Theorem hpgbr

Description: Half-planes : property for points A and B to belong to the same open half plane delimited by line D . Definition 9.7 of Schwabhauser p. 71. (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 )
Assertion	hpgbr	⊢ ( 𝜑 → ( 𝐴 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐵 ↔ ∃ 𝑐 ∈ 𝑃 ( 𝐴 𝑂 𝑐 ∧ 𝐵 𝑂 𝑐 ) ) )

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	1 2 3 4 5 6	ishpg	⊢ ( 𝜑 → ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) = { ⟨ 𝑎 , 𝑏 ⟩ ∣ ∃ 𝑐 ∈ 𝑃 ( 𝑎 𝑂 𝑐 ∧ 𝑏 𝑂 𝑐 ) } )
10		simpl	⊢ ( ( 𝑎 = 𝑢 ∧ 𝑏 = 𝑣 ) → 𝑎 = 𝑢 )
11	10	breq1d	⊢ ( ( 𝑎 = 𝑢 ∧ 𝑏 = 𝑣 ) → ( 𝑎 𝑂 𝑐 ↔ 𝑢 𝑂 𝑐 ) )
12		simpr	⊢ ( ( 𝑎 = 𝑢 ∧ 𝑏 = 𝑣 ) → 𝑏 = 𝑣 )
13	12	breq1d	⊢ ( ( 𝑎 = 𝑢 ∧ 𝑏 = 𝑣 ) → ( 𝑏 𝑂 𝑐 ↔ 𝑣 𝑂 𝑐 ) )
14	11 13	anbi12d	⊢ ( ( 𝑎 = 𝑢 ∧ 𝑏 = 𝑣 ) → ( ( 𝑎 𝑂 𝑐 ∧ 𝑏 𝑂 𝑐 ) ↔ ( 𝑢 𝑂 𝑐 ∧ 𝑣 𝑂 𝑐 ) ) )
15	14	rexbidv	⊢ ( ( 𝑎 = 𝑢 ∧ 𝑏 = 𝑣 ) → ( ∃ 𝑐 ∈ 𝑃 ( 𝑎 𝑂 𝑐 ∧ 𝑏 𝑂 𝑐 ) ↔ ∃ 𝑐 ∈ 𝑃 ( 𝑢 𝑂 𝑐 ∧ 𝑣 𝑂 𝑐 ) ) )
16	15	cbvopabv	⊢ { ⟨ 𝑎 , 𝑏 ⟩ ∣ ∃ 𝑐 ∈ 𝑃 ( 𝑎 𝑂 𝑐 ∧ 𝑏 𝑂 𝑐 ) } = { ⟨ 𝑢 , 𝑣 ⟩ ∣ ∃ 𝑐 ∈ 𝑃 ( 𝑢 𝑂 𝑐 ∧ 𝑣 𝑂 𝑐 ) }
17	9 16	eqtrdi	⊢ ( 𝜑 → ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) = { ⟨ 𝑢 , 𝑣 ⟩ ∣ ∃ 𝑐 ∈ 𝑃 ( 𝑢 𝑂 𝑐 ∧ 𝑣 𝑂 𝑐 ) } )
18	17	breqd	⊢ ( 𝜑 → ( 𝐴 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐵 ↔ 𝐴 { ⟨ 𝑢 , 𝑣 ⟩ ∣ ∃ 𝑐 ∈ 𝑃 ( 𝑢 𝑂 𝑐 ∧ 𝑣 𝑂 𝑐 ) } 𝐵 ) )
19		simpl	⊢ ( ( 𝑢 = 𝐴 ∧ 𝑣 = 𝐵 ) → 𝑢 = 𝐴 )
20	19	breq1d	⊢ ( ( 𝑢 = 𝐴 ∧ 𝑣 = 𝐵 ) → ( 𝑢 𝑂 𝑐 ↔ 𝐴 𝑂 𝑐 ) )
21		simpr	⊢ ( ( 𝑢 = 𝐴 ∧ 𝑣 = 𝐵 ) → 𝑣 = 𝐵 )
22	21	breq1d	⊢ ( ( 𝑢 = 𝐴 ∧ 𝑣 = 𝐵 ) → ( 𝑣 𝑂 𝑐 ↔ 𝐵 𝑂 𝑐 ) )
23	20 22	anbi12d	⊢ ( ( 𝑢 = 𝐴 ∧ 𝑣 = 𝐵 ) → ( ( 𝑢 𝑂 𝑐 ∧ 𝑣 𝑂 𝑐 ) ↔ ( 𝐴 𝑂 𝑐 ∧ 𝐵 𝑂 𝑐 ) ) )
24	23	rexbidv	⊢ ( ( 𝑢 = 𝐴 ∧ 𝑣 = 𝐵 ) → ( ∃ 𝑐 ∈ 𝑃 ( 𝑢 𝑂 𝑐 ∧ 𝑣 𝑂 𝑐 ) ↔ ∃ 𝑐 ∈ 𝑃 ( 𝐴 𝑂 𝑐 ∧ 𝐵 𝑂 𝑐 ) ) )
25		eqid	⊢ { ⟨ 𝑢 , 𝑣 ⟩ ∣ ∃ 𝑐 ∈ 𝑃 ( 𝑢 𝑂 𝑐 ∧ 𝑣 𝑂 𝑐 ) } = { ⟨ 𝑢 , 𝑣 ⟩ ∣ ∃ 𝑐 ∈ 𝑃 ( 𝑢 𝑂 𝑐 ∧ 𝑣 𝑂 𝑐 ) }
26	24 25	brabga	⊢ ( ( 𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃 ) → ( 𝐴 { ⟨ 𝑢 , 𝑣 ⟩ ∣ ∃ 𝑐 ∈ 𝑃 ( 𝑢 𝑂 𝑐 ∧ 𝑣 𝑂 𝑐 ) } 𝐵 ↔ ∃ 𝑐 ∈ 𝑃 ( 𝐴 𝑂 𝑐 ∧ 𝐵 𝑂 𝑐 ) ) )
27	7 8 26	syl2anc	⊢ ( 𝜑 → ( 𝐴 { ⟨ 𝑢 , 𝑣 ⟩ ∣ ∃ 𝑐 ∈ 𝑃 ( 𝑢 𝑂 𝑐 ∧ 𝑣 𝑂 𝑐 ) } 𝐵 ↔ ∃ 𝑐 ∈ 𝑃 ( 𝐴 𝑂 𝑐 ∧ 𝐵 𝑂 𝑐 ) ) )
28	18 27	bitrd	⊢ ( 𝜑 → ( 𝐴 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐵 ↔ ∃ 𝑐 ∈ 𝑃 ( 𝐴 𝑂 𝑐 ∧ 𝐵 𝑂 𝑐 ) ) )