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Theorem tglng 24059
Description: Lines of a Tarski Geometry. This relates to both Definition 4.10 of [Schwabhauser] p. 36. and Definition 6.14 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 28-Mar-2019.)
Hypotheses
Ref Expression
tglng.p  |-  P  =  ( Base `  G
)
tglng.l  |-  L  =  (LineG `  G )
tglng.i  |-  I  =  (Itv `  G )
Assertion
Ref Expression
tglng  |-  ( G  e. TarskiG  ->  L  =  ( x  e.  P , 
y  e.  ( P 
\  { x }
)  |->  { z  e.  P  |  ( z  e.  ( x I y )  \/  x  e.  ( z I y )  \/  y  e.  ( x I z ) ) } ) )
Distinct variable groups:    x, y,
z, G    x, I,
y, z    x, P, y, z
Allowed substitution hints:    L( x, y, z)

Proof of Theorem tglng
Dummy variables  f 
i  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-trkg 23976 . . . 4  |- TarskiG  =  ( (TarskiGC  i^i TarskiGB )  i^i  (TarskiGCB  i^i  {
f  |  [. ( Base `  f )  /  p ]. [. (Itv `  f )  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  ( p  \  {
x } )  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } ) } ) )
2 inss2 3715 . . . . 5  |-  ( (TarskiGC  i^i TarskiGB )  i^i  (TarskiGCB  i^i  { f  | 
[. ( Base `  f
)  /  p ]. [. (Itv `  f )  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  ( p 
\  { x }
)  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } ) } ) )  C_  (TarskiGCB  i^i  { f  |  [. ( Base `  f )  /  p ]. [. (Itv `  f )  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  ( p  \  {
x } )  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } ) } )
3 inss2 3715 . . . . 5  |-  (TarskiGCB  i^i  {
f  |  [. ( Base `  f )  /  p ]. [. (Itv `  f )  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  ( p  \  {
x } )  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } ) } )  C_  { f  |  [. ( Base `  f
)  /  p ]. [. (Itv `  f )  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  ( p 
\  { x }
)  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } ) }
42, 3sstri 3508 . . . 4  |-  ( (TarskiGC  i^i TarskiGB )  i^i  (TarskiGCB  i^i  { f  | 
[. ( Base `  f
)  /  p ]. [. (Itv `  f )  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  ( p 
\  { x }
)  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } ) } ) )  C_  { f  |  [. ( Base `  f )  /  p ]. [. (Itv `  f )  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  ( p  \  {
x } )  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } ) }
51, 4eqsstri 3529 . . 3  |- TarskiG  C_  { f  |  [. ( Base `  f )  /  p ]. [. (Itv `  f
)  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  (
p  \  { x } )  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } ) }
65sseli 3495 . 2  |-  ( G  e. TarskiG  ->  G  e.  {
f  |  [. ( Base `  f )  /  p ]. [. (Itv `  f )  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  ( p  \  {
x } )  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } ) } )
7 tglng.l . . 3  |-  L  =  (LineG `  G )
8 tglng.p . . . . 5  |-  P  =  ( Base `  G
)
9 eqid 2457 . . . . 5  |-  ( dist `  G )  =  (
dist `  G )
10 tglng.i . . . . 5  |-  I  =  (Itv `  G )
118, 9, 10istrkgl 23981 . . . 4  |-  ( G  e.  { f  | 
[. ( Base `  f
)  /  p ]. [. (Itv `  f )  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  ( p 
\  { x }
)  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } ) }  <->  ( G  e. 
_V  /\  (LineG `  G
)  =  ( x  e.  P ,  y  e.  ( P  \  { x } ) 
|->  { z  e.  P  |  ( z  e.  ( x I y )  \/  x  e.  ( z I y )  \/  y  e.  ( x I z ) ) } ) ) )
1211simprbi 464 . . 3  |-  ( G  e.  { f  | 
[. ( Base `  f
)  /  p ]. [. (Itv `  f )  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  ( p 
\  { x }
)  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } ) }  ->  (LineG `  G
)  =  ( x  e.  P ,  y  e.  ( P  \  { x } ) 
|->  { z  e.  P  |  ( z  e.  ( x I y )  \/  x  e.  ( z I y )  \/  y  e.  ( x I z ) ) } ) )
137, 12syl5eq 2510 . 2  |-  ( G  e.  { f  | 
[. ( Base `  f
)  /  p ]. [. (Itv `  f )  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  ( p 
\  { x }
)  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } ) }  ->  L  =  ( x  e.  P ,  y  e.  ( P  \  { x }
)  |->  { z  e.  P  |  ( z  e.  ( x I y )  \/  x  e.  ( z I y )  \/  y  e.  ( x I z ) ) } ) )
146, 13syl 16 1  |-  ( G  e. TarskiG  ->  L  =  ( x  e.  P , 
y  e.  ( P 
\  { x }
)  |->  { z  e.  P  |  ( 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    \/ w3o 972    = wceq 1395    e. wcel 1819   {cab 2442   {crab 2811   _Vcvv 3109   [.wsbc 3327    \ cdif 3468    i^i cin 3470   {csn 4032   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   Basecbs 14644   distcds 14721  TarskiGcstrkg 23951  TarskiGCcstrkgc 23952  TarskiGBcstrkgb 23953  TarskiGCBcstrkgcb 23954  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-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586
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-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-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-trkg 23976
This theorem is referenced by:  tglnfn  24060  tglnunirn  24061  tglngval  24064  tgisline  24133
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