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Theorem lnrot1 23178
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 2454 . . . . . 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 23087 . . . . 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 23087 . . . . 5  |-  ( ph  ->  ( Y  e.  ( Z I X )  <-> 
Y  e.  ( X I Z ) ) )
129, 10, 113orbi123d 1289 . . . 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 971 . . . . 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 23133 . . . 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 23133 . . 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 964    = wceq 1370    e. wcel 1758    =/= wne 2648   ` cfv 5529  (class class class)co 6203   Basecbs 14296   distcds 14370  TarskiGcstrkg 23032  Itvcitv 23039  LineGclng 23040
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-trkgc 23051  df-trkgb 23052  df-trkgcb 23053  df-trkg 23057
This theorem is referenced by:  tglineelsb2  23187  tglineneq  23198  coltr3  23203  lmieu  23283
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