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Theorem tglnne 26425
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 729 . . 3 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝐺 ∈ TarskiG)
6 tglnne.x . . . 4 (𝜑𝑋𝐵)
76ad3antrrr 729 . . 3 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑋𝐵)
8 tglnne.y . . . 4 (𝜑𝑌𝐵)
98ad3antrrr 729 . . 3 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑌𝐵)
10 simpllr 775 . . . . 5 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥𝐵)
11 simplr 768 . . . . 5 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑦𝐵)
12 simprr 772 . . . . 5 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥𝑦)
13 eqid 2824 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
141, 13, 3, 5, 10, 11tgbtwntriv1 26288 . . . . 5 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝑥𝐼𝑦))
151, 3, 2, 5, 10, 11, 10, 12, 14btwnlng1 26416 . . . 4 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝑥𝐿𝑦))
16 simprl 770 . . . 4 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → (𝑋𝐿𝑌) = (𝑥𝐿𝑦))
1715, 16eleqtrrd 2919 . . 3 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝑋𝐿𝑌))
181, 2, 3, 5, 7, 9, 17tglngne 26347 . 2 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑋𝑌)
19 tglnne.1 . . 3 (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)
201, 3, 2, 4, 19tgisline 26424 . 2 (𝜑 → ∃𝑥𝐵𝑦𝐵 ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦))
2118, 20r19.29vva 3327 1 (𝜑𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wne 3014  ran crn 5543  cfv 6343  (class class class)co 7149  Basecbs 16483  distcds 16574  TarskiGcstrkg 26227  Itvcitv 26233  LineGclng 26234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-trkgc 26245  df-trkgb 26246  df-trkgcb 26247  df-trkg 26250
This theorem is referenced by:  footne  26520  footeq  26521  hlperpnel  26522  colperp  26526  mideulem2  26531  opphllem  26532  midex  26534  opphllem3  26546  opphllem6  26549  opphl  26551  lmieu  26581  lnperpex  26600  trgcopy  26601
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