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Theorem tgelrnln 28653
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 28570 . . . 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 5728 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11 tgelrnln.d . . . . 5 (𝜑𝑋𝑌)
12 df-br 5149 . . . . . . . 8 (𝑋 I 𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ I )
13 ideqg 5865 . . . . . . . 8 (𝑌𝐵 → (𝑋 I 𝑌𝑋 = 𝑌))
1412, 13bitr3id 285 . . . . . . 7 (𝑌𝐵 → (⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋 = 𝑌))
1514necon3bbid 2976 . . . . . 6 (𝑌𝐵 → (¬ ⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋𝑌))
1615biimpar 477 . . . . 5 ((𝑌𝐵𝑋𝑌) → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
179, 11, 16syl2anc 584 . . . 4 (𝜑 → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
1810, 17eldifd 3974 . . 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 2843 1 (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  wne 2938  cdif 3960  cop 4637   class class class wbr 5148   I cid 5582   × cxp 5687  ran crn 5690   Fn wfn 6558  cfv 6563  (class class class)co 7431  Basecbs 17245  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  ax-un 7754
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-csb 3909  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-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-trkg 28476
This theorem is referenced by:  tghilberti1  28660  tglineinteq  28668  colline  28672  tglowdim2ln  28674  footexALT  28741  footexlem2  28743  foot  28745  perprag  28749  colperpexlem3  28755  mideulem2  28757  midex  28760  outpasch  28778  lnopp2hpgb  28786  colopp  28792  lmieu  28807  lmimid  28817  hypcgrlem1  28822  hypcgrlem2  28823  lnperpex  28826  trgcopy  28827  trgcopyeulem  28828  acopy  28856  acopyeu  28857  tgasa1  28881
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