Theorem hpgne2 27801

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 2731	. . 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 724	. . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐷 ∈ ran 𝐿)
8		ishpg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
9	8	ad2antrr 724	. . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐺 ∈ TarskiG)
10		hpgbr.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
11	10	ad2antrr 724	. . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐵 ∈ 𝑃)
12		simplr 767	. . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝑐 ∈ 𝑃)
13		simprr 771	. . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐵𝑂𝑐)
14	1, 2, 3, 4, 5, 7, 9, 11, 12, 13	oppne1 27780	. 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 27799	. . 3 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)))
18	15, 17	mpbid 231	. 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))
19	14, 18	r19.29a 3161	1 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3069   ∖ cdif 3925   class class class wbr 5125  {copab 5187  ran crn 5654  ‘cfv 6516  (class class class)co 7377  Basecbs 17109  distcds 17171  TarskiGcstrkg 27466  Itvcitv 27472  LineGclng 27473  hpGchpg 27796

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7380  df-hpg 27797

This theorem is referenced by: (None)