Theorem oppne3 28829

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 28827	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
12	7	ad3antrrr 731	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG)
13	8	ad3antrrr 731	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 ∈ 𝑃)
14	6	ad3antrrr 731	. . . . . . 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 28635	. . . . . 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 7378	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → (𝐴𝐼𝐴) = (𝐴𝐼𝐵))
20	17, 19	eleqtrrd 2840	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐴))
21	1, 2, 3, 12, 13, 16, 20	axtgbtwnid 28552	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 = 𝑡)
22	21, 15	eqeltrd 2837	. . . 4 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 ∈ 𝐷)
23	1, 2, 3, 4, 8, 9	islnopp 28825	. . . . . . 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 3146	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐷)
28	11, 27	mtand 816	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
29	28	neqned 2940	1 ⊢ (𝜑 → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062   ∖ cdif 3887   class class class wbr 5086  {copab 5148  ran crn 5627  ‘cfv 6494  (class class class)co 7362  Basecbs 17174  distcds 17224  TarskiGcstrkg 28513  Itvcitv 28519  LineGclng 28520

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-sep 5232  ax-nul 5242  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-cnv 5634  df-dm 5636  df-rn 5637  df-iota 6450  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-trkgb 28535  df-trkg 28539

This theorem is referenced by:  colopp  28855  trgcopyeulem  28891