Definition df-hpg

Description: Define the open half plane relation for a geometry G . Definition 9.7 of Schwabhauser p. 71. See hpgbr to find the same formulation. (Contributed by Thierry Arnoux, 4-Mar-2020)

	Ref	Expression
Assertion	df-hpg	⊢ hpG = ( 𝑔 ∈ V ↦ ( 𝑑 ∈ ran ( LineG ‘ 𝑔 ) ↦ { ⟨ 𝑎 , 𝑏 ⟩ ∣ [ ( Base ‘ 𝑔 ) / 𝑝 ] [ ( Itv ‘ 𝑔 ) / 𝑖 ] ∃ 𝑐 ∈ 𝑝 ( ( ( 𝑎 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑎 𝑖 𝑐 ) ) ∧ ( ( 𝑏 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑏 𝑖 𝑐 ) ) ) } ) )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		chpg	⊢ hpG
1		vg	⊢ 𝑔
2		cvv	⊢ V
3		vd	⊢ 𝑑
4		clng	⊢ LineG
5	1	cv	⊢ 𝑔
6	5 4	cfv	⊢ ( LineG ‘ 𝑔 )
7	6	crn	⊢ ran ( LineG ‘ 𝑔 )
8		va	⊢ 𝑎
9		vb	⊢ 𝑏
10		cbs	⊢ Base
11	5 10	cfv	⊢ ( Base ‘ 𝑔 )
12		vp	⊢ 𝑝
13		citv	⊢ Itv
14	5 13	cfv	⊢ ( Itv ‘ 𝑔 )
15		vi	⊢ 𝑖
16		vc	⊢ 𝑐
17	12	cv	⊢ 𝑝
18	8	cv	⊢ 𝑎
19	3	cv	⊢ 𝑑
20	17 19	cdif	⊢ ( 𝑝 ∖ 𝑑 )
21	18 20	wcel	⊢ 𝑎 ∈ ( 𝑝 ∖ 𝑑 )
22	16	cv	⊢ 𝑐
23	22 20	wcel	⊢ 𝑐 ∈ ( 𝑝 ∖ 𝑑 )
24	21 23	wa	⊢ ( 𝑎 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) )
25		vt	⊢ 𝑡
26	25	cv	⊢ 𝑡
27	15	cv	⊢ 𝑖
28	18 22 27	co	⊢ ( 𝑎 𝑖 𝑐 )
29	26 28	wcel	⊢ 𝑡 ∈ ( 𝑎 𝑖 𝑐 )
30	29 25 19	wrex	⊢ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑎 𝑖 𝑐 )
31	24 30	wa	⊢ ( ( 𝑎 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑎 𝑖 𝑐 ) )
32	9	cv	⊢ 𝑏
33	32 20	wcel	⊢ 𝑏 ∈ ( 𝑝 ∖ 𝑑 )
34	33 23	wa	⊢ ( 𝑏 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) )
35	32 22 27	co	⊢ ( 𝑏 𝑖 𝑐 )
36	26 35	wcel	⊢ 𝑡 ∈ ( 𝑏 𝑖 𝑐 )
37	36 25 19	wrex	⊢ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑏 𝑖 𝑐 )
38	34 37	wa	⊢ ( ( 𝑏 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑏 𝑖 𝑐 ) )
39	31 38	wa	⊢ ( ( ( 𝑎 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑎 𝑖 𝑐 ) ) ∧ ( ( 𝑏 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑏 𝑖 𝑐 ) ) )
40	39 16 17	wrex	⊢ ∃ 𝑐 ∈ 𝑝 ( ( ( 𝑎 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑎 𝑖 𝑐 ) ) ∧ ( ( 𝑏 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑏 𝑖 𝑐 ) ) )
41	40 15 14	wsbc	⊢ [ ( Itv ‘ 𝑔 ) / 𝑖 ] ∃ 𝑐 ∈ 𝑝 ( ( ( 𝑎 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑎 𝑖 𝑐 ) ) ∧ ( ( 𝑏 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑏 𝑖 𝑐 ) ) )
42	41 12 11	wsbc	⊢ [ ( Base ‘ 𝑔 ) / 𝑝 ] [ ( Itv ‘ 𝑔 ) / 𝑖 ] ∃ 𝑐 ∈ 𝑝 ( ( ( 𝑎 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑎 𝑖 𝑐 ) ) ∧ ( ( 𝑏 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑏 𝑖 𝑐 ) ) )
43	42 8 9	copab	⊢ { ⟨ 𝑎 , 𝑏 ⟩ ∣ [ ( Base ‘ 𝑔 ) / 𝑝 ] [ ( Itv ‘ 𝑔 ) / 𝑖 ] ∃ 𝑐 ∈ 𝑝 ( ( ( 𝑎 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑎 𝑖 𝑐 ) ) ∧ ( ( 𝑏 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑏 𝑖 𝑐 ) ) ) }
44	3 7 43	cmpt	⊢ ( 𝑑 ∈ ran ( LineG ‘ 𝑔 ) ↦ { ⟨ 𝑎 , 𝑏 ⟩ ∣ [ ( Base ‘ 𝑔 ) / 𝑝 ] [ ( Itv ‘ 𝑔 ) / 𝑖 ] ∃ 𝑐 ∈ 𝑝 ( ( ( 𝑎 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑎 𝑖 𝑐 ) ) ∧ ( ( 𝑏 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑏 𝑖 𝑐 ) ) ) } )
45	1 2 44	cmpt	⊢ ( 𝑔 ∈ V ↦ ( 𝑑 ∈ ran ( LineG ‘ 𝑔 ) ↦ { ⟨ 𝑎 , 𝑏 ⟩ ∣ [ ( Base ‘ 𝑔 ) / 𝑝 ] [ ( Itv ‘ 𝑔 ) / 𝑖 ] ∃ 𝑐 ∈ 𝑝 ( ( ( 𝑎 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑎 𝑖 𝑐 ) ) ∧ ( ( 𝑏 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑏 𝑖 𝑐 ) ) ) } ) )
46	0 45	wceq	⊢ hpG = ( 𝑔 ∈ V ↦ ( 𝑑 ∈ ran ( LineG ‘ 𝑔 ) ↦ { ⟨ 𝑎 , 𝑏 ⟩ ∣ [ ( Base ‘ 𝑔 ) / 𝑝 ] [ ( Itv ‘ 𝑔 ) / 𝑖 ] ∃ 𝑐 ∈ 𝑝 ( ( ( 𝑎 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑎 𝑖 𝑐 ) ) ∧ ( ( 𝑏 ∈ ( 𝑝 ∖ 𝑑 ) ∧ 𝑐 ∈ ( 𝑝 ∖ 𝑑 ) ) ∧ ∃ 𝑡 ∈ 𝑑 𝑡 ∈ ( 𝑏 𝑖 𝑐 ) ) ) } ) )