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Theorem bosser 26167
Description: "Being on the same side of  M " is an equivalence relation among points that are not on  M. (Contributed by FL, 10-Aug-2016.)
Hypotheses
Ref Expression
isside.1  |-  P  =  (PPoints `  G )
isside.2  |-  L  =  (PLines `  G )
isside.3  |-  .~  =  ( (ss `  G ) `
 M )
isside.4  |-  ( ph  ->  G  e. Ibg )
isside.5  |-  ( ph  ->  M  e.  L )
Assertion
Ref Expression
bosser  |-  ( ph  ->  .~  Er  ( P 
\  M ) )

Proof of Theorem bosser
Dummy variables  a 
b  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 4812 . . . 4  |-  Rel  { <. x ,  y >.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) }
21a1i 10 . . 3  |-  ( ph  ->  Rel  { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) } )
3 isside.1 . . . . 5  |-  P  =  (PPoints `  G )
4 isside.2 . . . . 5  |-  L  =  (PLines `  G )
5 isside.3 . . . . 5  |-  .~  =  ( (ss `  G ) `
 M )
6 isside.4 . . . . 5  |-  ( ph  ->  G  e. Ibg )
7 isside.5 . . . . 5  |-  ( ph  ->  M  e.  L )
8 eqid 2283 . . . . 5  |-  ( seg `  G )  =  ( seg `  G )
93, 4, 5, 6, 7, 8isside0 26164 . . . 4  |-  ( ph  ->  .~  =  { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) } )
109releqd 4773 . . 3  |-  ( ph  ->  ( Rel  .~  <->  Rel  { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) } ) )
112, 10mpbird 223 . 2  |-  ( ph  ->  Rel  .~  )
129dmeqd 4881 . . 3  |-  ( ph  ->  dom  .~  =  dom  {
<. x ,  y >.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) } )
13 3anass 938 . . . . . 6  |-  ( ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) )  <->  ( x  e.  ( P  \  M
)  /\  ( y  e.  ( P  \  M
)  /\  ( (
x ( seg `  G
) y )  i^i 
M )  =  (/) ) ) )
1413opabbii 4083 . . . . 5  |-  { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) }  =  { <. x ,  y >.  |  ( x  e.  ( P  \  M
)  /\  ( y  e.  ( P  \  M
)  /\  ( (
x ( seg `  G
) y )  i^i 
M )  =  (/) ) ) }
1514a1i 10 . . . 4  |-  ( ph  ->  { <. x ,  y
>.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) }  =  { <. x ,  y >.  |  ( x  e.  ( P  \  M
)  /\  ( y  e.  ( P  \  M
)  /\  ( (
x ( seg `  G
) y )  i^i 
M )  =  (/) ) ) } )
1615dmeqd 4881 . . 3  |-  ( ph  ->  dom  { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) }  =  dom  {
<. x ,  y >.  |  ( x  e.  ( P  \  M
)  /\  ( y  e.  ( P  \  M
)  /\  ( (
x ( seg `  G
) y )  i^i 
M )  =  (/) ) ) } )
17 simpr 447 . . . . . 6  |-  ( (
ph  /\  x  e.  ( P  \  M ) )  ->  x  e.  ( P  \  M ) )
186adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( P  \  M ) )  ->  G  e. Ibg )
19 eldifi 3298 . . . . . . . . . 10  |-  ( x  e.  ( P  \  M )  ->  x  e.  P )
2019adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( P  \  M ) )  ->  x  e.  P )
213, 8, 18, 20sgplpte22 26138 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( P  \  M ) )  ->  ( x
( seg `  G
) x )  =  { x } )
2221ineq1d 3369 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( P  \  M ) )  ->  ( (
x ( seg `  G
) x )  i^i 
M )  =  ( { x }  i^i  M ) )
23 incom 3361 . . . . . . . 8  |-  ( { x }  i^i  M
)  =  ( M  i^i  { x }
)
2423a1i 10 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( P  \  M ) )  ->  ( {
x }  i^i  M
)  =  ( M  i^i  { x }
) )
25 eldifn 3299 . . . . . . . . 9  |-  ( x  e.  ( P  \  M )  ->  -.  x  e.  M )
2625adantl 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( P  \  M ) )  ->  -.  x  e.  M )
27 disjsn 3693 . . . . . . . 8  |-  ( ( M  i^i  { x } )  =  (/)  <->  -.  x  e.  M )
2826, 27sylibr 203 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( P  \  M ) )  ->  ( M  i^i  { x } )  =  (/) )
2922, 24, 283eqtrd 2319 . . . . . 6  |-  ( (
ph  /\  x  e.  ( P  \  M ) )  ->  ( (
x ( seg `  G
) x )  i^i 
M )  =  (/) )
30 oveq2 5866 . . . . . . . . . 10  |-  ( y  =  x  ->  (
x ( seg `  G
) y )  =  ( x ( seg `  G ) x ) )
3130ineq1d 3369 . . . . . . . . 9  |-  ( y  =  x  ->  (
( x ( seg `  G ) y )  i^i  M )  =  ( ( x ( seg `  G ) x )  i^i  M
) )
3231eqeq1d 2291 . . . . . . . 8  |-  ( y  =  x  ->  (
( ( x ( seg `  G ) y )  i^i  M
)  =  (/)  <->  ( (
x ( seg `  G
) x )  i^i 
M )  =  (/) ) )
3332rspcev 2884 . . . . . . 7  |-  ( ( x  e.  ( P 
\  M )  /\  ( ( x ( seg `  G ) x )  i^i  M
)  =  (/) )  ->  E. y  e.  ( P  \  M ) ( ( x ( seg `  G ) y )  i^i  M )  =  (/) )
34 df-rex 2549 . . . . . . 7  |-  ( E. y  e.  ( P 
\  M ) ( ( x ( seg `  G ) y )  i^i  M )  =  (/) 
<->  E. y ( y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) )
3533, 34sylib 188 . . . . . 6  |-  ( ( x  e.  ( P 
\  M )  /\  ( ( x ( seg `  G ) x )  i^i  M
)  =  (/) )  ->  E. y ( y  e.  ( P  \  M
)  /\  ( (
x ( seg `  G
) y )  i^i 
M )  =  (/) ) )
3617, 29, 35syl2anc 642 . . . . 5  |-  ( (
ph  /\  x  e.  ( P  \  M ) )  ->  E. y
( y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) )
3736ralrimiva 2626 . . . 4  |-  ( ph  ->  A. x  e.  ( P  \  M ) E. y ( y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) )
38 dmopab3 4891 . . . 4  |-  ( A. x  e.  ( P  \  M ) E. y
( y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) 
<->  dom  { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  ( y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) ) }  =  ( P  \  M ) )
3937, 38sylib 188 . . 3  |-  ( ph  ->  dom  { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  ( y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) ) }  =  ( P  \  M ) )
4012, 16, 393eqtrd 2319 . 2  |-  ( ph  ->  dom  .~  =  ( P  \  M ) )
41 simpr2 962 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) )  ->  y  e.  ( P  \  M
) )
42 simpr1 961 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) )  ->  x  e.  ( P  \  M
) )
4363ad2ant3 978 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  ph )  ->  G  e. Ibg )
44193ad2ant1 976 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  ph )  ->  x  e.  P
)
45 eldifi 3298 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  ( P  \  M )  ->  y  e.  P )
46453ad2ant2 977 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  ph )  ->  y  e.  P
)
473, 8, 43, 44, 46xsyysx 26145 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  ph )  ->  ( x ( seg `  G ) y )  =  ( y ( seg `  G
) x ) )
4847ineq1d 3369 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  ph )  ->  ( ( x ( seg `  G
) y )  i^i 
M )  =  ( ( y ( seg `  G ) x )  i^i  M ) )
4948eqeq1d 2291 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  ph )  ->  ( ( ( x ( seg `  G
) y )  i^i 
M )  =  (/)  <->  (
( y ( seg `  G ) x )  i^i  M )  =  (/) ) )
5049biimpd 198 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  ph )  ->  ( ( ( x ( seg `  G
) y )  i^i 
M )  =  (/)  ->  ( ( y ( seg `  G ) x )  i^i  M
)  =  (/) ) )
51503exp 1150 . . . . . . . . . . . 12  |-  ( x  e.  ( P  \  M )  ->  (
y  e.  ( P 
\  M )  -> 
( ph  ->  ( ( ( x ( seg `  G ) y )  i^i  M )  =  (/)  ->  ( ( y ( seg `  G
) x )  i^i 
M )  =  (/) ) ) ) )
5251com34 77 . . . . . . . . . . 11  |-  ( x  e.  ( P  \  M )  ->  (
y  e.  ( P 
\  M )  -> 
( ( ( x ( seg `  G
) y )  i^i 
M )  =  (/)  ->  ( ph  ->  (
( y ( seg `  G ) x )  i^i  M )  =  (/) ) ) ) )
53523imp 1145 . . . . . . . . . 10  |-  ( ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) )  ->  ( ph  ->  ( ( y ( seg `  G ) x )  i^i  M
)  =  (/) ) )
5453impcom 419 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) )  ->  (
( y ( seg `  G ) x )  i^i  M )  =  (/) )
5541, 42, 543jca 1132 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) )  ->  (
y  e.  ( P 
\  M )  /\  x  e.  ( P  \  M )  /\  (
( y ( seg `  G ) x )  i^i  M )  =  (/) ) )
5655ex 423 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) )  ->  ( y  e.  ( P  \  M
)  /\  x  e.  ( P  \  M )  /\  ( ( y ( seg `  G
) x )  i^i 
M )  =  (/) ) ) )
57 df-br 4024 . . . . . . . . 9  |-  ( x { <. x ,  y
>.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) } y  <->  <. x ,  y >.  e.  { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) } )
58 opabid 4271 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) }  <->  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) )
5957, 58bitri 240 . . . . . . . 8  |-  ( x { <. x ,  y
>.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) } y  <->  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) )
6059a1i 10 . . . . . . 7  |-  ( ph  ->  ( x { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) } y  <->  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) ) )
61 vex 2791 . . . . . . . . 9  |-  y  e. 
_V
62 vex 2791 . . . . . . . . 9  |-  x  e. 
_V
63 eleq1 2343 . . . . . . . . . 10  |-  ( a  =  y  ->  (
a  e.  ( P 
\  M )  <->  y  e.  ( P  \  M ) ) )
64 oveq1 5865 . . . . . . . . . . . 12  |-  ( a  =  y  ->  (
a ( seg `  G
) b )  =  ( y ( seg `  G ) b ) )
6564ineq1d 3369 . . . . . . . . . . 11  |-  ( a  =  y  ->  (
( a ( seg `  G ) b )  i^i  M )  =  ( ( y ( seg `  G ) b )  i^i  M
) )
6665eqeq1d 2291 . . . . . . . . . 10  |-  ( a  =  y  ->  (
( ( a ( seg `  G ) b )  i^i  M
)  =  (/)  <->  ( (
y ( seg `  G
) b )  i^i 
M )  =  (/) ) )
6763, 663anbi13d 1254 . . . . . . . . 9  |-  ( a  =  y  ->  (
( a  e.  ( P  \  M )  /\  b  e.  ( P  \  M )  /\  ( ( a ( seg `  G
) b )  i^i 
M )  =  (/) ) 
<->  ( y  e.  ( P  \  M )  /\  b  e.  ( P  \  M )  /\  ( ( y ( seg `  G
) b )  i^i 
M )  =  (/) ) ) )
68 eleq1 2343 . . . . . . . . . 10  |-  ( b  =  x  ->  (
b  e.  ( P 
\  M )  <->  x  e.  ( P  \  M ) ) )
69 oveq2 5866 . . . . . . . . . . . 12  |-  ( b  =  x  ->  (
y ( seg `  G
) b )  =  ( y ( seg `  G ) x ) )
7069ineq1d 3369 . . . . . . . . . . 11  |-  ( b  =  x  ->  (
( y ( seg `  G ) b )  i^i  M )  =  ( ( y ( seg `  G ) x )  i^i  M
) )
7170eqeq1d 2291 . . . . . . . . . 10  |-  ( b  =  x  ->  (
( ( y ( seg `  G ) b )  i^i  M
)  =  (/)  <->  ( (
y ( seg `  G
) x )  i^i 
M )  =  (/) ) )
7268, 713anbi23d 1255 . . . . . . . . 9  |-  ( b  =  x  ->  (
( y  e.  ( P  \  M )  /\  b  e.  ( P  \  M )  /\  ( ( y ( seg `  G
) b )  i^i 
M )  =  (/) ) 
<->  ( y  e.  ( P  \  M )  /\  x  e.  ( P  \  M )  /\  ( ( y ( seg `  G
) x )  i^i 
M )  =  (/) ) ) )
73 eleq1 2343 . . . . . . . . . . . 12  |-  ( x  =  a  ->  (
x  e.  ( P 
\  M )  <->  a  e.  ( P  \  M ) ) )
7473adantr 451 . . . . . . . . . . 11  |-  ( ( x  =  a  /\  y  =  b )  ->  ( x  e.  ( P  \  M )  <-> 
a  e.  ( P 
\  M ) ) )
75 eleq1 2343 . . . . . . . . . . . 12  |-  ( y  =  b  ->  (
y  e.  ( P 
\  M )  <->  b  e.  ( P  \  M ) ) )
7675adantl 452 . . . . . . . . . . 11  |-  ( ( x  =  a  /\  y  =  b )  ->  ( y  e.  ( P  \  M )  <-> 
b  e.  ( P 
\  M ) ) )
77 oveq12 5867 . . . . . . . . . . . . 13  |-  ( ( x  =  a  /\  y  =  b )  ->  ( x ( seg `  G ) y )  =  ( a ( seg `  G ) b ) )
7877ineq1d 3369 . . . . . . . . . . . 12  |-  ( ( x  =  a  /\  y  =  b )  ->  ( ( x ( seg `  G ) y )  i^i  M
)  =  ( ( a ( seg `  G
) b )  i^i 
M ) )
7978eqeq1d 2291 . . . . . . . . . . 11  |-  ( ( x  =  a  /\  y  =  b )  ->  ( ( ( x ( seg `  G
) y )  i^i 
M )  =  (/)  <->  (
( a ( seg `  G ) b )  i^i  M )  =  (/) ) )
8074, 76, 793anbi123d 1252 . . . . . . . . . 10  |-  ( ( x  =  a  /\  y  =  b )  ->  ( ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) 
<->  ( a  e.  ( P  \  M )  /\  b  e.  ( P  \  M )  /\  ( ( a ( seg `  G
) b )  i^i 
M )  =  (/) ) ) )
8180cbvopabv 4088 . . . . . . . . 9  |-  { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) }  =  { <. a ,  b >.  |  ( a  e.  ( P  \  M
)  /\  b  e.  ( P  \  M )  /\  ( ( a ( seg `  G
) b )  i^i 
M )  =  (/) ) }
8261, 62, 67, 72, 81brab 4287 . . . . . . . 8  |-  ( y { <. x ,  y
>.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) } x  <->  ( y  e.  ( P  \  M
)  /\  x  e.  ( P  \  M )  /\  ( ( y ( seg `  G
) x )  i^i 
M )  =  (/) ) )
8382a1i 10 . . . . . . 7  |-  ( ph  ->  ( y { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) } x  <->  ( y  e.  ( P  \  M
)  /\  x  e.  ( P  \  M )  /\  ( ( y ( seg `  G
) x )  i^i 
M )  =  (/) ) ) )
8456, 60, 833imtr4d 259 . . . . . 6  |-  ( ph  ->  ( x { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) } y  -> 
y { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) } x ) )
859breqd 4034 . . . . . 6  |-  ( ph  ->  ( x  .~  y  <->  x { <. x ,  y
>.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) } y ) )
869breqd 4034 . . . . . 6  |-  ( ph  ->  ( y  .~  x  <->  y { <. x ,  y
>.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) } x ) )
8784, 85, 863imtr4d 259 . . . . 5  |-  ( ph  ->  ( x  .~  y  ->  y  .~  x ) )
88 simprl1 1000 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) )  /\  (
y  e.  ( P 
\  M )  /\  z  e.  ( P  \  M )  /\  (
( y ( seg `  G ) z )  i^i  M )  =  (/) ) ) )  ->  x  e.  ( P  \  M ) )
89 simprr2 1004 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) )  /\  (
y  e.  ( P 
\  M )  /\  z  e.  ( P  \  M )  /\  (
( y ( seg `  G ) z )  i^i  M )  =  (/) ) ) )  -> 
z  e.  ( P 
\  M ) )
906adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) )  /\  (
y  e.  ( P 
\  M )  /\  z  e.  ( P  \  M )  /\  (
( y ( seg `  G ) z )  i^i  M )  =  (/) ) ) )  ->  G  e. Ibg )
917adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) )  /\  (
y  e.  ( P 
\  M )  /\  z  e.  ( P  \  M )  /\  (
( y ( seg `  G ) z )  i^i  M )  =  (/) ) ) )  ->  M  e.  L )
92 simprl2 1001 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) )  /\  (
y  e.  ( P 
\  M )  /\  z  e.  ( P  \  M )  /\  (
( y ( seg `  G ) z )  i^i  M )  =  (/) ) ) )  -> 
y  e.  ( P 
\  M ) )
93 simprl3 1002 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) )  /\  (
y  e.  ( P 
\  M )  /\  z  e.  ( P  \  M )  /\  (
( y ( seg `  G ) z )  i^i  M )  =  (/) ) ) )  -> 
( ( x ( seg `  G ) y )  i^i  M
)  =  (/) )
94 simprr3 1005 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) )  /\  (
y  e.  ( P 
\  M )  /\  z  e.  ( P  \  M )  /\  (
( y ( seg `  G ) z )  i^i  M )  =  (/) ) ) )  -> 
( ( y ( seg `  G ) z )  i^i  M
)  =  (/) )
953, 4, 8, 90, 91, 88, 92, 89, 93, 94bsstrs 26146 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) )  /\  (
y  e.  ( P 
\  M )  /\  z  e.  ( P  \  M )  /\  (
( y ( seg `  G ) z )  i^i  M )  =  (/) ) ) )  -> 
( ( x ( seg `  G ) z )  i^i  M
)  =  (/) )
9688, 89, 953jca 1132 . . . . . . . 8  |-  ( (
ph  /\  ( (
x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) )  /\  (
y  e.  ( P 
\  M )  /\  z  e.  ( P  \  M )  /\  (
( y ( seg `  G ) z )  i^i  M )  =  (/) ) ) )  -> 
( x  e.  ( P  \  M )  /\  z  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) z )  i^i 
M )  =  (/) ) )
9796ex 423 . . . . . . 7  |-  ( ph  ->  ( ( ( x  e.  ( P  \  M )  /\  y  e.  ( P  \  M
)  /\  ( (
x ( seg `  G
) y )  i^i 
M )  =  (/) )  /\  ( y  e.  ( P  \  M
)  /\  z  e.  ( P  \  M )  /\  ( ( y ( seg `  G
) z )  i^i 
M )  =  (/) ) )  ->  (
x  e.  ( P 
\  M )  /\  z  e.  ( P  \  M )  /\  (
( x ( seg `  G ) z )  i^i  M )  =  (/) ) ) )
9881a1i 10 . . . . . . . . . 10  |-  ( ph  ->  { <. x ,  y
>.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) }  =  { <. a ,  b >.  |  ( a  e.  ( P  \  M
)  /\  b  e.  ( P  \  M )  /\  ( ( a ( seg `  G
) b )  i^i 
M )  =  (/) ) } )
9998breqd 4034 . . . . . . . . 9  |-  ( ph  ->  ( y { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) } z  <->  y { <. a ,  b >.  |  ( a  e.  ( P  \  M
)  /\  b  e.  ( P  \  M )  /\  ( ( a ( seg `  G
) b )  i^i 
M )  =  (/) ) } z ) )
100 vex 2791 . . . . . . . . . 10  |-  z  e. 
_V
101 eleq1 2343 . . . . . . . . . . 11  |-  ( b  =  z  ->  (
b  e.  ( P 
\  M )  <->  z  e.  ( P  \  M ) ) )
102 oveq2 5866 . . . . . . . . . . . . 13  |-  ( b  =  z  ->  (
y ( seg `  G
) b )  =  ( y ( seg `  G ) z ) )
103102ineq1d 3369 . . . . . . . . . . . 12  |-  ( b  =  z  ->  (
( y ( seg `  G ) b )  i^i  M )  =  ( ( y ( seg `  G ) z )  i^i  M
) )
104103eqeq1d 2291 . . . . . . . . . . 11  |-  ( b  =  z  ->  (
( ( y ( seg `  G ) b )  i^i  M
)  =  (/)  <->  ( (
y ( seg `  G
) z )  i^i 
M )  =  (/) ) )
105101, 1043anbi23d 1255 . . . . . . . . . 10  |-  ( b  =  z  ->  (
( y  e.  ( P  \  M )  /\  b  e.  ( P  \  M )  /\  ( ( y ( seg `  G
) b )  i^i 
M )  =  (/) ) 
<->  ( y  e.  ( P  \  M )  /\  z  e.  ( P  \  M )  /\  ( ( y ( seg `  G
) z )  i^i 
M )  =  (/) ) ) )
106 eqid 2283 . . . . . . . . . 10  |-  { <. a ,  b >.  |  ( a  e.  ( P 
\  M )  /\  b  e.  ( P  \  M )  /\  (
( a ( seg `  G ) b )  i^i  M )  =  (/) ) }  =  { <. a ,  b >.  |  ( a  e.  ( P  \  M
)  /\  b  e.  ( P  \  M )  /\  ( ( a ( seg `  G
) b )  i^i 
M )  =  (/) ) }
10761, 100, 67, 105, 106brab 4287 . . . . . . . . 9  |-  ( y { <. a ,  b
>.  |  ( a  e.  ( P  \  M
)  /\  b  e.  ( P  \  M )  /\  ( ( a ( seg `  G
) b )  i^i 
M )  =  (/) ) } z  <->  ( y  e.  ( P  \  M
)  /\  z  e.  ( P  \  M )  /\  ( ( y ( seg `  G
) z )  i^i 
M )  =  (/) ) )
10899, 107syl6bb 252 . . . . . . . 8  |-  ( ph  ->  ( y { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) } z  <->  ( y  e.  ( P  \  M
)  /\  z  e.  ( P  \  M )  /\  ( ( y ( seg `  G
) z )  i^i 
M )  =  (/) ) ) )
10960, 108anbi12d 691 . . . . . . 7  |-  ( ph  ->  ( ( x { <. x ,  y >.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) } y  /\  y { <. x ,  y
>.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) } z )  <->  ( (
x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) )  /\  (
y  e.  ( P 
\  M )  /\  z  e.  ( P  \  M )  /\  (
( y ( seg `  G ) z )  i^i  M )  =  (/) ) ) ) )
11098breqd 4034 . . . . . . . 8  |-  ( ph  ->  ( x { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) } z  <->  x { <. a ,  b >.  |  ( a  e.  ( P  \  M
)  /\  b  e.  ( P  \  M )  /\  ( ( a ( seg `  G
) b )  i^i 
M )  =  (/) ) } z ) )
111 eleq1 2343 . . . . . . . . . 10  |-  ( a  =  x  ->  (
a  e.  ( P 
\  M )  <->  x  e.  ( P  \  M ) ) )
112 oveq1 5865 . . . . . . . . . . . 12  |-  ( a  =  x  ->  (
a ( seg `  G
) b )  =  ( x ( seg `  G ) b ) )
113112ineq1d 3369 . . . . . . . . . . 11  |-  ( a  =  x  ->  (
( a ( seg `  G ) b )  i^i  M )  =  ( ( x ( seg `  G ) b )  i^i  M
) )
114113eqeq1d 2291 . . . . . . . . . 10  |-  ( a  =  x  ->  (
( ( a ( seg `  G ) b )  i^i  M
)  =  (/)  <->  ( (
x ( seg `  G
) b )  i^i 
M )  =  (/) ) )
115111, 1143anbi13d 1254 . . . . . . . . 9  |-  ( a  =  x  ->  (
( a  e.  ( P  \  M )  /\  b  e.  ( P  \  M )  /\  ( ( a ( seg `  G
) b )  i^i 
M )  =  (/) ) 
<->  ( x  e.  ( P  \  M )  /\  b  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) b )  i^i 
M )  =  (/) ) ) )
116 oveq2 5866 . . . . . . . . . . . 12  |-  ( b  =  z  ->  (
x ( seg `  G
) b )  =  ( x ( seg `  G ) z ) )
117116ineq1d 3369 . . . . . . . . . . 11  |-  ( b  =  z  ->  (
( x ( seg `  G ) b )  i^i  M )  =  ( ( x ( seg `  G ) z )  i^i  M
) )
118117eqeq1d 2291 . . . . . . . . . 10  |-  ( b  =  z  ->  (
( ( x ( seg `  G ) b )  i^i  M
)  =  (/)  <->  ( (
x ( seg `  G
) z )  i^i 
M )  =  (/) ) )
119101, 1183anbi23d 1255 . . . . . . . . 9  |-  ( b  =  z  ->  (
( x  e.  ( P  \  M )  /\  b  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) b )  i^i 
M )  =  (/) ) 
<->  ( x  e.  ( P  \  M )  /\  z  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) z )  i^i 
M )  =  (/) ) ) )
12062, 100, 115, 119, 106brab 4287 . . . . . . . 8  |-  ( x { <. a ,  b
>.  |  ( a  e.  ( P  \  M
)  /\  b  e.  ( P  \  M )  /\  ( ( a ( seg `  G
) b )  i^i 
M )  =  (/) ) } z  <->  ( x  e.  ( P  \  M
)  /\  z  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) z )  i^i 
M )  =  (/) ) )
121110, 120syl6bb 252 . . . . . . 7  |-  ( ph  ->  ( x { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) } z  <->  ( x  e.  ( P  \  M
)  /\  z  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) z )  i^i 
M )  =  (/) ) ) )
12297, 109, 1213imtr4d 259 . . . . . 6  |-  ( ph  ->  ( ( x { <. x ,  y >.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) } y  /\  y { <. x ,  y
>.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) } z )  ->  x { <. x ,  y
>.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) } z ) )
1239breqd 4034 . . . . . . 7  |-  ( ph  ->  ( y  .~  z  <->  y { <. x ,  y
>.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) } z ) )
12485, 123anbi12d 691 . . . . . 6  |-  ( ph  ->  ( ( x  .~  y  /\  y  .~  z
)  <->  ( x { <. x ,  y >.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) } y  /\  y { <. x ,  y
>.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) } z ) ) )
1259breqd 4034 . . . . . 6  |-  ( ph  ->  ( x  .~  z  <->  x { <. x ,  y
>.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x ( seg `  G
) y )  i^i 
M )  =  (/) ) } z ) )
126122, 124, 1253imtr4d 259 . . . . 5  |-  ( ph  ->  ( ( x  .~  y  /\  y  .~  z
)  ->  x  .~  z ) )
12787, 126jca 518 . . . 4  |-  ( ph  ->  ( ( x  .~  y  ->  y  .~  x
)  /\  ( (
x  .~  y  /\  y  .~  z )  ->  x  .~  z ) ) )
128127alrimiv 1617 . . 3  |-  ( ph  ->  A. z ( ( x  .~  y  -> 
y  .~  x )  /\  ( ( x  .~  y  /\  y  .~  z
)  ->  x  .~  z ) ) )
129128alrimivv 1618 . 2  |-  ( ph  ->  A. x A. y A. z ( ( x  .~  y  ->  y  .~  x )  /\  (
( x  .~  y  /\  y  .~  z
)  ->  x  .~  z ) ) )
130 dfer2 6661 . 2  |-  (  .~  Er  ( P  \  M
)  <->  ( Rel  .~  /\ 
dom  .~  =  ( P  \  M )  /\  A. x A. y A. z ( ( x  .~  y  ->  y  .~  x )  /\  (
( x  .~  y  /\  y  .~  z
)  ->  x  .~  z ) ) ) )
13111, 40, 129, 130syl3anbrc 1136 1  |-  ( ph  ->  .~  Er  ( P 
\  M ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    \ cdif 3149    i^i cin 3151   (/)c0 3455   {csn 3640   <.cop 3643   class class class wbr 4023   {copab 4076   dom cdm 4689   Rel wrel 4694   ` cfv 5255  (class class class)co 5858    Er wer 6657  PPointscpoints 26056  PLinescplines 26058  Ibgcibg 26107   segcseg 26130  sscsas 26162
This theorem is referenced by:  pdiveql  26168
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-er 6660  df-ig2 26061  df-ibg2 26109  df-seg2 26131  df-sside 26163
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