Theorem oppne3 28766

Description: Points lying on opposite sides of a line cannot be equal. (Contributed by Thierry Arnoux, 3-Aug-2020.)

Hypotheses

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

Assertion

Ref	Expression
oppne3	⊢ (𝜑 → 𝐴 ≠ 𝐵)

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

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

Proof of Theorem oppne3

Step	Hyp	Ref	Expression
1		hpg.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		hpg.d	. . . 4 ⊢ − = (dist‘𝐺)
3		hpg.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
4		hpg.o	. . . 4 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
5		opphl.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
6		opphl.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
7		opphl.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
8		oppcom.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9		oppcom.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
10		oppcom.o	. . . 4 ⊢ (𝜑 → 𝐴𝑂𝐵)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	oppne1 28764	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
12	7	ad3antrrr 730	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG)
13	8	ad3antrrr 730	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 ∈ 𝑃)
14	6	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐷 ∈ ran 𝐿)
15		simplr 769	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ 𝐷)
16	1, 5, 3, 12, 14, 15	tglnpt 28572	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ 𝑃)
17		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐵))
18		simpllr 776	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 = 𝐵)
19	18	oveq2d 7447	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → (𝐴𝐼𝐴) = (𝐴𝐼𝐵))
20	17, 19	eleqtrrd 2842	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐴))
21	1, 2, 3, 12, 13, 16, 20	axtgbtwnid 28489	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 = 𝑡)
22	21, 15	eqeltrd 2839	. . . 4 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 ∈ 𝐷)
23	1, 2, 3, 4, 8, 9	islnopp 28762	. . . . . . 7 ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
24	10, 23	mpbid 232	. . . . . 6 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
25	24	simprd 495	. . . . 5 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))
26	25	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))
27	22, 26	r19.29a 3160	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐷)
28	11, 27	mtand 816	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
29	28	neqned 2945	1 ⊢ (𝜑 → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∃wrex 3068   ∖ cdif 3960   class class class wbr 5148  {copab 5210  ran crn 5690  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  distcds 17307  TarskiGcstrkg 28450  Itvcitv 28456  LineGclng 28457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-iota 6516  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-trkgb 28472  df-trkg 28476

This theorem is referenced by:  colopp  28792  trgcopyeulem  28828