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Theorem ishlg 24647
Description: Rays : Definition 6.1 of [Schwabhauser] p. 43. With this definition,  A ( K `
 C ) B means that  A and  B are on the same ray with initial point  C. This follows the same notation as Schwabhauser where rays are first defined as a relation. It is possible to recover the ray itself using e.g.  ( ( K `  C ) " { A } ) (Contributed by Thierry Arnoux, 21-Dec-2019.)
Hypotheses
Ref Expression
ishlg.p  |-  P  =  ( Base `  G
)
ishlg.i  |-  I  =  (Itv `  G )
ishlg.k  |-  K  =  (hlG `  G )
ishlg.a  |-  ( ph  ->  A  e.  P )
ishlg.b  |-  ( ph  ->  B  e.  P )
ishlg.c  |-  ( ph  ->  C  e.  P )
ishlg.g  |-  ( ph  ->  G  e.  V )
Assertion
Ref Expression
ishlg  |-  ( ph  ->  ( A ( K `
 C ) B  <-> 
( A  =/=  C  /\  B  =/=  C  /\  ( A  e.  ( C I B )  \/  B  e.  ( C I A ) ) ) ) )

Proof of Theorem ishlg
Dummy variables  a 
b  c  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 459 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  a  =  A )
21neeq1d 2683 . . . . 5  |-  ( ( a  =  A  /\  b  =  B )  ->  ( a  =/=  C  <->  A  =/=  C ) )
3 simpr 463 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  b  =  B )
43neeq1d 2683 . . . . 5  |-  ( ( a  =  A  /\  b  =  B )  ->  ( b  =/=  C  <->  B  =/=  C ) )
53oveq2d 6306 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  B )  ->  ( C I b )  =  ( C I B ) )
61, 5eleq12d 2523 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  ( a  e.  ( C I b )  <-> 
A  e.  ( C I B ) ) )
71oveq2d 6306 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  B )  ->  ( C I a )  =  ( C I A ) )
83, 7eleq12d 2523 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  ( b  e.  ( C I a )  <-> 
B  e.  ( C I A ) ) )
96, 8orbi12d 716 . . . . 5  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( a  e.  ( C I b )  \/  b  e.  ( C I a ) )  <->  ( A  e.  ( C I B )  \/  B  e.  ( C I A ) ) ) )
102, 4, 93anbi123d 1339 . . . 4  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( a  =/= 
C  /\  b  =/=  C  /\  ( a  e.  ( C I b )  \/  b  e.  ( C I a ) ) )  <->  ( A  =/=  C  /\  B  =/= 
C  /\  ( A  e.  ( C I B )  \/  B  e.  ( C I A ) ) ) ) )
11 eqid 2451 . . . 4  |-  { <. a ,  b >.  |  ( ( a  e.  P  /\  b  e.  P
)  /\  ( a  =/=  C  /\  b  =/= 
C  /\  ( a  e.  ( C I b )  \/  b  e.  ( C I a ) ) ) ) }  =  { <. a ,  b >.  |  ( ( a  e.  P  /\  b  e.  P
)  /\  ( a  =/=  C  /\  b  =/= 
C  /\  ( a  e.  ( C I b )  \/  b  e.  ( C I a ) ) ) ) }
1210, 11brab2a 4884 . . 3  |-  ( A { <. a ,  b
>.  |  ( (
a  e.  P  /\  b  e.  P )  /\  ( a  =/=  C  /\  b  =/=  C  /\  ( a  e.  ( C I b )  \/  b  e.  ( C I a ) ) ) ) } B  <->  ( ( A  e.  P  /\  B  e.  P )  /\  ( A  =/=  C  /\  B  =/=  C  /\  ( A  e.  ( C I B )  \/  B  e.  ( C I A ) ) ) ) )
1312a1i 11 . 2  |-  ( ph  ->  ( A { <. a ,  b >.  |  ( ( a  e.  P  /\  b  e.  P
)  /\  ( a  =/=  C  /\  b  =/= 
C  /\  ( a  e.  ( C I b )  \/  b  e.  ( C I a ) ) ) ) } B  <->  ( ( A  e.  P  /\  B  e.  P )  /\  ( A  =/=  C  /\  B  =/=  C  /\  ( A  e.  ( C I B )  \/  B  e.  ( C I A ) ) ) ) ) )
14 ishlg.k . . . . 5  |-  K  =  (hlG `  G )
15 ishlg.g . . . . . 6  |-  ( ph  ->  G  e.  V )
16 elex 3054 . . . . . 6  |-  ( G  e.  V  ->  G  e.  _V )
17 fveq2 5865 . . . . . . . . 9  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
18 ishlg.p . . . . . . . . 9  |-  P  =  ( Base `  G
)
1917, 18syl6eqr 2503 . . . . . . . 8  |-  ( g  =  G  ->  ( Base `  g )  =  P )
2019eleq2d 2514 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
a  e.  ( Base `  g )  <->  a  e.  P ) )
2119eleq2d 2514 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
b  e.  ( Base `  g )  <->  b  e.  P ) )
2220, 21anbi12d 717 . . . . . . . . . 10  |-  ( g  =  G  ->  (
( a  e.  (
Base `  g )  /\  b  e.  ( Base `  g ) )  <-> 
( a  e.  P  /\  b  e.  P
) ) )
23 fveq2 5865 . . . . . . . . . . . . . . 15  |-  ( g  =  G  ->  (Itv `  g )  =  (Itv
`  G ) )
24 ishlg.i . . . . . . . . . . . . . . 15  |-  I  =  (Itv `  G )
2523, 24syl6eqr 2503 . . . . . . . . . . . . . 14  |-  ( g  =  G  ->  (Itv `  g )  =  I )
2625oveqd 6307 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  (
c (Itv `  g
) b )  =  ( c I b ) )
2726eleq2d 2514 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (
a  e.  ( c (Itv `  g )
b )  <->  a  e.  ( c I b ) ) )
2825oveqd 6307 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  (
c (Itv `  g
) a )  =  ( c I a ) )
2928eleq2d 2514 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (
b  e.  ( c (Itv `  g )
a )  <->  b  e.  ( c I a ) ) )
3027, 29orbi12d 716 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
( a  e.  ( c (Itv `  g
) b )  \/  b  e.  ( c (Itv `  g )
a ) )  <->  ( a  e.  ( c I b )  \/  b  e.  ( c I a ) ) ) )
31303anbi3d 1345 . . . . . . . . . 10  |-  ( g  =  G  ->  (
( a  =/=  c  /\  b  =/=  c  /\  ( a  e.  ( c (Itv `  g
) b )  \/  b  e.  ( c (Itv `  g )
a ) ) )  <-> 
( a  =/=  c  /\  b  =/=  c  /\  ( a  e.  ( c I b )  \/  b  e.  ( c I a ) ) ) ) )
3222, 31anbi12d 717 . . . . . . . . 9  |-  ( g  =  G  ->  (
( ( a  e.  ( Base `  g
)  /\  b  e.  ( Base `  g )
)  /\  ( a  =/=  c  /\  b  =/=  c  /\  (
a  e.  ( c (Itv `  g )
b )  \/  b  e.  ( c (Itv `  g ) a ) ) ) )  <->  ( (
a  e.  P  /\  b  e.  P )  /\  ( a  =/=  c  /\  b  =/=  c  /\  ( a  e.  ( c I b )  \/  b  e.  ( c I a ) ) ) ) ) )
3332opabbidv 4466 . . . . . . . 8  |-  ( g  =  G  ->  { <. a ,  b >.  |  ( ( a  e.  (
Base `  g )  /\  b  e.  ( Base `  g ) )  /\  ( a  =/=  c  /\  b  =/=  c  /\  ( a  e.  ( c (Itv
`  g ) b )  \/  b  e.  ( c (Itv `  g ) a ) ) ) ) }  =  { <. a ,  b >.  |  ( ( a  e.  P  /\  b  e.  P
)  /\  ( a  =/=  c  /\  b  =/=  c  /\  (
a  e.  ( c I b )  \/  b  e.  ( c I a ) ) ) ) } )
3419, 33mpteq12dv 4481 . . . . . . 7  |-  ( g  =  G  ->  (
c  e.  ( Base `  g )  |->  { <. a ,  b >.  |  ( ( a  e.  (
Base `  g )  /\  b  e.  ( Base `  g ) )  /\  ( a  =/=  c  /\  b  =/=  c  /\  ( a  e.  ( c (Itv
`  g ) b )  \/  b  e.  ( c (Itv `  g ) a ) ) ) ) } )  =  ( c  e.  P  |->  { <. a ,  b >.  |  ( ( a  e.  P  /\  b  e.  P
)  /\  ( a  =/=  c  /\  b  =/=  c  /\  (
a  e.  ( c I b )  \/  b  e.  ( c I a ) ) ) ) } ) )
35 df-hlg 24646 . . . . . . 7  |- hlG  =  ( g  e.  _V  |->  ( c  e.  ( Base `  g )  |->  { <. a ,  b >.  |  ( ( a  e.  (
Base `  g )  /\  b  e.  ( Base `  g ) )  /\  ( a  =/=  c  /\  b  =/=  c  /\  ( a  e.  ( c (Itv
`  g ) b )  \/  b  e.  ( c (Itv `  g ) a ) ) ) ) } ) )
36 fvex 5875 . . . . . . . . 9  |-  ( Base `  G )  e.  _V
3718, 36eqeltri 2525 . . . . . . . 8  |-  P  e. 
_V
3837mptex 6136 . . . . . . 7  |-  ( c  e.  P  |->  { <. a ,  b >.  |  ( ( a  e.  P  /\  b  e.  P
)  /\  ( a  =/=  c  /\  b  =/=  c  /\  (
a  e.  ( c I b )  \/  b  e.  ( c I a ) ) ) ) } )  e.  _V
3934, 35, 38fvmpt 5948 . . . . . 6  |-  ( G  e.  _V  ->  (hlG `  G )  =  ( c  e.  P  |->  {
<. a ,  b >.  |  ( ( a  e.  P  /\  b  e.  P )  /\  (
a  =/=  c  /\  b  =/=  c  /\  (
a  e.  ( c I b )  \/  b  e.  ( c I a ) ) ) ) } ) )
4015, 16, 393syl 18 . . . . 5  |-  ( ph  ->  (hlG `  G )  =  ( c  e.  P  |->  { <. a ,  b >.  |  ( ( a  e.  P  /\  b  e.  P
)  /\  ( a  =/=  c  /\  b  =/=  c  /\  (
a  e.  ( c I b )  \/  b  e.  ( c I a ) ) ) ) } ) )
4114, 40syl5eq 2497 . . . 4  |-  ( ph  ->  K  =  ( c  e.  P  |->  { <. a ,  b >.  |  ( ( a  e.  P  /\  b  e.  P
)  /\  ( a  =/=  c  /\  b  =/=  c  /\  (
a  e.  ( c I b )  \/  b  e.  ( c I a ) ) ) ) } ) )
42 neeq2 2687 . . . . . . . 8  |-  ( c  =  C  ->  (
a  =/=  c  <->  a  =/=  C ) )
43 neeq2 2687 . . . . . . . 8  |-  ( c  =  C  ->  (
b  =/=  c  <->  b  =/=  C ) )
44 oveq1 6297 . . . . . . . . . 10  |-  ( c  =  C  ->  (
c I b )  =  ( C I b ) )
4544eleq2d 2514 . . . . . . . . 9  |-  ( c  =  C  ->  (
a  e.  ( c I b )  <->  a  e.  ( C I b ) ) )
46 oveq1 6297 . . . . . . . . . 10  |-  ( c  =  C  ->  (
c I a )  =  ( C I a ) )
4746eleq2d 2514 . . . . . . . . 9  |-  ( c  =  C  ->  (
b  e.  ( c I a )  <->  b  e.  ( C I a ) ) )
4845, 47orbi12d 716 . . . . . . . 8  |-  ( c  =  C  ->  (
( a  e.  ( c I b )  \/  b  e.  ( c I a ) )  <->  ( a  e.  ( C I b )  \/  b  e.  ( C I a ) ) ) )
4942, 43, 483anbi123d 1339 . . . . . . 7  |-  ( c  =  C  ->  (
( a  =/=  c  /\  b  =/=  c  /\  ( a  e.  ( c I b )  \/  b  e.  ( c I a ) ) )  <->  ( a  =/=  C  /\  b  =/= 
C  /\  ( a  e.  ( C I b )  \/  b  e.  ( C I a ) ) ) ) )
5049anbi2d 710 . . . . . 6  |-  ( c  =  C  ->  (
( ( a  e.  P  /\  b  e.  P )  /\  (
a  =/=  c  /\  b  =/=  c  /\  (
a  e.  ( c I b )  \/  b  e.  ( c I a ) ) ) )  <->  ( (
a  e.  P  /\  b  e.  P )  /\  ( a  =/=  C  /\  b  =/=  C  /\  ( a  e.  ( C I b )  \/  b  e.  ( C I a ) ) ) ) ) )
5150opabbidv 4466 . . . . 5  |-  ( c  =  C  ->  { <. a ,  b >.  |  ( ( a  e.  P  /\  b  e.  P
)  /\  ( a  =/=  c  /\  b  =/=  c  /\  (
a  e.  ( c I b )  \/  b  e.  ( c I a ) ) ) ) }  =  { <. a ,  b
>.  |  ( (
a  e.  P  /\  b  e.  P )  /\  ( a  =/=  C  /\  b  =/=  C  /\  ( a  e.  ( C I b )  \/  b  e.  ( C I a ) ) ) ) } )
5251adantl 468 . . . 4  |-  ( (
ph  /\  c  =  C )  ->  { <. a ,  b >.  |  ( ( a  e.  P  /\  b  e.  P
)  /\  ( a  =/=  c  /\  b  =/=  c  /\  (
a  e.  ( c I b )  \/  b  e.  ( c I a ) ) ) ) }  =  { <. a ,  b
>.  |  ( (
a  e.  P  /\  b  e.  P )  /\  ( a  =/=  C  /\  b  =/=  C  /\  ( a  e.  ( C I b )  \/  b  e.  ( C I a ) ) ) ) } )
53 ishlg.c . . . 4  |-  ( ph  ->  C  e.  P )
5437, 37xpex 6595 . . . . . 6  |-  ( P  X.  P )  e. 
_V
55 opabssxp 4909 . . . . . 6  |-  { <. a ,  b >.  |  ( ( a  e.  P  /\  b  e.  P
)  /\  ( a  =/=  C  /\  b  =/= 
C  /\  ( a  e.  ( C I b )  \/  b  e.  ( C I a ) ) ) ) }  C_  ( P  X.  P )
5654, 55ssexi 4548 . . . . 5  |-  { <. a ,  b >.  |  ( ( a  e.  P  /\  b  e.  P
)  /\  ( a  =/=  C  /\  b  =/= 
C  /\  ( a  e.  ( C I b )  \/  b  e.  ( C I a ) ) ) ) }  e.  _V
5756a1i 11 . . . 4  |-  ( ph  ->  { <. a ,  b
>.  |  ( (
a  e.  P  /\  b  e.  P )  /\  ( a  =/=  C  /\  b  =/=  C  /\  ( a  e.  ( C I b )  \/  b  e.  ( C I a ) ) ) ) }  e.  _V )
5841, 52, 53, 57fvmptd 5954 . . 3  |-  ( ph  ->  ( K `  C
)  =  { <. a ,  b >.  |  ( ( a  e.  P  /\  b  e.  P
)  /\  ( a  =/=  C  /\  b  =/= 
C  /\  ( a  e.  ( C I b )  \/  b  e.  ( C I a ) ) ) ) } )
5958breqd 4413 . 2  |-  ( ph  ->  ( A ( K `
 C ) B  <-> 
A { <. a ,  b >.  |  ( ( a  e.  P  /\  b  e.  P
)  /\  ( a  =/=  C  /\  b  =/= 
C  /\  ( a  e.  ( C I b )  \/  b  e.  ( C I a ) ) ) ) } B ) )
60 ishlg.a . . . 4  |-  ( ph  ->  A  e.  P )
61 ishlg.b . . . 4  |-  ( ph  ->  B  e.  P )
6260, 61jca 535 . . 3  |-  ( ph  ->  ( A  e.  P  /\  B  e.  P
) )
6362biantrurd 511 . 2  |-  ( ph  ->  ( ( A  =/= 
C  /\  B  =/=  C  /\  ( A  e.  ( C I B )  \/  B  e.  ( C I A ) ) )  <->  ( ( A  e.  P  /\  B  e.  P )  /\  ( A  =/=  C  /\  B  =/=  C  /\  ( A  e.  ( C I B )  \/  B  e.  ( C I A ) ) ) ) ) )
6413, 59, 633bitr4d 289 1  |-  ( ph  ->  ( A ( K `
 C ) B  <-> 
( A  =/=  C  /\  B  =/=  C  /\  ( A  e.  ( C I B )  \/  B  e.  ( C I A ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   _Vcvv 3045   class class class wbr 4402   {copab 4460    |-> cmpt 4461    X. cxp 4832   ` cfv 5582  (class class class)co 6290   Basecbs 15121  Itvcitv 24484  hlGchlg 24645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-hlg 24646
This theorem is referenced by:  hlcomb  24648  hlne1  24650  hlne2  24651  hlln  24652  hlid  24654  hltr  24655  hlbtwn  24656  btwnhl1  24657  btwnhl2  24658  btwnhl  24659  lnhl  24660  hlcgrex  24661  mirhl  24724  mirbtwnhl  24725  mirhl2  24726  hlperpnel  24767  opphllem4  24792  opphl  24796  hlpasch  24798  lnopp2hpgb  24805  cgracgr  24860  cgraswap  24862  cgrahl  24868  cgracol  24869
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