𝑂 𝑐 ∧ 𝐵 𝑂 𝑐 ) ) )
18	9 17	mpbid	⊢ ( 𝜑 → ∃ 𝑐 ∈ 𝑃 ( 𝐴 𝑂 𝑐 ∧ 𝐵 𝑂 𝑐 ) )
19	16 18	r19.29a	⊢ ( 𝜑 → ¬ 𝐴 ∈ 𝐷 )