Theorem islnopp 26528

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 2903	. . . . . 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 7166	. . . . . . 7 ⊢ (𝑢 = 𝐴 → (𝑢𝐼𝑣) = (𝐴𝐼𝑣))
6	5	eleq2d 2901	. . . . . 6 ⊢ (𝑢 = 𝐴 → (𝑡 ∈ (𝑢𝐼𝑣) ↔ 𝑡 ∈ (𝐴𝐼𝑣)))
7	6	rexbidv 3300	. . . . 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 2903	. . . . . 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 7167	. . . . . . 7 ⊢ (𝑣 = 𝐵 → (𝐴𝐼𝑣) = (𝐴𝐼𝐵))
12	11	eleq2d 2901	. . . . . 6 ⊢ (𝑣 = 𝐵 → (𝑡 ∈ (𝐴𝐼𝑣) ↔ 𝑡 ∈ (𝐴𝐼𝐵)))
13	12	rexbidv 3300	. . . . 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 485	. . . . . . . . 9 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑎 = 𝑢)
17	16	eleq1d 2900	. . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑎 ∈ (𝑃 ∖ 𝐷) ↔ 𝑢 ∈ (𝑃 ∖ 𝐷)))
18		simpr 487	. . . . . . . . 9 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑏 = 𝑣)
19	18	eleq1d 2900	. . . . . . . 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 7168	. . . . . . . . 9 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑎𝐼𝑏) = (𝑢𝐼𝑣))
22	21	eleq2d 2901	. . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑡 ∈ (𝑢𝐼𝑣)))
23	22	rexbidv 3300	. . . . . . 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 5141	. . . . 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 2847	. . . 4 ⊢ 𝑂 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝑃 ∖ 𝐷) ∧ 𝑣 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑢𝐼𝑣))}
27	8, 14, 26	brabg 5429	. . 3 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) → (𝐴𝑂𝐵 ↔ ((𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝐵 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
28	1, 2, 27	syl2anc 586	. 2 ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((𝐴 ∈ (𝑃 ∖ 𝐷) ∧ 𝐵 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
29	1	biantrurd 535	. . . . 5 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝐷 ↔ (𝐴 ∈ 𝑃 ∧ ¬ 𝐴 ∈ 𝐷)))
30		eldif 3949	. . . . 5 ⊢ (𝐴 ∈ (𝑃 ∖ 𝐷) ↔ (𝐴 ∈ 𝑃 ∧ ¬ 𝐴 ∈ 𝐷))
31	29, 30	syl6bbr 291	. . . 4 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝐷 ↔ 𝐴 ∈ (𝑃 ∖ 𝐷)))
32	2	biantrurd 535	. . . . 5 ⊢ (𝜑 → (¬ 𝐵 ∈ 𝐷 ↔ (𝐵 ∈ 𝑃 ∧ ¬ 𝐵 ∈ 𝐷)))
33		eldif 3949	. . . . 5 ⊢ (𝐵 ∈ (𝑃 ∖ 𝐷) ↔ (𝐵 ∈ 𝑃 ∧ ¬ 𝐵 ∈ 𝐷))
34	32, 33	syl6bbr 291	. . . 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 284	1 ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∃wrex 3142   ∖ cdif 3936   class class class wbr 5069  {copab 5131  ‘cfv 6358  (class class class)co 7159  Basecbs 16486  distcds 16577  Itvcitv 26225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-iota 6317  df-fv 6366  df-ov 7162

This theorem is referenced by:  islnoppd  26529  oppne1  26530  oppne2  26531  oppne3  26532  oppcom  26533  oppnid  26535  opphllem1  26536  opphllem3  26538  opphllem5  26540  opphllem6  26541  oppperpex  26542  outpasch  26544  lnopp2hpgb  26552  hpgerlem  26554  colopp  26558  colhp  26559  trgcopyeulem  26594