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Theorem tglnne 25940
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 721 . . 3 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝐺 ∈ TarskiG)
6 tglnne.x . . . 4 (𝜑𝑋𝐵)
76ad3antrrr 721 . . 3 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑋𝐵)
8 tglnne.y . . . 4 (𝜑𝑌𝐵)
98ad3antrrr 721 . . 3 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑌𝐵)
10 simpllr 793 . . . . 5 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥𝐵)
11 simplr 785 . . . . 5 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑦𝐵)
12 simprr 789 . . . . 5 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥𝑦)
13 eqid 2825 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
141, 13, 3, 5, 10, 11tgbtwntriv1 25803 . . . . 5 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝑥𝐼𝑦))
151, 3, 2, 5, 10, 11, 10, 12, 14btwnlng1 25931 . . . 4 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝑥𝐿𝑦))
16 simprl 787 . . . 4 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → (𝑋𝐿𝑌) = (𝑥𝐿𝑦))
1715, 16eleqtrrd 2909 . . 3 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝑋𝐿𝑌))
181, 2, 3, 5, 7, 9, 17tglngne 25862 . 2 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑋𝑌)
19 tglnne.1 . . 3 (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)
201, 3, 2, 4, 19tgisline 25939 . 2 (𝜑 → ∃𝑥𝐵𝑦𝐵 ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦))
2118, 20r19.29vva 3291 1 (𝜑𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  wne 2999  ran crn 5343  cfv 6123  (class class class)co 6905  Basecbs 16222  distcds 16314  TarskiGcstrkg 25742  Itvcitv 25748  LineGclng 25749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-trkgc 25760  df-trkgb 25761  df-trkgcb 25762  df-trkg 25765
This theorem is referenced by:  footne  26032  footeq  26033  hlperpnel  26034  colperp  26038  mideulem2  26043  opphllem  26044  midex  26046  opphllem3  26058  opphllem6  26061  opphl  26063  lmieu  26093  lnperpex  26112  trgcopy  26113
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