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

Theorem tgelrnln 26102
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 7026 . 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 26019 . . . 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 5488 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11 tgelrnln.d . . . . 5 (𝜑𝑋𝑌)
12 df-br 4969 . . . . . . . 8 (𝑋 I 𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ I )
13 ideqg 5615 . . . . . . . 8 (𝑌𝐵 → (𝑋 I 𝑌𝑋 = 𝑌))
1412, 13syl5bbr 286 . . . . . . 7 (𝑌𝐵 → (⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋 = 𝑌))
1514necon3bbid 3023 . . . . . 6 (𝑌𝐵 → (¬ ⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋𝑌))
1615biimpar 478 . . . . 5 ((𝑌𝐵𝑋𝑌) → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
179, 11, 16syl2anc 584 . . . 4 (𝜑 → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
1810, 17eldifd 3876 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((𝐵 × 𝐵) ∖ I ))
19 fnfvelrn 6720 . . 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, 20syl5eqel 2889 1 (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1525  wcel 2083  wne 2986  cdif 3862  cop 4484   class class class wbr 4968   I cid 5354   × cxp 5448  ran crn 5451   Fn wfn 6227  cfv 6232  (class class class)co 7023  Basecbs 16316  TarskiGcstrkg 25902  Itvcitv 25908  LineGclng 25909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-1st 7552  df-2nd 7553  df-trkg 25925
This theorem is referenced by:  tghilberti1  26109  tglineinteq  26117  colline  26121  tglowdim2ln  26123  footexALT  26190  footexlem2  26192  foot  26194  perprag  26198  colperpexlem3  26204  mideulem2  26206  midex  26209  outpasch  26227  lnopp2hpgb  26235  colopp  26241  lmieu  26256  lmimid  26266  hypcgrlem1  26271  hypcgrlem2  26272  lnperpex  26275  trgcopy  26276  trgcopyeulem  26277  acopy  26306  acopyeu  26307  tgasa1  26331
  Copyright terms: Public domain W3C validator