Theorem hpgne2 28665

Description: Points on the open half plane cannot lie on its border. (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)
hpgne1.1	⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐵)

Assertion

Ref	Expression
hpgne2	⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷)

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

Allowed substitution hints:   𝜑(𝑎,𝑏)   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐿(𝑎,𝑏)

Proof of Theorem hpgne2

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ishpg.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		eqid 2729	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
3		ishpg.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		ishpg.o	. . 3 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
5		ishpg.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
6		ishpg.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
7	6	ad2antrr 726	. . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐷 ∈ ran 𝐿)
8		ishpg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
9	8	ad2antrr 726	. . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐺 ∈ TarskiG)
10		hpgbr.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
11	10	ad2antrr 726	. . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐵 ∈ 𝑃)
12		simplr 768	. . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝑐 ∈ 𝑃)
13		simprr 772	. . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐵𝑂𝑐)
14	1, 2, 3, 4, 5, 7, 9, 11, 12, 13	oppne1 28644	. 2 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → ¬ 𝐵 ∈ 𝐷)
15		hpgne1.1	. . 3 ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐵)
16		hpgbr.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
17	1, 3, 5, 4, 8, 6, 16, 10	hpgbr 28663	. . 3 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)))
18	15, 17	mpbid 232	. 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))
19	14, 18	r19.29a 3141	1 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ∖ cdif 3908   class class class wbr 5102  {copab 5164  ran crn 5632  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  distcds 17205  TarskiGcstrkg 28330  Itvcitv 28336  LineGclng 28337  hpGchpg 28660

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-hpg 28661

This theorem is referenced by: (None)