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Theorem lncom 25562
Description: Swapping the points defining a line keeps it unchanged. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.)
Hypotheses
Ref Expression
btwnlng1.p 𝑃 = (Base‘𝐺)
btwnlng1.i 𝐼 = (Itv‘𝐺)
btwnlng1.l 𝐿 = (LineG‘𝐺)
btwnlng1.g (𝜑𝐺 ∈ TarskiG)
btwnlng1.x (𝜑𝑋𝑃)
btwnlng1.y (𝜑𝑌𝑃)
btwnlng1.z (𝜑𝑍𝑃)
btwnlng1.d (𝜑𝑋𝑌)
lncom.1 (𝜑𝑍 ∈ (𝑌𝐿𝑋))
Assertion
Ref Expression
lncom (𝜑𝑍 ∈ (𝑋𝐿𝑌))

Proof of Theorem lncom
StepHypRef Expression
1 lncom.1 . 2 (𝜑𝑍 ∈ (𝑌𝐿𝑋))
2 3orcomb 1065 . . . 4 ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌)))
3 btwnlng1.p . . . . . 6 𝑃 = (Base‘𝐺)
4 eqid 2651 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
5 btwnlng1.i . . . . . 6 𝐼 = (Itv‘𝐺)
6 btwnlng1.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
7 btwnlng1.x . . . . . 6 (𝜑𝑋𝑃)
8 btwnlng1.z . . . . . 6 (𝜑𝑍𝑃)
9 btwnlng1.y . . . . . 6 (𝜑𝑌𝑃)
103, 4, 5, 6, 7, 8, 9tgbtwncomb 25429 . . . . 5 (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑌𝐼𝑋)))
113, 4, 5, 6, 7, 9, 8tgbtwncomb 25429 . . . . 5 (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ 𝑌 ∈ (𝑍𝐼𝑋)))
123, 4, 5, 6, 8, 7, 9tgbtwncomb 25429 . . . . 5 (𝜑 → (𝑋 ∈ (𝑍𝐼𝑌) ↔ 𝑋 ∈ (𝑌𝐼𝑍)))
1310, 11, 123orbi123d 1438 . . . 4 (𝜑 → ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌)) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍))))
142, 13syl5bb 272 . . 3 (𝜑 → ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍))))
15 btwnlng1.l . . . 4 𝐿 = (LineG‘𝐺)
16 btwnlng1.d . . . 4 (𝜑𝑋𝑌)
173, 15, 5, 6, 7, 9, 16, 8tgellng 25493 . . 3 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
1816necomd 2878 . . . 4 (𝜑𝑌𝑋)
193, 15, 5, 6, 9, 7, 18, 8tgellng 25493 . . 3 (𝜑 → (𝑍 ∈ (𝑌𝐿𝑋) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍))))
2014, 17, 193bitr4d 300 . 2 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ 𝑍 ∈ (𝑌𝐿𝑋)))
211, 20mpbird 247 1 (𝜑𝑍 ∈ (𝑋𝐿𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3o 1053   = wceq 1523  wcel 2030  wne 2823  cfv 5926  (class class class)co 6690  Basecbs 15904  distcds 15997  TarskiGcstrkg 25374  Itvcitv 25380  LineGclng 25381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-trkgc 25392  df-trkgb 25393  df-trkgcb 25394  df-trkg 25397
This theorem is referenced by:  tglineelsb2  25572  tglinecom  25575  ncolncol  25586  coltr  25587  midexlem  25632  footex  25658  opphllem1  25684  opphllem2  25685  outpasch  25692  hlpasch  25693  trgcopy  25741  trgcopyeulem  25742  cgracgr  25755  tgasa1  25784
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