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Theorem istrkgl 24053
Description: Building lines from the segment property (Contributed by Thierry Arnoux, 14-Mar-2019.)
Hypotheses
Ref Expression
istrkg.p  |-  P  =  ( Base `  G
)
istrkg.d  |-  .-  =  ( dist `  G )
istrkg.i  |-  I  =  (Itv `  G )
Assertion
Ref Expression
istrkgl  |-  ( 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 ) ) } ) ) )
Distinct variable groups:    f, i, p, G    x, f, y, z, I, i, p    P, f, i, p, x, y, z    .- , f,
i, p, x, y, z
Allowed substitution hints:    G( x, y, z)

Proof of Theorem istrkgl
StepHypRef Expression
1 istrkg.p . . . 4  |-  P  =  ( Base `  G
)
2 istrkg.i . . . 4  |-  I  =  (Itv `  G )
3 simpl 455 . . . . . . 7  |-  ( ( p  =  P  /\  i  =  I )  ->  p  =  P )
43eqcomd 2462 . . . . . 6  |-  ( ( p  =  P  /\  i  =  I )  ->  P  =  p )
54adantr 463 . . . . . . 7  |-  ( ( ( p  =  P  /\  i  =  I )  /\  x  e.  P )  ->  P  =  p )
65difeq1d 3607 . . . . . 6  |-  ( ( ( p  =  P  /\  i  =  I )  /\  x  e.  P )  ->  ( P  \  { x }
)  =  ( p 
\  { x }
) )
7 simpr 459 . . . . . . . . . . . 12  |-  ( ( p  =  P  /\  i  =  I )  ->  i  =  I )
87eqcomd 2462 . . . . . . . . . . 11  |-  ( ( p  =  P  /\  i  =  I )  ->  I  =  i )
98oveqd 6287 . . . . . . . . . 10  |-  ( ( p  =  P  /\  i  =  I )  ->  ( x I y )  =  ( x i y ) )
109eleq2d 2524 . . . . . . . . 9  |-  ( ( p  =  P  /\  i  =  I )  ->  ( z  e.  ( x I y )  <-> 
z  e.  ( x i y ) ) )
118oveqd 6287 . . . . . . . . . 10  |-  ( ( p  =  P  /\  i  =  I )  ->  ( z I y )  =  ( z i y ) )
1211eleq2d 2524 . . . . . . . . 9  |-  ( ( p  =  P  /\  i  =  I )  ->  ( x  e.  ( z I y )  <-> 
x  e.  ( z i y ) ) )
138oveqd 6287 . . . . . . . . . 10  |-  ( ( p  =  P  /\  i  =  I )  ->  ( x I z )  =  ( x i z ) )
1413eleq2d 2524 . . . . . . . . 9  |-  ( ( p  =  P  /\  i  =  I )  ->  ( y  e.  ( x I z )  <-> 
y  e.  ( x i z ) ) )
1510, 12, 143orbi123d 1296 . . . . . . . 8  |-  ( ( p  =  P  /\  i  =  I )  ->  ( ( 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 ) ) ) )
164, 15rabeqbidv 3101 . . . . . . 7  |-  ( ( p  =  P  /\  i  =  I )  ->  { 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 ) ) } )
1716adantr 463 . . . . . 6  |-  ( ( ( p  =  P  /\  i  =  I )  /\  ( 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 ) ) }  =  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } )
184, 6, 17mpt2eq123dva 6331 . . . . 5  |-  ( ( p  =  P  /\  i  =  I )  ->  ( 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 ) ) } )  =  ( 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 ) ) } ) )
1918eqeq2d 2468 . . . 4  |-  ( ( p  =  P  /\  i  =  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 `  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 ) ) } ) ) )
201, 2, 19sbcie2s 14761 . . 3  |-  ( f  =  G  ->  ( [. ( 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 `  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 ) ) } ) ) )
21 fveq2 5848 . . . 4  |-  ( f  =  G  ->  (LineG `  f )  =  (LineG `  G ) )
2221eqeq1d 2456 . . 3  |-  ( f  =  G  ->  (
(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 ) ) } ) ) )
2320, 22bitrd 253 . 2  |-  ( f  =  G  ->  ( [. ( 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 ) ) } ) ) )
24 eqid 2454 . 2  |-  { 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 ) ) } ) }  =  { 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 ) ) } ) }
2523, 24elab4g 3247 1  |-  ( 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 ) ) } ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    \/ w3o 970    = wceq 1398    e. wcel 1823   {cab 2439   {crab 2808   _Vcvv 3106   [.wsbc 3324    \ cdif 3458   {csn 4016   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   Basecbs 14716   distcds 14793  Itvcitv 24030  LineGclng 24031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275
This theorem is referenced by:  tglng  24134  f1otrg  24376  eengtrkg  24490
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