Theorem hpgbr 29072

Description: Half-planes : property for points 𝐴 and 𝐵 to belong to the same open half plane delimited by line 𝐷. Definition 9.7 of [Schwabhauser] p. 71. (Contributed by Thierry Arnoux, 4-Mar-2020.)

Hypotheses

Ref	Expression
ishpg.p	⊢ 𝑃 = (Base‘𝐺)
ishpg.i	⊢ 𝐼 = (Itv‘𝐺)
ishpg.l	⊢ 𝐿 = (LineG‘𝐺)
ishpg.o	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
ishpg.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
ishpg.d	⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
hpgbr.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
hpgbr.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)

Assertion

Ref	Expression
hpgbr	⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)))

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

Allowed substitution hints:   𝜑(𝑡, 𝑎, 𝑏, 𝑐)   𝐴(𝑡, 𝑎, 𝑏)   𝐵(𝑡, 𝑎, 𝑏)   𝐺(𝑡, 𝑐)   𝐿(𝑡, 𝑎, 𝑏, 𝑐)   𝑂(𝑡, 𝑐)

Proof of Theorem hpgbr

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

Step	Hyp	Ref	Expression
1		ishpg.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		ishpg.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
3		ishpg.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
4		ishpg.o	. . . . 5 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
5		ishpg.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6		ishpg.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
7	1, 2, 3, 4, 5, 6	ishpg 29071	. . . 4 ⊢ (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐)})
8		simpl 488	. . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑎 = 𝑢)
9	8	breq1d 5124	. . . . . . 7 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑎𝑂𝑐 ↔ 𝑢𝑂𝑐))
10		simpr 490	. . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑏 = 𝑣)
11	10	breq1d 5124	. . . . . . 7 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑏𝑂𝑐 ↔ 𝑣𝑂𝑐))
12	9, 11	anbi12d 644	. . . . . 6 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ((𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐) ↔ (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)))
13	12	rexbidv 3192	. . . . 5 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐) ↔ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)))
14	13	cbvopabv 5189	. . . 4 ⊢ {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)}
15	7, 14	eqtrdi 2817	. . 3 ⊢ (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)})
16	15	breqd 5125	. 2 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ 𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)}𝐵))
17		hpgbr.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
18		hpgbr.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
19		simpl 488	. . . . . . 7 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → 𝑢 = 𝐴)
20	19	breq1d 5124	. . . . . 6 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑢𝑂𝑐 ↔ 𝐴𝑂𝑐))
21		simpr 490	. . . . . . 7 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → 𝑣 = 𝐵)
22	21	breq1d 5124	. . . . . 6 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑣𝑂𝑐 ↔ 𝐵𝑂𝑐))
23	20, 22	anbi12d 644	. . . . 5 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → ((𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐) ↔ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)))
24	23	rexbidv 3192	. . . 4 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → (∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐) ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)))
25		eqid 2766	. . . 4 ⊢ {⟨𝑢, 𝑣⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)}
26	24, 25	brabga 5523	. . 3 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) → (𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)}𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)))
27	17, 18, 26	syl2anc 596	. 2 ⊢ (𝜑 → (𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)}𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)))
28	16, 27	bitrd 282	1 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3092   ∖ cdif 3905   class class class wbr 5114  {copab 5178  ran crn 5667  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  TarskiGcstrkg 28726  Itvcitv 28732  LineGclng 28733  hpGchpg 29069

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-hpg 29070

This theorem is used by:  hpgne1  29073  hpgne2  29074  lnopp2hpgb  29075  hpgid  29078  hpgcom  29079  hpgtr  29080