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Theorem colrot1 23122
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
tglngval.p  |-  P  =  ( Base `  G
)
tglngval.l  |-  L  =  (LineG `  G )
tglngval.i  |-  I  =  (Itv `  G )
tglngval.g  |-  ( ph  ->  G  e. TarskiG )
tglngval.x  |-  ( ph  ->  X  e.  P )
tglngval.y  |-  ( ph  ->  Y  e.  P )
tgcolg.z  |-  ( ph  ->  Z  e.  P )
colrot  |-  ( ph  ->  ( Z  e.  ( X L Y )  \/  X  =  Y ) )
Assertion
Ref Expression
colrot1  |-  ( ph  ->  ( X  e.  ( Y L Z )  \/  Y  =  Z ) )

Proof of Theorem colrot1
StepHypRef Expression
1 colrot . 2  |-  ( ph  ->  ( Z  e.  ( X L Y )  \/  X  =  Y ) )
2 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 ) ) )
3 tglngval.p . . . . . 6  |-  P  =  ( Base `  G
)
4 eqid 2451 . . . . . 6  |-  ( dist `  G )  =  (
dist `  G )
5 tglngval.i . . . . . 6  |-  I  =  (Itv `  G )
6 tglngval.g . . . . . 6  |-  ( ph  ->  G  e. TarskiG )
7 tgcolg.z . . . . . 6  |-  ( ph  ->  Z  e.  P )
8 tglngval.x . . . . . 6  |-  ( ph  ->  X  e.  P )
9 tglngval.y . . . . . 6  |-  ( ph  ->  Y  e.  P )
103, 4, 5, 6, 7, 8, 9tgbtwncomb 23070 . . . . 5  |-  ( ph  ->  ( X  e.  ( Z I Y )  <-> 
X  e.  ( Y I Z ) ) )
11 biidd 237 . . . . 5  |-  ( ph  ->  ( Y  e.  ( X I Z )  <-> 
Y  e.  ( X I Z ) ) )
123, 4, 5, 6, 8, 7, 9tgbtwncomb 23070 . . . . 5  |-  ( ph  ->  ( Z  e.  ( X I Y )  <-> 
Z  e.  ( Y I X ) ) )
1310, 11, 123orbi123d 1289 . . . 4  |-  ( ph  ->  ( ( X  e.  ( Z I Y )  \/  Y  e.  ( X I Z )  \/  Z  e.  ( X I Y ) )  <->  ( X  e.  ( Y I Z )  \/  Y  e.  ( X I Z )  \/  Z  e.  ( Y I X ) ) ) )
142, 13syl5bb 257 . . 3  |-  ( ph  ->  ( ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) )  <->  ( X  e.  ( Y I Z )  \/  Y  e.  ( X I Z )  \/  Z  e.  ( Y I X ) ) ) )
15 tglngval.l . . . 4  |-  L  =  (LineG `  G )
163, 15, 5, 6, 8, 9, 7tgcolg 23117 . . 3  |-  ( ph  ->  ( ( Z  e.  ( X L Y )  \/  X  =  Y )  <->  ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) )
173, 15, 5, 6, 9, 7, 8tgcolg 23117 . . 3  |-  ( ph  ->  ( ( X  e.  ( Y L Z )  \/  Y  =  Z )  <->  ( X  e.  ( Y I Z )  \/  Y  e.  ( X I Z )  \/  Z  e.  ( Y I X ) ) ) )
1814, 16, 173bitr4d 285 . 2  |-  ( ph  ->  ( ( Z  e.  ( X L Y )  \/  X  =  Y )  <->  ( X  e.  ( Y L Z )  \/  Y  =  Z ) ) )
191, 18mpbid 210 1  |-  ( ph  ->  ( X  e.  ( Y L Z )  \/  Y  =  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    \/ w3o 964    = wceq 1370    e. wcel 1758   ` cfv 5519  (class class class)co 6193   Basecbs 14285   distcds 14358  TarskiGcstrkg 23015  Itvcitv 23022  LineGclng 23023
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 4514  ax-nul 4522  ax-pr 4632
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 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-iota 5482  df-fun 5521  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-trkgc 23034  df-trkgb 23035  df-trkgcb 23036  df-trkg 23040
This theorem is referenced by:  colrot2  23123  ncolrot2  23126  ncolncol  23183  coltr2  23185  midexlem  23222  ragflat3  23236  mideulem  23254
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