MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgcolg Structured version   Visualization version   GIF version

Theorem tgcolg 27785
Description: We choose the notation (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) instead of "colinear" in order to avoid defining an additional symbol for colinearity because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 25-May-2019.)
Hypotheses
Ref Expression
tglngval.p 𝑃 = (Base‘𝐺)
tglngval.l 𝐿 = (LineG‘𝐺)
tglngval.i 𝐼 = (Itv‘𝐺)
tglngval.g (𝜑𝐺 ∈ TarskiG)
tglngval.x (𝜑𝑋𝑃)
tglngval.y (𝜑𝑌𝑃)
tgcolg.z (𝜑𝑍𝑃)
Assertion
Ref Expression
tgcolg (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))

Proof of Theorem tgcolg
StepHypRef Expression
1 animorr 978 . . 3 ((𝜑𝑋 = 𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
2 tglngval.p . . . . . 6 𝑃 = (Base‘𝐺)
3 eqid 2733 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
4 tglngval.i . . . . . 6 𝐼 = (Itv‘𝐺)
5 tglngval.g . . . . . . 7 (𝜑𝐺 ∈ TarskiG)
65adantr 482 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝐺 ∈ TarskiG)
7 tgcolg.z . . . . . . 7 (𝜑𝑍𝑃)
87adantr 482 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑍𝑃)
9 tglngval.x . . . . . . 7 (𝜑𝑋𝑃)
109adantr 482 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋𝑃)
112, 3, 4, 6, 8, 10tgbtwntriv2 27718 . . . . 5 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑍𝐼𝑋))
12 simpr 486 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋 = 𝑌)
1312oveq2d 7420 . . . . 5 ((𝜑𝑋 = 𝑌) → (𝑍𝐼𝑋) = (𝑍𝐼𝑌))
1411, 13eleqtrd 2836 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑍𝐼𝑌))
15143mix2d 1338 . . 3 ((𝜑𝑋 = 𝑌) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
161, 152thd 265 . 2 ((𝜑𝑋 = 𝑌) → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
17 simpr 486 . . . . . 6 ((𝜑𝑋𝑌) → 𝑋𝑌)
1817neneqd 2946 . . . . 5 ((𝜑𝑋𝑌) → ¬ 𝑋 = 𝑌)
19 biorf 936 . . . . 5 𝑋 = 𝑌 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑋 = 𝑌𝑍 ∈ (𝑋𝐿𝑌))))
2018, 19syl 17 . . . 4 ((𝜑𝑋𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑋 = 𝑌𝑍 ∈ (𝑋𝐿𝑌))))
21 orcom 869 . . . 4 ((𝑋 = 𝑌𝑍 ∈ (𝑋𝐿𝑌)) ↔ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
2220, 21bitrdi 287 . . 3 ((𝜑𝑋𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)))
23 tglngval.l . . . 4 𝐿 = (LineG‘𝐺)
245adantr 482 . . . 4 ((𝜑𝑋𝑌) → 𝐺 ∈ TarskiG)
259adantr 482 . . . 4 ((𝜑𝑋𝑌) → 𝑋𝑃)
26 tglngval.y . . . . 5 (𝜑𝑌𝑃)
2726adantr 482 . . . 4 ((𝜑𝑋𝑌) → 𝑌𝑃)
287adantr 482 . . . 4 ((𝜑𝑋𝑌) → 𝑍𝑃)
292, 23, 4, 24, 25, 27, 17, 28tgellng 27784 . . 3 ((𝜑𝑋𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
3022, 29bitr3d 281 . 2 ((𝜑𝑋𝑌) → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
3116, 30pm2.61dane 3030 1 (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846  w3o 1087   = wceq 1542  wcel 2107  wne 2941  cfv 6540  (class class class)co 7404  Basecbs 17140  distcds 17202  TarskiGcstrkg 27658  Itvcitv 27664  LineGclng 27665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-trkgc 27679  df-trkgcb 27681  df-trkg 27684
This theorem is referenced by:  btwncolg1  27786  btwncolg2  27787  btwncolg3  27788  colcom  27789  colrot1  27790  lnxfr  27797  lnext  27798  tgfscgr  27799  tglowdim2l  27881  outpasch  27986
  Copyright terms: Public domain W3C validator