Theorem islnopp 28811

Description: The property for two points 𝐴 and 𝐵 to lie on the opposite sides of a set 𝐷 Definition 9.1 of [Schwabhauser] p. 67. (Contributed by Thierry Arnoux, 19-Dec-2019.)

Hypotheses

Ref	Expression
hpg.p	⊢ 𝑃 = (Base‘𝐺)
hpg.d	⊢ − = (dist‘𝐺)
hpg.i	⊢ 𝐼 = (Itv‘𝐺)
hpg.o	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
islnopp.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
islnopp.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)

Assertion

Ref	Expression
islnopp	⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))

Distinct variable groups:   𝐷,𝑎,𝑏   𝐼,𝑎,𝑏   𝑃,𝑎,𝑏   𝑡,𝐴   𝑡,𝐵   𝑡,𝑎,𝑏

Allowed substitution hints:   𝜑(𝑡,𝑎,𝑏)   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐷(𝑡)   𝑃(𝑡)   𝐺(𝑡,𝑎,𝑏)   𝐼(𝑡)   − (𝑡,𝑎,𝑏)   𝑂(𝑡,𝑎,𝑏)

Proof of Theorem islnopp

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		islnopp.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
2		islnopp.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
3		eleq1 2824	. . . . . 6 ⊢ (𝑢 = 𝐴 → (𝑢 ∈ (𝑃 ∖ 𝐷) ↔ 𝐴 ∈ (𝑃 ∖ 𝐷)))
4	3	anbi1d 631	. . . . 5 ⊢ (𝑢 = 𝐴 → ((𝑢 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷)) ↔ (𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷))))
5		oveq1 7365	. . . . . . 7 ⊢ (𝑢 = 𝐴 → (𝑢𝐼𝑣) = (𝐴𝐼𝑣))
6	5	eleq2d 2822	. . . . . 6 ⊢ (𝑢 = 𝐴 → (𝑡 ∈ (𝑢𝐼𝑣) ↔ 𝑡 ∈ (𝐴𝐼𝑣)))
7	6	rexbidv 3160	. . . . 5 ⊢ (𝑢 = 𝐴 → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑢𝐼𝑣) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝑣)))
8	4, 7	anbi12d 632	. . . 4 ⊢ (𝑢 = 𝐴 → (((𝑢 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑢𝐼𝑣)) ↔ ((𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝑣))))
9		eleq1 2824	. . . . . 6 ⊢ (𝑣 = 𝐵 → (𝑣 ∈ (𝑃 ∖ 𝐷) ↔ 𝐵 ∈ (𝑃 ∖ 𝐷)))
10	9	anbi2d 630	. . . . 5 ⊢ (𝑣 = 𝐵 → ((𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷)) ↔ (𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝐵 ∈ (𝑃 ∖ 𝐷))))
11		oveq2 7366	. . . . . . 7 ⊢ (𝑣 = 𝐵 → (𝐴𝐼𝑣) = (𝐴𝐼𝐵))
12	11	eleq2d 2822	. . . . . 6 ⊢ (𝑣 = 𝐵 → (𝑡 ∈ (𝐴𝐼𝑣) ↔ 𝑡 ∈ (𝐴𝐼𝐵)))
13	12	rexbidv 3160	. . . . 5 ⊢ (𝑣 = 𝐵 → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝑣) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
14	10, 13	anbi12d 632	. . . 4 ⊢ (𝑣 = 𝐵 → (((𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝑣)) ↔ ((𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝐵 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
15		hpg.o	. . . . 5 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
16		simpl 482	. . . . . . . . 9 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑎 = 𝑢)
17	16	eleq1d 2821	. . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑎 ∈ (𝑃 ∖ 𝐷) ↔ 𝑢 ∈ (𝑃 ∖ 𝐷)))
18		simpr 484	. . . . . . . . 9 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑏 = 𝑣)
19	18	eleq1d 2821	. . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑏 ∈ (𝑃 ∖ 𝐷) ↔ 𝑣 ∈ (𝑃 ∖ 𝐷)))
20	17, 19	anbi12d 632	. . . . . . 7 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ↔ (𝑢 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷))))
21		oveq12 7367	. . . . . . . . 9 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑎𝐼𝑏) = (𝑢𝐼𝑣))
22	21	eleq2d 2822	. . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑡 ∈ (𝑢𝐼𝑣)))
23	22	rexbidv 3160	. . . . . . 7 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑢𝐼𝑣)))
24	20, 23	anbi12d 632	. . . . . 6 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏)) ↔ ((𝑢 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑢𝐼𝑣))))
25	24	cbvopabv 5171	. . . . 5 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑢𝐼𝑣))}
26	15, 25	eqtri 2759	. . . 4 ⊢ 𝑂 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑢𝐼𝑣))}
27	8, 14, 26	brabg 5487	. . 3 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) → (𝐴𝑂𝐵 ↔ ((𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝐵 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
28	1, 2, 27	syl2anc 584	. 2 ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝐵 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
29	1	biantrurd 532	. . . . 5 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝐷 ↔ (𝐴 ∈ 𝑃 ∧ ¬ 𝐴 ∈ 𝐷)))
30		eldif 3911	. . . . 5 ⊢ (𝐴 ∈ (𝑃 ∖ 𝐷) ↔ (𝐴 ∈ 𝑃 ∧ ¬ 𝐴 ∈ 𝐷))
31	29, 30	bitr4di 289	. . . 4 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝐷 ↔ 𝐴 ∈ (𝑃 ∖ 𝐷)))
32	2	biantrurd 532	. . . . 5 ⊢ (𝜑 → (¬ 𝐵 ∈ 𝐷 ↔ (𝐵 ∈ 𝑃 ∧ ¬ 𝐵 ∈ 𝐷)))
33		eldif 3911	. . . . 5 ⊢ (𝐵 ∈ (𝑃 ∖ 𝐷) ↔ (𝐵 ∈ 𝑃 ∧ ¬ 𝐵 ∈ 𝐷))
34	32, 33	bitr4di 289	. . . 4 ⊢ (𝜑 → (¬ 𝐵 ∈ 𝐷 ↔ 𝐵 ∈ (𝑃 ∖ 𝐷)))
35	31, 34	anbi12d 632	. . 3 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ↔ (𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝐵 ∈ (𝑃 ∖ 𝐷))))
36	35	anbi1d 631	. 2 ⊢ (𝜑 → (((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)) ↔ ((𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝐵 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
37	28, 36	bitr4d 282	1 ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3060   ∖ cdif 3898   class class class wbr 5098  {copab 5160  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  distcds 17186  Itvcitv 28505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-iota 6448  df-fv 6500  df-ov 7361

This theorem is referenced by:  islnoppd  28812  oppne1  28813  oppne2  28814  oppne3  28815  oppcom  28816  oppnid  28818  opphllem1  28819  opphllem3  28821  opphllem5  28823  opphllem6  28824  oppperpex  28825  outpasch  28827  lnopp2hpgb  28835  hpgerlem  28837  colopp  28841  colhp  28842  trgcopyeulem  28877