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

Theorem tgelrnln 28638
Description: The property of being a proper line, generated by two distinct points. (Contributed by Thierry Arnoux, 25-May-2019.)
Hypotheses
Ref Expression
tglineelsb2.p 𝐵 = (Base‘𝐺)
tglineelsb2.i 𝐼 = (Itv‘𝐺)
tglineelsb2.l 𝐿 = (LineG‘𝐺)
tglineelsb2.g (𝜑𝐺 ∈ TarskiG)
tgelrnln.x (𝜑𝑋𝐵)
tgelrnln.y (𝜑𝑌𝐵)
tgelrnln.d (𝜑𝑋𝑌)
Assertion
Ref Expression
tgelrnln (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)

Proof of Theorem tgelrnln
StepHypRef Expression
1 df-ov 7434 . 2 (𝑋𝐿𝑌) = (𝐿‘⟨𝑋, 𝑌⟩)
2 tglineelsb2.g . . . 4 (𝜑𝐺 ∈ TarskiG)
3 tglineelsb2.p . . . . 5 𝐵 = (Base‘𝐺)
4 tglineelsb2.l . . . . 5 𝐿 = (LineG‘𝐺)
5 tglineelsb2.i . . . . 5 𝐼 = (Itv‘𝐺)
63, 4, 5tglnfn 28555 . . . 4 (𝐺 ∈ TarskiG → 𝐿 Fn ((𝐵 × 𝐵) ∖ I ))
72, 6syl 17 . . 3 (𝜑𝐿 Fn ((𝐵 × 𝐵) ∖ I ))
8 tgelrnln.x . . . . 5 (𝜑𝑋𝐵)
9 tgelrnln.y . . . . 5 (𝜑𝑌𝐵)
108, 9opelxpd 5724 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11 tgelrnln.d . . . . 5 (𝜑𝑋𝑌)
12 df-br 5144 . . . . . . . 8 (𝑋 I 𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ I )
13 ideqg 5862 . . . . . . . 8 (𝑌𝐵 → (𝑋 I 𝑌𝑋 = 𝑌))
1412, 13bitr3id 285 . . . . . . 7 (𝑌𝐵 → (⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋 = 𝑌))
1514necon3bbid 2978 . . . . . 6 (𝑌𝐵 → (¬ ⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋𝑌))
1615biimpar 477 . . . . 5 ((𝑌𝐵𝑋𝑌) → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
179, 11, 16syl2anc 584 . . . 4 (𝜑 → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
1810, 17eldifd 3962 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((𝐵 × 𝐵) ∖ I ))
19 fnfvelrn 7100 . . 3 ((𝐿 Fn ((𝐵 × 𝐵) ∖ I ) ∧ ⟨𝑋, 𝑌⟩ ∈ ((𝐵 × 𝐵) ∖ I )) → (𝐿‘⟨𝑋, 𝑌⟩) ∈ ran 𝐿)
207, 18, 19syl2anc 584 . 2 (𝜑 → (𝐿‘⟨𝑋, 𝑌⟩) ∈ ran 𝐿)
211, 20eqeltrid 2845 1 (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108  wne 2940  cdif 3948  cop 4632   class class class wbr 5143   I cid 5577   × cxp 5683  ran crn 5686   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:  tghilberti1  28645  tglineinteq  28653  colline  28657  tglowdim2ln  28659  footexALT  28726  footexlem2  28728  foot  28730  perprag  28734  colperpexlem3  28740  mideulem2  28742  midex  28745  outpasch  28763  lnopp2hpgb  28771  colopp  28777  lmieu  28792  lmimid  28802  hypcgrlem1  28807  hypcgrlem2  28808  lnperpex  28811  trgcopy  28812  trgcopyeulem  28813  acopy  28841  acopyeu  28842  tgasa1  28866
  Copyright terms: Public domain W3C validator