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Theorem tgellng 23696
Description: Property of lying on the line going through points  X and  Y. Definition 4.10 of [Schwabhauser] p. 36. We choose the notation  Z  e.  ( X (LineG `  G
) Y ) instead of "colinear" because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 28-Mar-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 )
tglngval.z  |-  ( ph  ->  X  =/=  Y )
tgellng.z  |-  ( ph  ->  Z  e.  P )
Assertion
Ref Expression
tgellng  |-  ( ph  ->  ( Z  e.  ( X L Y )  <-> 
( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) )

Proof of Theorem tgellng
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 tglngval.p . . . . 5  |-  P  =  ( Base `  G
)
2 tglngval.l . . . . 5  |-  L  =  (LineG `  G )
3 tglngval.i . . . . 5  |-  I  =  (Itv `  G )
4 tglngval.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
5 tglngval.x . . . . 5  |-  ( ph  ->  X  e.  P )
6 tglngval.y . . . . 5  |-  ( ph  ->  Y  e.  P )
7 tglngval.z . . . . 5  |-  ( ph  ->  X  =/=  Y )
81, 2, 3, 4, 5, 6, 7tglngval 23694 . . . 4  |-  ( ph  ->  ( X L Y )  =  { z  e.  P  |  ( z  e.  ( X I Y )  \/  X  e.  ( z I Y )  \/  Y  e.  ( X I z ) ) } )
98eleq2d 2537 . . 3  |-  ( ph  ->  ( Z  e.  ( X L Y )  <-> 
Z  e.  { z  e.  P  |  ( z  e.  ( X I Y )  \/  X  e.  ( z I Y )  \/  Y  e.  ( X I z ) ) } ) )
10 eleq1 2539 . . . . 5  |-  ( z  =  Z  ->  (
z  e.  ( X I Y )  <->  Z  e.  ( X I Y ) ) )
11 oveq1 6291 . . . . . 6  |-  ( z  =  Z  ->  (
z I Y )  =  ( Z I Y ) )
1211eleq2d 2537 . . . . 5  |-  ( z  =  Z  ->  ( X  e.  ( z
I Y )  <->  X  e.  ( Z I Y ) ) )
13 oveq2 6292 . . . . . 6  |-  ( z  =  Z  ->  ( X I z )  =  ( X I Z ) )
1413eleq2d 2537 . . . . 5  |-  ( z  =  Z  ->  ( Y  e.  ( X I z )  <->  Y  e.  ( X I Z ) ) )
1510, 12, 143orbi123d 1298 . . . 4  |-  ( z  =  Z  ->  (
( z  e.  ( X I Y )  \/  X  e.  ( z I Y )  \/  Y  e.  ( X I z ) )  <->  ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) )
1615elrab 3261 . . 3  |-  ( Z  e.  { z  e.  P  |  ( z  e.  ( X I Y )  \/  X  e.  ( z I Y )  \/  Y  e.  ( X I z ) ) }  <->  ( Z  e.  P  /\  ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) )
179, 16syl6bb 261 . 2  |-  ( ph  ->  ( Z  e.  ( X L Y )  <-> 
( Z  e.  P  /\  ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) ) )
18 tgellng.z . . 3  |-  ( ph  ->  Z  e.  P )
1918biantrurd 508 . 2  |-  ( ph  ->  ( ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) )  <->  ( Z  e.  P  /\  ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) ) )
2017, 19bitr4d 256 1  |-  ( ph  ->  ( Z  e.  ( X L Y )  <-> 
( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 972    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2818   ` cfv 5588  (class class class)co 6284   Basecbs 14490  TarskiGcstrkg 23581  Itvcitv 23588  LineGclng 23589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-trkg 23606
This theorem is referenced by:  tgcolg  23697  btwnlng1  23741  btwnlng2  23742  btwnlng3  23743  lncom  23744  lnrot1  23745  lnrot2  23746  tglineeltr  23753  colmid  23801
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