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Theorem tglnne 28452
Description: It takes two different points to form a line. (Contributed by Thierry Arnoux, 27-Nov-2019.)
Hypotheses
Ref Expression
tglineelsb2.p 𝐵 = (Base‘𝐺)
tglineelsb2.i 𝐼 = (Itv‘𝐺)
tglineelsb2.l 𝐿 = (LineG‘𝐺)
tglineelsb2.g (𝜑𝐺 ∈ TarskiG)
tglnne.x (𝜑𝑋𝐵)
tglnne.y (𝜑𝑌𝐵)
tglnne.1 (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)
Assertion
Ref Expression
tglnne (𝜑𝑋𝑌)

Proof of Theorem tglnne
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tglineelsb2.p . . 3 𝐵 = (Base‘𝐺)
2 tglineelsb2.l . . 3 𝐿 = (LineG‘𝐺)
3 tglineelsb2.i . . 3 𝐼 = (Itv‘𝐺)
4 tglineelsb2.g . . . 4 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 728 . . 3 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝐺 ∈ TarskiG)
6 tglnne.x . . . 4 (𝜑𝑋𝐵)
76ad3antrrr 728 . . 3 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑋𝐵)
8 tglnne.y . . . 4 (𝜑𝑌𝐵)
98ad3antrrr 728 . . 3 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑌𝐵)
10 simpllr 774 . . . . 5 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥𝐵)
11 simplr 767 . . . . 5 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑦𝐵)
12 simprr 771 . . . . 5 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥𝑦)
13 eqid 2728 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
141, 13, 3, 5, 10, 11tgbtwntriv1 28315 . . . . 5 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝑥𝐼𝑦))
151, 3, 2, 5, 10, 11, 10, 12, 14btwnlng1 28443 . . . 4 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝑥𝐿𝑦))
16 simprl 769 . . . 4 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → (𝑋𝐿𝑌) = (𝑥𝐿𝑦))
1715, 16eleqtrrd 2832 . . 3 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝑋𝐿𝑌))
181, 2, 3, 5, 7, 9, 17tglngne 28374 . 2 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑋𝑌)
19 tglnne.1 . . 3 (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)
201, 3, 2, 4, 19tgisline 28451 . 2 (𝜑 → ∃𝑥𝐵𝑦𝐵 ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦))
2118, 20r19.29vva 3211 1 (𝜑𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wne 2937  ran crn 5683  cfv 6553  (class class class)co 7426  Basecbs 17187  distcds 17249  TarskiGcstrkg 28251  Itvcitv 28257  LineGclng 28258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-trkgc 28272  df-trkgb 28273  df-trkgcb 28274  df-trkg 28277
This theorem is referenced by:  footne  28547  footeq  28548  hlperpnel  28549  colperp  28553  mideulem2  28558  opphllem  28559  midex  28561  opphllem3  28573  opphllem6  28576  opphl  28578  lmieu  28608  lnperpex  28627  trgcopy  28628
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