Theorem islnopp 28821

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 2825	. . . . . 6 ⊢ (𝑢 = 𝐴 → (𝑢 ∈ (𝑃 ∖ 𝐷) ↔ 𝐴 ∈ (𝑃 ∖ 𝐷)))
4	3	anbi1d 632	. . . . 5 ⊢ (𝑢 = 𝐴 → ((𝑢 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷)) ↔ (𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷))))
5		oveq1 7367	. . . . . . 7 ⊢ (𝑢 = 𝐴 → (𝑢𝐼𝑣) = (𝐴𝐼𝑣))
6	5	eleq2d 2823	. . . . . 6 ⊢ (𝑢 = 𝐴 → (𝑡 ∈ (𝑢𝐼𝑣) ↔ 𝑡 ∈ (𝐴𝐼𝑣)))
7	6	rexbidv 3162	. . . . 5 ⊢ (𝑢 = 𝐴 → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑢𝐼𝑣) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝑣)))
8	4, 7	anbi12d 633	. . . 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 2825	. . . . . 6 ⊢ (𝑣 = 𝐵 → (𝑣 ∈ (𝑃 ∖ 𝐷) ↔ 𝐵 ∈ (𝑃 ∖ 𝐷)))
10	9	anbi2d 631	. . . . 5 ⊢ (𝑣 = 𝐵 → ((𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷)) ↔ (𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝐵 ∈ (𝑃 ∖ 𝐷))))
11		oveq2 7368	. . . . . . 7 ⊢ (𝑣 = 𝐵 → (𝐴𝐼𝑣) = (𝐴𝐼𝐵))
12	11	eleq2d 2823	. . . . . 6 ⊢ (𝑣 = 𝐵 → (𝑡 ∈ (𝐴𝐼𝑣) ↔ 𝑡 ∈ (𝐴𝐼𝐵)))
13	12	rexbidv 3162	. . . . 5 ⊢ (𝑣 = 𝐵 → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝑣) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
14	10, 13	anbi12d 633	. . . 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 2822	. . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑎 ∈ (𝑃 ∖ 𝐷) ↔ 𝑢 ∈ (𝑃 ∖ 𝐷)))
18		simpr 484	. . . . . . . . 9 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑏 = 𝑣)
19	18	eleq1d 2822	. . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑏 ∈ (𝑃 ∖ 𝐷) ↔ 𝑣 ∈ (𝑃 ∖ 𝐷)))
20	17, 19	anbi12d 633	. . . . . . 7 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ↔ (𝑢 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷))))
21		oveq12 7369	. . . . . . . . 9 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑎𝐼𝑏) = (𝑢𝐼𝑣))
22	21	eleq2d 2823	. . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑡 ∈ (𝑢𝐼𝑣)))
23	22	rexbidv 3162	. . . . . . 7 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑢𝐼𝑣)))
24	20, 23	anbi12d 633	. . . . . 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 5159	. . . . 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 2760	. . . 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 585	. 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 3900	. . . . 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 3900	. . . . 5 ⊢ (𝐵 ∈ (𝑃 ∖ 𝐷) ↔ (𝐵 ∈ 𝑃 ∧ ¬ 𝐵 ∈ 𝐷))
34	32, 33	bitr4di 289	. . . 4 ⊢ (𝜑 → (¬ 𝐵 ∈ 𝐷 ↔ 𝐵 ∈ (𝑃 ∖ 𝐷)))
35	31, 34	anbi12d 633	. . 3 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ↔ (𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝐵 ∈ (𝑃 ∖ 𝐷))))
36	35	anbi1d 632	. 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 1542   ∈ wcel 2114  ∃wrex 3062   ∖ cdif 3887   class class class wbr 5086  {copab 5148  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  distcds 17220  Itvcitv 28515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-iota 6448  df-fv 6500  df-ov 7363

This theorem is referenced by:  islnoppd  28822  oppne1  28823  oppne2  28824  oppne3  28825  oppcom  28826  oppnid  28828  opphllem1  28829  opphllem3  28831  opphllem5  28833  opphllem6  28834  oppperpex  28835  outpasch  28837  lnopp2hpgb  28845  hpgerlem  28847  colopp  28851  colhp  28852  trgcopyeulem  28887