MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgellng Structured version   Unicode version

Theorem tgellng 24458
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 24456 . . . 4  |-  ( ph  ->  ( X L Y )  =  { z  e.  P  |  ( z  e.  ( X I Y )  \/  X  e.  ( z I Y )  \/  Y  e.  ( X I z ) ) } )
98eleq2d 2499 . . 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 2501 . . . . 5  |-  ( z  =  Z  ->  (
z  e.  ( X I Y )  <->  Z  e.  ( X I Y ) ) )
11 oveq1 6312 . . . . . 6  |-  ( z  =  Z  ->  (
z I Y )  =  ( Z I Y ) )
1211eleq2d 2499 . . . . 5  |-  ( z  =  Z  ->  ( X  e.  ( z
I Y )  <->  X  e.  ( Z I Y ) ) )
13 oveq2 6313 . . . . . 6  |-  ( z  =  Z  ->  ( X I z )  =  ( X I Z ) )
1413eleq2d 2499 . . . . 5  |-  ( z  =  Z  ->  ( Y  e.  ( X I z )  <->  Y  e.  ( X I Z ) ) )
1510, 12, 143orbi123d 1334 . . . 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 3235 . . 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 264 . 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 510 . 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 259 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 187    /\ wa 370    \/ w3o 981    = wceq 1437    e. wcel 1870    =/= wne 2625   {crab 2786   ` cfv 5601  (class class class)co 6305   Basecbs 15084  TarskiGcstrkg 24341  Itvcitv 24347  LineGclng 24348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-trkg 24364
This theorem is referenced by:  tgcolg  24459  hlln  24512  btwnlng1  24523  btwnlng2  24524  btwnlng3  24525  lncom  24526  lnrot1  24527  lnrot2  24528  tglineeltr  24535  colmid  24590  cgracol  24724
  Copyright terms: Public domain W3C validator