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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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