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Theorem tglnne 25440
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 765 . . 3 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝐺 ∈ TarskiG)
6 tglnne.x . . . 4 (𝜑𝑋𝐵)
76ad3antrrr 765 . . 3 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑋𝐵)
8 tglnne.y . . . 4 (𝜑𝑌𝐵)
98ad3antrrr 765 . . 3 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑌𝐵)
10 simpllr 798 . . . . 5 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥𝐵)
11 simplr 791 . . . . 5 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑦𝐵)
12 simprr 795 . . . . 5 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥𝑦)
13 eqid 2621 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
141, 13, 3, 5, 10, 11tgbtwntriv1 25303 . . . . 5 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝑥𝐼𝑦))
151, 3, 2, 5, 10, 11, 10, 12, 14btwnlng1 25431 . . . 4 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝑥𝐿𝑦))
16 simprl 793 . . . 4 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → (𝑋𝐿𝑌) = (𝑥𝐿𝑦))
1715, 16eleqtrrd 2701 . . 3 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝑋𝐿𝑌))
181, 2, 3, 5, 7, 9, 17tglngne 25362 . 2 ((((𝜑𝑥𝐵) ∧ 𝑦𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑋𝑌)
19 tglnne.1 . . 3 (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)
201, 3, 2, 4, 19tgisline 25439 . 2 (𝜑 → ∃𝑥𝐵𝑦𝐵 ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥𝑦))
2118, 20r19.29vva 3074 1 (𝜑𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  ran crn 5080  cfv 5852  (class class class)co 6610  Basecbs 15792  distcds 15882  TarskiGcstrkg 25246  Itvcitv 25252  LineGclng 25253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-trkgc 25264  df-trkgb 25265  df-trkgcb 25266  df-trkg 25269
This theorem is referenced by:  footne  25532  footeq  25533  hlperpnel  25534  colperp  25538  mideulem2  25543  opphllem  25544  midex  25546  opphllem3  25558  opphllem6  25561  opphl  25563  lmieu  25593  lnperpex  25612  trgcopy  25613
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