MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tglngne Structured version   Visualization version   GIF version

Theorem tglngne 28558
Description: It takes two different points to form a line. (Contributed by Thierry Arnoux, 6-Aug-2019.)
Hypotheses
Ref Expression
tglngval.p 𝑃 = (Base‘𝐺)
tglngval.l 𝐿 = (LineG‘𝐺)
tglngval.i 𝐼 = (Itv‘𝐺)
tglngval.g (𝜑𝐺 ∈ TarskiG)
tglngval.x (𝜑𝑋𝑃)
tglngval.y (𝜑𝑌𝑃)
tglngne.1 (𝜑𝑍 ∈ (𝑋𝐿𝑌))
Assertion
Ref Expression
tglngne (𝜑𝑋𝑌)

Proof of Theorem tglngne
StepHypRef Expression
1 tglngne.1 . . . . . 6 (𝜑𝑍 ∈ (𝑋𝐿𝑌))
2 df-ov 7434 . . . . . 6 (𝑋𝐿𝑌) = (𝐿‘⟨𝑋, 𝑌⟩)
31, 2eleqtrdi 2851 . . . . 5 (𝜑𝑍 ∈ (𝐿‘⟨𝑋, 𝑌⟩))
4 elfvdm 6943 . . . . 5 (𝑍 ∈ (𝐿‘⟨𝑋, 𝑌⟩) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐿)
53, 4syl 17 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom 𝐿)
6 tglngval.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
7 tglngval.p . . . . . 6 𝑃 = (Base‘𝐺)
8 tglngval.l . . . . . 6 𝐿 = (LineG‘𝐺)
9 tglngval.i . . . . . 6 𝐼 = (Itv‘𝐺)
107, 8, 9tglnfn 28555 . . . . 5 (𝐺 ∈ TarskiG → 𝐿 Fn ((𝑃 × 𝑃) ∖ I ))
11 fndm 6671 . . . . 5 (𝐿 Fn ((𝑃 × 𝑃) ∖ I ) → dom 𝐿 = ((𝑃 × 𝑃) ∖ I ))
126, 10, 113syl 18 . . . 4 (𝜑 → dom 𝐿 = ((𝑃 × 𝑃) ∖ I ))
135, 12eleqtrd 2843 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((𝑃 × 𝑃) ∖ I ))
1413eldifbd 3964 . 2 (𝜑 → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
15 df-br 5144 . . . 4 (𝑋 I 𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ I )
16 tglngval.y . . . . 5 (𝜑𝑌𝑃)
17 ideqg 5862 . . . . 5 (𝑌𝑃 → (𝑋 I 𝑌𝑋 = 𝑌))
1816, 17syl 17 . . . 4 (𝜑 → (𝑋 I 𝑌𝑋 = 𝑌))
1915, 18bitr3id 285 . . 3 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋 = 𝑌))
2019necon3bbid 2978 . 2 (𝜑 → (¬ ⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋𝑌))
2114, 20mpbid 232 1 (𝜑𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1540  wcel 2108  wne 2940  cdif 3948  cop 4632   class class class wbr 5143   I cid 5577   × cxp 5683  dom cdm 5685   Fn wfn 6556  cfv 6561  (class class class)co 7431  Basecbs 17247  TarskiGcstrkg 28435  Itvcitv 28441  LineGclng 28442
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-trkg 28461
This theorem is referenced by:  lnhl  28623  tglnne  28636  tglineneq  28652  tglineinteq  28653  ncolncol  28654  coltr  28655  coltr3  28656  perprag  28734
  Copyright terms: Public domain W3C validator