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Theorem lnrot1 24388
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  |-  ( ph  ->  G  e. TarskiG )
btwnlng1.x  |-  ( ph  ->  X  e.  P )
btwnlng1.y  |-  ( ph  ->  Y  e.  P )
btwnlng1.z  |-  ( ph  ->  Z  e.  P )
btwnlng1.d  |-  ( ph  ->  X  =/=  Y )
lnrot1.1  |-  ( ph  ->  Y  e.  ( Z L X ) )
lnrot1.2  |-  ( ph  ->  Z  =/=  X )
Assertion
Ref Expression
lnrot1  |-  ( ph  ->  Z  e.  ( X L Y ) )

Proof of Theorem lnrot1
StepHypRef Expression
1 lnrot1.1 . 2  |-  ( ph  ->  Y  e.  ( Z L X ) )
2 btwnlng1.p . . . . . 6  |-  P  =  ( Base `  G
)
3 eqid 2402 . . . . . 6  |-  ( dist `  G )  =  (
dist `  G )
4 btwnlng1.i . . . . . 6  |-  I  =  (Itv `  G )
5 btwnlng1.g . . . . . 6  |-  ( ph  ->  G  e. TarskiG )
6 btwnlng1.y . . . . . 6  |-  ( ph  ->  Y  e.  P )
7 btwnlng1.z . . . . . 6  |-  ( ph  ->  Z  e.  P )
8 btwnlng1.x . . . . . 6  |-  ( ph  ->  X  e.  P )
92, 3, 4, 5, 6, 7, 8tgbtwncomb 24261 . . . . 5  |-  ( ph  ->  ( Z  e.  ( Y I X )  <-> 
Z  e.  ( X I Y ) ) )
10 biidd 237 . . . . 5  |-  ( ph  ->  ( X  e.  ( Z I Y )  <-> 
X  e.  ( Z I Y ) ) )
112, 3, 4, 5, 7, 6, 8tgbtwncomb 24261 . . . . 5  |-  ( ph  ->  ( Y  e.  ( Z I X )  <-> 
Y  e.  ( X I Z ) ) )
129, 10, 113orbi123d 1300 . . . 4  |-  ( ph  ->  ( ( Z  e.  ( Y I X )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( Z I X ) )  <->  ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) )
13 3orrot 980 . . . . 5  |-  ( ( Y  e.  ( Z I X )  \/  Z  e.  ( Y I X )  \/  X  e.  ( Z I Y ) )  <-> 
( Z  e.  ( Y I X )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( Z I X ) ) )
1413a1i 11 . . . 4  |-  ( ph  ->  ( ( Y  e.  ( Z I X )  \/  Z  e.  ( Y I X )  \/  X  e.  ( Z I Y ) )  <->  ( Z  e.  ( Y I X )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( Z I X ) ) ) )
15 btwnlng1.l . . . . 5  |-  L  =  (LineG `  G )
16 btwnlng1.d . . . . 5  |-  ( ph  ->  X  =/=  Y )
172, 15, 4, 5, 8, 6, 16, 7tgellng 24323 . . . 4  |-  ( ph  ->  ( Z  e.  ( X L Y )  <-> 
( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) )
1812, 14, 173bitr4rd 286 . . 3  |-  ( ph  ->  ( Z  e.  ( X L Y )  <-> 
( Y  e.  ( Z I X )  \/  Z  e.  ( Y I X )  \/  X  e.  ( Z I Y ) ) ) )
19 lnrot1.2 . . . 4  |-  ( ph  ->  Z  =/=  X )
202, 15, 4, 5, 7, 8, 19, 6tgellng 24323 . . 3  |-  ( ph  ->  ( Y  e.  ( Z L X )  <-> 
( Y  e.  ( Z I X )  \/  Z  e.  ( Y I X )  \/  X  e.  ( Z I Y ) ) ) )
2118, 20bitr4d 256 . 2  |-  ( ph  ->  ( Z  e.  ( X L Y )  <-> 
Y  e.  ( Z L X ) ) )
221, 21mpbird 232 1  |-  ( ph  ->  Z  e.  ( X L Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ w3o 973    = wceq 1405    e. wcel 1842    =/= wne 2598   ` cfv 5569  (class class class)co 6278   Basecbs 14841   distcds 14918  TarskiGcstrkg 24206  Itvcitv 24212  LineGclng 24213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-trkgc 24224  df-trkgb 24225  df-trkgcb 24226  df-trkg 24229
This theorem is referenced by:  tglineelsb2  24397  tglineneq  24409  coltr3  24414  hlperpnel  24485  opphllem4  24509  lmieu  24540
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