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Theorem ishlg 24726
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 464 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  a  =  A )
21neeq1d 2702 . . . . 5  |-  ( ( a  =  A  /\  b  =  B )  ->  ( a  =/=  C  <->  A  =/=  C ) )
3 simpr 468 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  b  =  B )
43neeq1d 2702 . . . . 5  |-  ( ( a  =  A  /\  b  =  B )  ->  ( b  =/=  C  <->  B  =/=  C ) )
53oveq2d 6324 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  B )  ->  ( C I b )  =  ( C I B ) )
61, 5eleq12d 2543 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  ( a  e.  ( C I b )  <-> 
A  e.  ( C I B ) ) )
71oveq2d 6324 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  B )  ->  ( C I a )  =  ( C I A ) )
83, 7eleq12d 2543 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  ( b  e.  ( C I a )  <-> 
B  e.  ( C I A ) ) )
96, 8orbi12d 724 . . . . 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 1365 . . . 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 2471 . . . 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 4889 . . 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 3040 . . . . . 6  |-  ( G  e.  V  ->  G  e.  _V )
17 fveq2 5879 . . . . . . . . 9  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
18 ishlg.p . . . . . . . . 9  |-  P  =  ( Base `  G
)
1917, 18syl6eqr 2523 . . . . . . . 8  |-  ( g  =  G  ->  ( Base `  g )  =  P )
2019eleq2d 2534 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
a  e.  ( Base `  g )  <->  a  e.  P ) )
2119eleq2d 2534 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
b  e.  ( Base `  g )  <->  b  e.  P ) )
2220, 21anbi12d 725 . . . . . . . . . 10  |-  ( g  =  G  ->  (
( a  e.  (
Base `  g )  /\  b  e.  ( Base `  g ) )  <-> 
( a  e.  P  /\  b  e.  P
) ) )
23 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( g  =  G  ->  (Itv `  g )  =  (Itv
`  G ) )
24 ishlg.i . . . . . . . . . . . . . . 15  |-  I  =  (Itv `  G )
2523, 24syl6eqr 2523 . . . . . . . . . . . . . 14  |-  ( g  =  G  ->  (Itv `  g )  =  I )
2625oveqd 6325 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  (
c (Itv `  g
) b )  =  ( c I b ) )
2726eleq2d 2534 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (
a  e.  ( c (Itv `  g )
b )  <->  a  e.  ( c I b ) ) )
2825oveqd 6325 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  (
c (Itv `  g
) a )  =  ( c I a ) )
2928eleq2d 2534 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (
b  e.  ( c (Itv `  g )
a )  <->  b  e.  ( c I a ) ) )
3027, 29orbi12d 724 . . . . . . . . . . 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 1371 . . . . . . . . . 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 725 . . . . . . . . 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 4459 . . . . . . . 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 4474 . . . . . . 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 24725 . . . . . . 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 5889 . . . . . . . . 9  |-  ( Base `  G )  e.  _V
3718, 36eqeltri 2545 . . . . . . . 8  |-  P  e. 
_V
3837mptex 6152 . . . . . . 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 5963 . . . . . 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 2517 . . . 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 2706 . . . . . . . 8  |-  ( c  =  C  ->  (
a  =/=  c  <->  a  =/=  C ) )
43 neeq2 2706 . . . . . . . 8  |-  ( c  =  C  ->  (
b  =/=  c  <->  b  =/=  C ) )
44 oveq1 6315 . . . . . . . . . 10  |-  ( c  =  C  ->  (
c I b )  =  ( C I b ) )
4544eleq2d 2534 . . . . . . . . 9  |-  ( c  =  C  ->  (
a  e.  ( c I b )  <->  a  e.  ( C I b ) ) )
46 oveq1 6315 . . . . . . . . . 10  |-  ( c  =  C  ->  (
c I a )  =  ( C I a ) )
4746eleq2d 2534 . . . . . . . . 9  |-  ( c  =  C  ->  (
b  e.  ( c I a )  <->  b  e.  ( C I a ) ) )
4845, 47orbi12d 724 . . . . . . . 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 1365 . . . . . . 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 718 . . . . . 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 4459 . . . . 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 473 . . . 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 6614 . . . . . 6  |-  ( P  X.  P )  e. 
_V
55 opabssxp 4914 . . . . . 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 4541 . . . . 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 5969 . . 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 4406 . 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 541 . . 3  |-  ( ph  ->  ( A  e.  P  /\  B  e.  P
) )
6362biantrurd 516 . 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 293 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 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031   class class class wbr 4395   {copab 4453    |-> cmpt 4454    X. cxp 4837   ` cfv 5589  (class class class)co 6308   Basecbs 15199  Itvcitv 24563  hlGchlg 24724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-hlg 24725
This theorem is referenced by:  hlcomb  24727  hlne1  24729  hlne2  24730  hlln  24731  hlid  24733  hltr  24734  hlbtwn  24735  btwnhl1  24736  btwnhl2  24737  btwnhl  24738  lnhl  24739  hlcgrex  24740  mirhl  24803  mirbtwnhl  24804  mirhl2  24805  hlperpnel  24846  opphllem4  24871  opphl  24875  hlpasch  24877  lnopp2hpgb  24884  cgracgr  24939  cgraswap  24941  cgrahl  24947  cgracol  24948
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