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Theorem tgelrnln 25716
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 6808 . 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 25633 . . . 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 5297 . . . . 5 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
118, 9, 10syl2anc 696 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
12 tgelrnln.d . . . . 5 (𝜑𝑋𝑌)
13 df-br 4797 . . . . . . . 8 (𝑋 I 𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ I )
14 ideqg 5421 . . . . . . . 8 (𝑌𝐵 → (𝑋 I 𝑌𝑋 = 𝑌))
1513, 14syl5bbr 274 . . . . . . 7 (𝑌𝐵 → (⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋 = 𝑌))
1615necon3bbid 2961 . . . . . 6 (𝑌𝐵 → (¬ ⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋𝑌))
1716biimpar 503 . . . . 5 ((𝑌𝐵𝑋𝑌) → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
189, 12, 17syl2anc 696 . . . 4 (𝜑 → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
1911, 18eldifd 3718 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((𝐵 × 𝐵) ∖ I ))
20 fnfvelrn 6511 . . 3 ((𝐿 Fn ((𝐵 × 𝐵) ∖ I ) ∧ ⟨𝑋, 𝑌⟩ ∈ ((𝐵 × 𝐵) ∖ I )) → (𝐿‘⟨𝑋, 𝑌⟩) ∈ ran 𝐿)
217, 19, 20syl2anc 696 . 2 (𝜑 → (𝐿‘⟨𝑋, 𝑌⟩) ∈ ran 𝐿)
221, 21syl5eqel 2835 1 (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1624  wcel 2131  wne 2924  cdif 3704  cop 4319   class class class wbr 4796   I cid 5165   × cxp 5256  ran crn 5259   Fn wfn 6036  cfv 6041  (class class class)co 6805  Basecbs 16051  TarskiGcstrkg 25520  Itvcitv 25526  LineGclng 25527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326  df-trkg 25543
This theorem is referenced by:  tghilberti1  25723  tglineinteq  25731  colline  25735  tglowdim2ln  25737  footex  25804  foot  25805  perprag  25809  colperpexlem3  25815  mideulem2  25817  midex  25820  opphllem5  25834  opphllem6  25835  outpasch  25838  lnopp2hpgb  25846  colopp  25852  lmieu  25867  lmimid  25877  hypcgrlem1  25882  hypcgrlem2  25883  lnperpex  25886  trgcopy  25887  trgcopyeulem  25888  acopy  25915  acopyeu  25916  tgasa1  25930
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