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Theorem istrkgl 22943
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:    x, p, y, z    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 457 . . . . . . 7  |-  ( ( p  =  P  /\  i  =  I )  ->  p  =  P )
43eqcomd 2448 . . . . . 6  |-  ( ( p  =  P  /\  i  =  I )  ->  P  =  p )
54adantr 465 . . . . . . 7  |-  ( ( ( p  =  P  /\  i  =  I )  /\  x  e.  P )  ->  P  =  p )
65difeq1d 3494 . . . . . 6  |-  ( ( ( p  =  P  /\  i  =  I )  /\  x  e.  P )  ->  ( P  \  { x }
)  =  ( p 
\  { x }
) )
7 simpr 461 . . . . . . . . . . . 12  |-  ( ( p  =  P  /\  i  =  I )  ->  i  =  I )
87eqcomd 2448 . . . . . . . . . . 11  |-  ( ( p  =  P  /\  i  =  I )  ->  I  =  i )
98oveqd 6129 . . . . . . . . . 10  |-  ( ( p  =  P  /\  i  =  I )  ->  ( x I y )  =  ( x i y ) )
109eleq2d 2510 . . . . . . . . 9  |-  ( ( p  =  P  /\  i  =  I )  ->  ( z  e.  ( x I y )  <-> 
z  e.  ( x i y ) ) )
118oveqd 6129 . . . . . . . . . 10  |-  ( ( p  =  P  /\  i  =  I )  ->  ( z I y )  =  ( z i y ) )
1211eleq2d 2510 . . . . . . . . 9  |-  ( ( p  =  P  /\  i  =  I )  ->  ( x  e.  ( z I y )  <-> 
x  e.  ( z i y ) ) )
138oveqd 6129 . . . . . . . . . 10  |-  ( ( p  =  P  /\  i  =  I )  ->  ( x I z )  =  ( x i z ) )
1413eleq2d 2510 . . . . . . . . 9  |-  ( ( p  =  P  /\  i  =  I )  ->  ( y  e.  ( x I z )  <-> 
y  e.  ( x i z ) ) )
1510, 12, 143orbi123d 1288 . . . . . . . 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 2988 . . . . . . 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 465 . . . . . 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 6168 . . . . 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 2454 . . . 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 14238 . . 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 5712 . . . 4  |-  ( f  =  G  ->  (LineG `  f )  =  (LineG `  G ) )
2221eqeq1d 2451 . . 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 2443 . 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 3131 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 369    \/ w3o 964    = wceq 1369    e. wcel 1756   {cab 2429   {crab 2740   _Vcvv 2993   [.wsbc 3207    \ cdif 3346   {csn 3898   ` cfv 5439  (class class class)co 6112    e. cmpt2 6114   Basecbs 14195   distcds 14268  Itvcitv 22919  LineGclng 22920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4442
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-iota 5402  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117
This theorem is referenced by:  tglng  23002  f1otrg  23139  eengtrkg  23253
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