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Theorem lncom 25738
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 1078 . . . 4 ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌)))
3 btwnlng1.p . . . . . 6 𝑃 = (Base‘𝐺)
4 eqid 2771 . . . . . 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 25605 . . . . 5 (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑌𝐼𝑋)))
113, 4, 5, 6, 7, 9, 8tgbtwncomb 25605 . . . . 5 (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ 𝑌 ∈ (𝑍𝐼𝑋)))
123, 4, 5, 6, 8, 7, 9tgbtwncomb 25605 . . . . 5 (𝜑 → (𝑋 ∈ (𝑍𝐼𝑌) ↔ 𝑋 ∈ (𝑌𝐼𝑍)))
1310, 11, 123orbi123d 1546 . . . 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 25669 . . 3 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
1816necomd 2998 . . . 4 (𝜑𝑌𝑋)
193, 15, 5, 6, 9, 7, 18, 8tgellng 25669 . . 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 1070   = wceq 1631  wcel 2145  wne 2943  cfv 6031  (class class class)co 6793  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556  LineGclng 25557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-trkgc 25568  df-trkgb 25569  df-trkgcb 25570  df-trkg 25573
This theorem is referenced by:  tglineelsb2  25748  tglinecom  25751  ncolncol  25762  coltr  25763  midexlem  25808  footex  25834  opphllem1  25860  opphllem2  25861  outpasch  25868  hlpasch  25869  trgcopy  25917  trgcopyeulem  25918  cgracgr  25931  tgasa1  25960
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