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Theorem lnrot1 26342
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 2826 . . . . . 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 26208 . . . . 5 (𝜑 → (𝑍 ∈ (𝑌𝐼𝑋) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
10 biidd 263 . . . . 5 (𝜑 → (𝑋 ∈ (𝑍𝐼𝑌) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
112, 3, 4, 5, 7, 6, 8tgbtwncomb 26208 . . . . 5 (𝜑 → (𝑌 ∈ (𝑍𝐼𝑋) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
129, 10, 113orbi123d 1428 . . . 4 (𝜑 → ((𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑍𝐼𝑋)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
13 3orrot 1086 . . . . 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 26272 . . . 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 26272 . . 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 1080   = wceq 1530  wcel 2107  wne 3021  cfv 6354  (class class class)co 7150  Basecbs 16478  distcds 16569  TarskiGcstrkg 26149  Itvcitv 26155  LineGclng 26156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-trkgc 26167  df-trkgb 26168  df-trkgcb 26169  df-trkg 26172
This theorem is referenced by:  tglineelsb2  26351  tglineneq  26363  coltr3  26367  hlperpnel  26444  opphllem4  26469  lmieu  26503
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