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Theorem lnrot2 23149
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 )
lnrot2.1  |-  ( ph  ->  X  e.  ( Y L Z ) )
lnrot2.2  |-  ( ph  ->  Y  =/=  Z )
Assertion
Ref Expression
lnrot2  |-  ( ph  ->  Z  e.  ( X L Y ) )

Proof of Theorem lnrot2
StepHypRef Expression
1 lnrot2.1 . 2  |-  ( ph  ->  X  e.  ( Y L Z ) )
2 btwnlng1.p . . . . . 6  |-  P  =  ( Base `  G
)
3 eqid 2451 . . . . . 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.x . . . . . 6  |-  ( ph  ->  X  e.  P )
8 btwnlng1.z . . . . . 6  |-  ( ph  ->  Z  e.  P )
92, 3, 4, 5, 6, 7, 8tgbtwncomb 23057 . . . . 5  |-  ( ph  ->  ( X  e.  ( Y I Z )  <-> 
X  e.  ( Z I Y ) ) )
10 biidd 237 . . . . 5  |-  ( ph  ->  ( Y  e.  ( X I Z )  <-> 
Y  e.  ( X I Z ) ) )
112, 3, 4, 5, 6, 8, 7tgbtwncomb 23057 . . . . 5  |-  ( ph  ->  ( Z  e.  ( Y I X )  <-> 
Z  e.  ( X I Y ) ) )
129, 10, 113orbi123d 1289 . . . 4  |-  ( ph  ->  ( ( X  e.  ( Y I Z )  \/  Y  e.  ( X I Z )  \/  Z  e.  ( Y I X ) )  <->  ( X  e.  ( Z I Y )  \/  Y  e.  ( X I Z )  \/  Z  e.  ( X I Y ) ) ) )
13 3orrot 971 . . . 4  |-  ( ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) )  <-> 
( X  e.  ( Z I Y )  \/  Y  e.  ( X I Z )  \/  Z  e.  ( X I Y ) ) )
1412, 13syl6bbr 263 . . 3  |-  ( ph  ->  ( ( X  e.  ( Y I Z )  \/  Y  e.  ( X I Z )  \/  Z  e.  ( Y I X ) )  <->  ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) )
15 btwnlng1.l . . . 4  |-  L  =  (LineG `  G )
16 lnrot2.2 . . . 4  |-  ( ph  ->  Y  =/=  Z )
172, 15, 4, 5, 6, 8, 16, 7tgellng 23103 . . 3  |-  ( ph  ->  ( X  e.  ( Y L Z )  <-> 
( X  e.  ( Y I Z )  \/  Y  e.  ( X I Z )  \/  Z  e.  ( Y I X ) ) ) )
18 btwnlng1.d . . . 4  |-  ( ph  ->  X  =/=  Y )
192, 15, 4, 5, 7, 6, 18, 8tgellng 23103 . . 3  |-  ( ph  ->  ( Z  e.  ( X L Y )  <-> 
( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) )
2014, 17, 193bitr4d 285 . 2  |-  ( ph  ->  ( X  e.  ( Y L Z )  <-> 
Z  e.  ( X L Y ) ) )
211, 20mpbid 210 1  |-  ( ph  ->  Z  e.  ( X L Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ w3o 964    = wceq 1370    e. wcel 1758    =/= wne 2642   ` cfv 5513  (class class class)co 6187   Basecbs 14273   distcds 14346  TarskiGcstrkg 23002  Itvcitv 23009  LineGclng 23010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-iota 5476  df-fun 5515  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-trkgc 23021  df-trkgb 23022  df-trkgcb 23023  df-trkg 23027
This theorem is referenced by:  coltr  23171  mideulem  23241
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