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Theorem lnrot2 24130
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 2457 . . . . . 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 24006 . . . . 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 24006 . . . . 5  |-  ( ph  ->  ( Z  e.  ( Y I X )  <-> 
Z  e.  ( X I Y ) ) )
129, 10, 113orbi123d 1298 . . . 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 979 . . . 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 24066 . . 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 24066 . . 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 972    = wceq 1395    e. wcel 1819    =/= wne 2652   ` cfv 5594  (class class class)co 6296   Basecbs 14644   distcds 14721  TarskiGcstrkg 23951  Itvcitv 23958  LineGclng 23959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-trkgc 23970  df-trkgb 23971  df-trkgcb 23972  df-trkg 23976
This theorem is referenced by:  coltr  24153  mideulem2  24234
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