Theorem islnopp 27004

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 2826	. . . . . 6 ⊢ (𝑢 = 𝐴 → (𝑢 ∈ (𝑃 ∖ 𝐷) ↔ 𝐴 ∈ (𝑃 ∖ 𝐷)))
4	3	anbi1d 629	. . . . 5 ⊢ (𝑢 = 𝐴 → ((𝑢 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷)) ↔ (𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷))))
5		oveq1 7262	. . . . . . 7 ⊢ (𝑢 = 𝐴 → (𝑢𝐼𝑣) = (𝐴𝐼𝑣))
6	5	eleq2d 2824	. . . . . 6 ⊢ (𝑢 = 𝐴 → (𝑡 ∈ (𝑢𝐼𝑣) ↔ 𝑡 ∈ (𝐴𝐼𝑣)))
7	6	rexbidv 3225	. . . . 5 ⊢ (𝑢 = 𝐴 → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑢𝐼𝑣) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝑣)))
8	4, 7	anbi12d 630	. . . 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 2826	. . . . . 6 ⊢ (𝑣 = 𝐵 → (𝑣 ∈ (𝑃 ∖ 𝐷) ↔ 𝐵 ∈ (𝑃 ∖ 𝐷)))
10	9	anbi2d 628	. . . . 5 ⊢ (𝑣 = 𝐵 → ((𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷)) ↔ (𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝐵 ∈ (𝑃 ∖ 𝐷))))
11		oveq2 7263	. . . . . . 7 ⊢ (𝑣 = 𝐵 → (𝐴𝐼𝑣) = (𝐴𝐼𝐵))
12	11	eleq2d 2824	. . . . . 6 ⊢ (𝑣 = 𝐵 → (𝑡 ∈ (𝐴𝐼𝑣) ↔ 𝑡 ∈ (𝐴𝐼𝐵)))
13	12	rexbidv 3225	. . . . 5 ⊢ (𝑣 = 𝐵 → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝑣) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
14	10, 13	anbi12d 630	. . . 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 2823	. . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑎 ∈ (𝑃 ∖ 𝐷) ↔ 𝑢 ∈ (𝑃 ∖ 𝐷)))
18		simpr 484	. . . . . . . . 9 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑏 = 𝑣)
19	18	eleq1d 2823	. . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑏 ∈ (𝑃 ∖ 𝐷) ↔ 𝑣 ∈ (𝑃 ∖ 𝐷)))
20	17, 19	anbi12d 630	. . . . . . 7 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ↔ (𝑢 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷))))
21		oveq12 7264	. . . . . . . . 9 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑎𝐼𝑏) = (𝑢𝐼𝑣))
22	21	eleq2d 2824	. . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑡 ∈ (𝑢𝐼𝑣)))
23	22	rexbidv 3225	. . . . . . 7 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑢𝐼𝑣)))
24	20, 23	anbi12d 630	. . . . . 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 5143	. . . . 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 2766	. . . 4 ⊢ 𝑂 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑢𝐼𝑣))}
27	8, 14, 26	brabg 5445	. . 3 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) → (𝐴𝑂𝐵 ↔ ((𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝐵 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
28	1, 2, 27	syl2anc 583	. 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 3893	. . . . 5 ⊢ (𝐴 ∈ (𝑃 ∖ 𝐷) ↔ (𝐴 ∈ 𝑃 ∧ ¬ 𝐴 ∈ 𝐷))
31	29, 30	bitr4di 288	. . . 4 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝐷 ↔ 𝐴 ∈ (𝑃 ∖ 𝐷)))
32	2	biantrurd 532	. . . . 5 ⊢ (𝜑 → (¬ 𝐵 ∈ 𝐷 ↔ (𝐵 ∈ 𝑃 ∧ ¬ 𝐵 ∈ 𝐷)))
33		eldif 3893	. . . . 5 ⊢ (𝐵 ∈ (𝑃 ∖ 𝐷) ↔ (𝐵 ∈ 𝑃 ∧ ¬ 𝐵 ∈ 𝐷))
34	32, 33	bitr4di 288	. . . 4 ⊢ (𝜑 → (¬ 𝐵 ∈ 𝐷 ↔ 𝐵 ∈ (𝑃 ∖ 𝐷)))
35	31, 34	anbi12d 630	. . 3 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ↔ (𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝐵 ∈ (𝑃 ∖ 𝐷))))
36	35	anbi1d 629	. 2 ⊢ (𝜑 → (((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)) ↔ ((𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝐵 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
37	28, 36	bitr4d 281	1 ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃wrex 3064   ∖ cdif 3880   class class class wbr 5070  {copab 5132  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  distcds 16897  Itvcitv 26699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-iota 6376  df-fv 6426  df-ov 7258

This theorem is referenced by:  islnoppd  27005  oppne1  27006  oppne2  27007  oppne3  27008  oppcom  27009  oppnid  27011  opphllem1  27012  opphllem3  27014  opphllem5  27016  opphllem6  27017  oppperpex  27018  outpasch  27020  lnopp2hpgb  27028  hpgerlem  27030  colopp  27034  colhp  27035  trgcopyeulem  27070