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Theorem lnrot1 26337
Description: Rotating the points defining a line. 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)
lnrot1.1 (𝜑𝑌 ∈ (𝑍𝐿𝑋))
lnrot1.2 (𝜑𝑍𝑋)
Assertion
Ref Expression
lnrot1 (𝜑𝑍 ∈ (𝑋𝐿𝑌))

Proof of Theorem lnrot1
StepHypRef Expression
1 lnrot1.1 . 2 (𝜑𝑌 ∈ (𝑍𝐿𝑋))
2 btwnlng1.p . . . . . 6 𝑃 = (Base‘𝐺)
3 eqid 2821 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
4 btwnlng1.i . . . . . 6 𝐼 = (Itv‘𝐺)
5 btwnlng1.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
6 btwnlng1.y . . . . . 6 (𝜑𝑌𝑃)
7 btwnlng1.z . . . . . 6 (𝜑𝑍𝑃)
8 btwnlng1.x . . . . . 6 (𝜑𝑋𝑃)
92, 3, 4, 5, 6, 7, 8tgbtwncomb 26203 . . . . 5 (𝜑 → (𝑍 ∈ (𝑌𝐼𝑋) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
10 biidd 263 . . . . 5 (𝜑 → (𝑋 ∈ (𝑍𝐼𝑌) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
112, 3, 4, 5, 7, 6, 8tgbtwncomb 26203 . . . . 5 (𝜑 → (𝑌 ∈ (𝑍𝐼𝑋) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
129, 10, 113orbi123d 1426 . . . 4 (𝜑 → ((𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑍𝐼𝑋)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
13 3orrot 1084 . . . . 5 ((𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌)) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑍𝐼𝑋)))
1413a1i 11 . . . 4 (𝜑 → ((𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌)) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑍𝐼𝑋))))
15 btwnlng1.l . . . . 5 𝐿 = (LineG‘𝐺)
16 btwnlng1.d . . . . 5 (𝜑𝑋𝑌)
172, 15, 4, 5, 8, 6, 16, 7tgellng 26267 . . . 4 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
1812, 14, 173bitr4rd 313 . . 3 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌))))
19 lnrot1.2 . . . 4 (𝜑𝑍𝑋)
202, 15, 4, 5, 7, 8, 19, 6tgellng 26267 . . 3 (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ↔ (𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌))))
2118, 20bitr4d 283 . 2 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ 𝑌 ∈ (𝑍𝐿𝑋)))
221, 21mpbird 258 1 (𝜑𝑍 ∈ (𝑋𝐿𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  w3o 1078   = wceq 1528  wcel 2105  wne 3016  cfv 6349  (class class class)co 7145  Basecbs 16473  distcds 16564  TarskiGcstrkg 26144  Itvcitv 26150  LineGclng 26151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-ov 7148  df-oprab 7149  df-mpo 7150  df-trkgc 26162  df-trkgb 26163  df-trkgcb 26164  df-trkg 26167
This theorem is referenced by:  tglineelsb2  26346  tglineneq  26358  coltr3  26362  hlperpnel  26439  opphllem4  26464  lmieu  26498
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