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Theorem lnrot2 26522
 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)
lnrot2.1 (𝜑𝑋 ∈ (𝑌𝐿𝑍))
lnrot2.2 (𝜑𝑌𝑍)
Assertion
Ref Expression
lnrot2 (𝜑𝑍 ∈ (𝑋𝐿𝑌))

Proof of Theorem lnrot2
StepHypRef Expression
1 lnrot2.1 . 2 (𝜑𝑋 ∈ (𝑌𝐿𝑍))
2 btwnlng1.p . . . . . 6 𝑃 = (Base‘𝐺)
3 eqid 2758 . . . . . 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.x . . . . . 6 (𝜑𝑋𝑃)
8 btwnlng1.z . . . . . 6 (𝜑𝑍𝑃)
92, 3, 4, 5, 6, 7, 8tgbtwncomb 26387 . . . . 5 (𝜑 → (𝑋 ∈ (𝑌𝐼𝑍) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
10 biidd 265 . . . . 5 (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
112, 3, 4, 5, 6, 8, 7tgbtwncomb 26387 . . . . 5 (𝜑 → (𝑍 ∈ (𝑌𝐼𝑋) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
129, 10, 113orbi123d 1432 . . . 4 (𝜑 → ((𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑌𝐼𝑋)) ↔ (𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌))))
13 3orrot 1089 . . . 4 ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌)))
1412, 13bitr4di 292 . . 3 (𝜑 → ((𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑌𝐼𝑋)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
15 btwnlng1.l . . . 4 𝐿 = (LineG‘𝐺)
16 lnrot2.2 . . . 4 (𝜑𝑌𝑍)
172, 15, 4, 5, 6, 8, 16, 7tgellng 26451 . . 3 (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ↔ (𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑌𝐼𝑋))))
18 btwnlng1.d . . . 4 (𝜑𝑋𝑌)
192, 15, 4, 5, 7, 6, 18, 8tgellng 26451 . . 3 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
2014, 17, 193bitr4d 314 . 2 (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ↔ 𝑍 ∈ (𝑋𝐿𝑌)))
211, 20mpbid 235 1 (𝜑𝑍 ∈ (𝑋𝐿𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ w3o 1083   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ‘cfv 6339  (class class class)co 7155  Basecbs 16546  distcds 16637  TarskiGcstrkg 26328  Itvcitv 26334  LineGclng 26335 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-iota 6298  df-fun 6341  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-trkgc 26346  df-trkgb 26347  df-trkgcb 26348  df-trkg 26351 This theorem is referenced by:  coltr  26545  mideulem2  26632
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