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Theorem tgelrnln 25506
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 6638 . 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 25423 . . . 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)
10 opelxpi 5138 . . . . 5 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
118, 9, 10syl2anc 692 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
12 tgelrnln.d . . . . 5 (𝜑𝑋𝑌)
13 df-br 4645 . . . . . . . 8 (𝑋 I 𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ I )
14 ideqg 5262 . . . . . . . 8 (𝑌𝐵 → (𝑋 I 𝑌𝑋 = 𝑌))
1513, 14syl5bbr 274 . . . . . . 7 (𝑌𝐵 → (⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋 = 𝑌))
1615necon3bbid 2828 . . . . . 6 (𝑌𝐵 → (¬ ⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋𝑌))
1716biimpar 502 . . . . 5 ((𝑌𝐵𝑋𝑌) → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
189, 12, 17syl2anc 692 . . . 4 (𝜑 → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
1911, 18eldifd 3578 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((𝐵 × 𝐵) ∖ I ))
20 fnfvelrn 6342 . . 3 ((𝐿 Fn ((𝐵 × 𝐵) ∖ I ) ∧ ⟨𝑋, 𝑌⟩ ∈ ((𝐵 × 𝐵) ∖ I )) → (𝐿‘⟨𝑋, 𝑌⟩) ∈ ran 𝐿)
217, 19, 20syl2anc 692 . 2 (𝜑 → (𝐿‘⟨𝑋, 𝑌⟩) ∈ ran 𝐿)
221, 21syl5eqel 2703 1 (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1481  wcel 1988  wne 2791  cdif 3564  cop 4174   class class class wbr 4644   I cid 5013   × cxp 5102  ran crn 5105   Fn wfn 5871  cfv 5876  (class class class)co 6635  Basecbs 15838  TarskiGcstrkg 25310  Itvcitv 25316  LineGclng 25317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-trkg 25333
This theorem is referenced by:  tghilberti1  25513  tglineinteq  25521  colline  25525  tglowdim2ln  25527  footex  25594  foot  25595  perprag  25599  colperpexlem3  25605  mideulem2  25607  midex  25610  opphllem5  25624  opphllem6  25625  outpasch  25628  lnopp2hpgb  25636  colopp  25642  lmieu  25657  lmimid  25667  hypcgrlem1  25672  hypcgrlem2  25673  lnperpex  25676  trgcopy  25677  trgcopyeulem  25678  acopy  25705  acopyeu  25706  tgasa1  25720
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