Theorem hpgbr 28844

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 28843	. . . 4 ⊢ (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐)})
8		simpl 482	. . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑎 = 𝑢)
9	8	breq1d 5110	. . . . . . 7 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑎𝑂𝑐 ↔ 𝑢𝑂𝑐))
10		simpr 484	. . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑏 = 𝑣)
11	10	breq1d 5110	. . . . . . 7 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑏𝑂𝑐 ↔ 𝑣𝑂𝑐))
12	9, 11	anbi12d 633	. . . . . 6 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ((𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐) ↔ (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)))
13	12	rexbidv 3162	. . . . 5 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐) ↔ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)))
14	13	cbvopabv 5173	. . . 4 ⊢ {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)}
15	7, 14	eqtrdi 2788	. . 3 ⊢ (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)})
16	15	breqd 5111	. 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 482	. . . . . . 7 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → 𝑢 = 𝐴)
20	19	breq1d 5110	. . . . . 6 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑢𝑂𝑐 ↔ 𝐴𝑂𝑐))
21		simpr 484	. . . . . . 7 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → 𝑣 = 𝐵)
22	21	breq1d 5110	. . . . . 6 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑣𝑂𝑐 ↔ 𝐵𝑂𝑐))
23	20, 22	anbi12d 633	. . . . 5 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → ((𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐) ↔ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)))
24	23	rexbidv 3162	. . . 4 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → (∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐) ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)))
25		eqid 2737	. . . 4 ⊢ {⟨𝑢, 𝑣⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)}
26	24, 25	brabga 5490	. . 3 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) → (𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)}𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)))
27	17, 18, 26	syl2anc 585	. 2 ⊢ (𝜑 → (𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)}𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)))
28	16, 27	bitrd 279	1 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ∖ cdif 3900   class class class wbr 5100  {copab 5162  ran crn 5633  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  TarskiGcstrkg 28511  Itvcitv 28517  LineGclng 28518  hpGchpg 28841

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-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-hpg 28842

This theorem is referenced by:  hpgne1  28845  hpgne2  28846  lnopp2hpgb  28847  hpgid  28850  hpgcom  28851  hpgtr  28852