Theorem hpgne2 28771

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 2736	. . 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 28750	. 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 28769	. . 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 3161	1 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∃wrex 3069   ∖ cdif 3947   class class class wbr 5142  {copab 5204  ran crn 5685  ‘cfv 6560  (class class class)co 7432  Basecbs 17248  distcds 17307  TarskiGcstrkg 28436  Itvcitv 28442  LineGclng 28443  hpGchpg 28766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-hpg 28767

This theorem is referenced by: (None)