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Theorem lnrot1 27565
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 2736 . . . . . 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 27431 . . . . 5 (𝜑 → (𝑍 ∈ (𝑌𝐼𝑋) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
10 biidd 261 . . . . 5 (𝜑 → (𝑋 ∈ (𝑍𝐼𝑌) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
112, 3, 4, 5, 7, 6, 8tgbtwncomb 27431 . . . . 5 (𝜑 → (𝑌 ∈ (𝑍𝐼𝑋) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
129, 10, 113orbi123d 1435 . . . 4 (𝜑 → ((𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑍𝐼𝑋)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
13 3orrot 1092 . . . . 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 27495 . . . 4 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
1812, 14, 173bitr4rd 311 . . 3 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌))))
19 lnrot1.2 . . . 4 (𝜑𝑍𝑋)
202, 15, 4, 5, 7, 8, 19, 6tgellng 27495 . . 3 (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ↔ (𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌))))
2118, 20bitr4d 281 . 2 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ 𝑌 ∈ (𝑍𝐿𝑋)))
221, 21mpbird 256 1 (𝜑𝑍 ∈ (𝑋𝐿𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3o 1086   = wceq 1541  wcel 2106  wne 2943  cfv 6496  (class class class)co 7357  Basecbs 17083  distcds 17142  TarskiGcstrkg 27369  Itvcitv 27375  LineGclng 27376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-trkgc 27390  df-trkgb 27391  df-trkgcb 27392  df-trkg 27395
This theorem is referenced by:  tglineelsb2  27574  tglineneq  27586  coltr3  27590  hlperpnel  27667  opphllem4  27692  lmieu  27726
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