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Theorem islmib 24354
Description: Property of the line mirror. (Contributed by Thierry Arnoux, 11-Dec-2019.)
Hypotheses
Ref Expression
ismid.p  |-  P  =  ( Base `  G
)
ismid.d  |-  .-  =  ( dist `  G )
ismid.i  |-  I  =  (Itv `  G )
ismid.g  |-  ( ph  ->  G  e. TarskiG )
ismid.1  |-  ( ph  ->  GDimTarskiG 2 )
lmif.m  |-  M  =  ( (lInvG `  G
) `  D )
lmif.l  |-  L  =  (LineG `  G )
lmif.d  |-  ( ph  ->  D  e.  ran  L
)
lmicl.1  |-  ( ph  ->  A  e.  P )
islmib.b  |-  ( ph  ->  B  e.  P )
Assertion
Ref Expression
islmib  |-  ( ph  ->  ( B  =  ( M `  A )  <-> 
( ( A (midG `  G ) B )  e.  D  /\  ( D (⟂G `  G )
( A L B )  \/  A  =  B ) ) ) )

Proof of Theorem islmib
Dummy variables  a 
b  g  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmif.m . . . . 5  |-  M  =  ( (lInvG `  G
) `  D )
2 df-lmi 24342 . . . . . . . 8  |- lInvG  =  ( g  e.  _V  |->  ( d  e.  ran  (LineG `  g )  |->  ( a  e.  ( Base `  g
)  |->  ( iota_ b  e.  ( Base `  g
) ( ( a (midG `  g )
b )  e.  d  /\  ( d (⟂G `  g ) ( a (LineG `  g )
b )  \/  a  =  b ) ) ) ) ) )
32a1i 11 . . . . . . 7  |-  ( ph  -> lInvG  =  ( g  e. 
_V  |->  ( d  e. 
ran  (LineG `  g )  |->  ( a  e.  (
Base `  g )  |->  ( iota_ b  e.  (
Base `  g )
( ( a (midG `  g ) b )  e.  d  /\  (
d (⟂G `  g )
( a (LineG `  g ) b )  \/  a  =  b ) ) ) ) ) ) )
4 fveq2 5848 . . . . . . . . . . 11  |-  ( g  =  G  ->  (LineG `  g )  =  (LineG `  G ) )
5 lmif.l . . . . . . . . . . 11  |-  L  =  (LineG `  G )
64, 5syl6eqr 2513 . . . . . . . . . 10  |-  ( g  =  G  ->  (LineG `  g )  =  L )
76rneqd 5219 . . . . . . . . 9  |-  ( g  =  G  ->  ran  (LineG `  g )  =  ran  L )
8 fveq2 5848 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
9 ismid.p . . . . . . . . . . 11  |-  P  =  ( Base `  G
)
108, 9syl6eqr 2513 . . . . . . . . . 10  |-  ( g  =  G  ->  ( Base `  g )  =  P )
11 fveq2 5848 . . . . . . . . . . . . . 14  |-  ( g  =  G  ->  (midG `  g )  =  (midG `  G ) )
1211oveqd 6287 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  (
a (midG `  g
) b )  =  ( a (midG `  G ) b ) )
1312eleq1d 2523 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (
( a (midG `  g ) b )  e.  d  <->  ( a
(midG `  G )
b )  e.  d ) )
14 eqidd 2455 . . . . . . . . . . . . . 14  |-  ( g  =  G  ->  d  =  d )
15 fveq2 5848 . . . . . . . . . . . . . 14  |-  ( g  =  G  ->  (⟂G `  g )  =  (⟂G `  G ) )
166oveqd 6287 . . . . . . . . . . . . . 14  |-  ( g  =  G  ->  (
a (LineG `  g
) b )  =  ( a L b ) )
1714, 15, 16breq123d 4453 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  (
d (⟂G `  g )
( a (LineG `  g ) b )  <-> 
d (⟂G `  G )
( a L b ) ) )
1817orbi1d 700 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (
( d (⟂G `  g
) ( a (LineG `  g ) b )  \/  a  =  b )  <->  ( d (⟂G `  G ) ( a L b )  \/  a  =  b ) ) )
1913, 18anbi12d 708 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
( ( a (midG `  g ) b )  e.  d  /\  (
d (⟂G `  g )
( a (LineG `  g ) b )  \/  a  =  b ) )  <->  ( (
a (midG `  G
) b )  e.  d  /\  ( d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) )
2010, 19riotaeqbidv 6235 . . . . . . . . . 10  |-  ( g  =  G  ->  ( iota_ b  e.  ( Base `  g ) ( ( a (midG `  g
) b )  e.  d  /\  ( d (⟂G `  g )
( a (LineG `  g ) b )  \/  a  =  b ) ) )  =  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) )
2110, 20mpteq12dv 4517 . . . . . . . . 9  |-  ( g  =  G  ->  (
a  e.  ( Base `  g )  |->  ( iota_ b  e.  ( Base `  g
) ( ( a (midG `  g )
b )  e.  d  /\  ( d (⟂G `  g ) ( a (LineG `  g )
b )  \/  a  =  b ) ) ) )  =  ( a  e.  P  |->  (
iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) )
227, 21mpteq12dv 4517 . . . . . . . 8  |-  ( g  =  G  ->  (
d  e.  ran  (LineG `  g )  |->  ( a  e.  ( Base `  g
)  |->  ( iota_ b  e.  ( Base `  g
) ( ( a (midG `  g )
b )  e.  d  /\  ( d (⟂G `  g ) ( a (LineG `  g )
b )  \/  a  =  b ) ) ) ) )  =  ( d  e.  ran  L 
|->  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) ) )
2322adantl 464 . . . . . . 7  |-  ( (
ph  /\  g  =  G )  ->  (
d  e.  ran  (LineG `  g )  |->  ( a  e.  ( Base `  g
)  |->  ( iota_ b  e.  ( Base `  g
) ( ( a (midG `  g )
b )  e.  d  /\  ( d (⟂G `  g ) ( a (LineG `  g )
b )  \/  a  =  b ) ) ) ) )  =  ( d  e.  ran  L 
|->  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) ) )
24 ismid.g . . . . . . . 8  |-  ( ph  ->  G  e. TarskiG )
25 elex 3115 . . . . . . . 8  |-  ( G  e. TarskiG  ->  G  e.  _V )
2624, 25syl 16 . . . . . . 7  |-  ( ph  ->  G  e.  _V )
27 fvex 5858 . . . . . . . . . 10  |-  (LineG `  G )  e.  _V
285, 27eqeltri 2538 . . . . . . . . 9  |-  L  e. 
_V
29 rnexg 6705 . . . . . . . . 9  |-  ( L  e.  _V  ->  ran  L  e.  _V )
30 mptexg 6117 . . . . . . . . 9  |-  ( ran 
L  e.  _V  ->  ( d  e.  ran  L  |->  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) )  e.  _V )
3128, 29, 30mp2b 10 . . . . . . . 8  |-  ( d  e.  ran  L  |->  ( a  e.  P  |->  (
iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) )  e.  _V
3231a1i 11 . . . . . . 7  |-  ( ph  ->  ( d  e.  ran  L 
|->  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) )  e.  _V )
333, 23, 26, 32fvmptd 5936 . . . . . 6  |-  ( ph  ->  (lInvG `  G )  =  ( d  e. 
ran  L  |->  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G
) b )  e.  d  /\  ( d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) ) )
34 eleq2 2527 . . . . . . . . . 10  |-  ( d  =  D  ->  (
( a (midG `  G ) b )  e.  d  <->  ( a
(midG `  G )
b )  e.  D
) )
35 breq1 4442 . . . . . . . . . . 11  |-  ( d  =  D  ->  (
d (⟂G `  G )
( a L b )  <->  D (⟂G `  G
) ( a L b ) ) )
3635orbi1d 700 . . . . . . . . . 10  |-  ( d  =  D  ->  (
( d (⟂G `  G
) ( a L b )  \/  a  =  b )  <->  ( D
(⟂G `  G ) ( a L b )  \/  a  =  b ) ) )
3734, 36anbi12d 708 . . . . . . . . 9  |-  ( d  =  D  ->  (
( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) )  <->  ( (
a (midG `  G
) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) )
3837riotabidv 6234 . . . . . . . 8  |-  ( d  =  D  ->  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) )  =  ( iota_ b  e.  P  ( ( a (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( a L b )  \/  a  =  b ) ) ) )
3938mpteq2dv 4526 . . . . . . 7  |-  ( d  =  D  ->  (
a  e.  P  |->  (
iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) )  =  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G
) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) )
4039adantl 464 . . . . . 6  |-  ( (
ph  /\  d  =  D )  ->  (
a  e.  P  |->  (
iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) )  =  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G
) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) )
41 lmif.d . . . . . 6  |-  ( ph  ->  D  e.  ran  L
)
42 fvex 5858 . . . . . . . . 9  |-  ( Base `  G )  e.  _V
439, 42eqeltri 2538 . . . . . . . 8  |-  P  e. 
_V
4443mptex 6118 . . . . . . 7  |-  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G
) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) )  e.  _V
4544a1i 11 . . . . . 6  |-  ( ph  ->  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) )  e.  _V )
4633, 40, 41, 45fvmptd 5936 . . . . 5  |-  ( ph  ->  ( (lInvG `  G
) `  D )  =  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( a L b )  \/  a  =  b ) ) ) ) )
471, 46syl5eq 2507 . . . 4  |-  ( ph  ->  M  =  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G
) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) )
48 oveq1 6277 . . . . . . . 8  |-  ( a  =  A  ->  (
a (midG `  G
) b )  =  ( A (midG `  G ) b ) )
4948eleq1d 2523 . . . . . . 7  |-  ( a  =  A  ->  (
( a (midG `  G ) b )  e.  D  <->  ( A
(midG `  G )
b )  e.  D
) )
50 oveq1 6277 . . . . . . . . 9  |-  ( a  =  A  ->  (
a L b )  =  ( A L b ) )
5150breq2d 4451 . . . . . . . 8  |-  ( a  =  A  ->  ( D (⟂G `  G )
( a L b )  <->  D (⟂G `  G
) ( A L b ) ) )
52 eqeq1 2458 . . . . . . . 8  |-  ( a  =  A  ->  (
a  =  b  <->  A  =  b ) )
5351, 52orbi12d 707 . . . . . . 7  |-  ( a  =  A  ->  (
( D (⟂G `  G
) ( a L b )  \/  a  =  b )  <->  ( D
(⟂G `  G ) ( A L b )  \/  A  =  b ) ) )
5449, 53anbi12d 708 . . . . . 6  |-  ( a  =  A  ->  (
( ( a (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) )  <->  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) ) )
5554riotabidv 6234 . . . . 5  |-  ( a  =  A  ->  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) )  =  ( iota_ b  e.  P  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) ) )
5655adantl 464 . . . 4  |-  ( (
ph  /\  a  =  A )  ->  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) )  =  ( iota_ b  e.  P  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) ) )
57 lmicl.1 . . . 4  |-  ( ph  ->  A  e.  P )
58 ismid.d . . . . . 6  |-  .-  =  ( dist `  G )
59 ismid.i . . . . . 6  |-  I  =  (Itv `  G )
60 ismid.1 . . . . . 6  |-  ( ph  ->  GDimTarskiG 2 )
619, 58, 59, 24, 60, 5, 41, 57lmieu 24351 . . . . 5  |-  ( ph  ->  E! b  e.  P  ( ( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) )
62 riotacl 6246 . . . . 5  |-  ( E! b  e.  P  ( ( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) )  -> 
( iota_ b  e.  P  ( ( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) )  e.  P )
6361, 62syl 16 . . . 4  |-  ( ph  ->  ( iota_ b  e.  P  ( ( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) )  e.  P )
6447, 56, 57, 63fvmptd 5936 . . 3  |-  ( ph  ->  ( M `  A
)  =  ( iota_ b  e.  P  ( ( A (midG `  G
) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) ) )
6564eqeq2d 2468 . 2  |-  ( ph  ->  ( B  =  ( M `  A )  <-> 
B  =  ( iota_ b  e.  P  ( ( A (midG `  G
) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) ) ) )
66 islmib.b . . . 4  |-  ( ph  ->  B  e.  P )
67 oveq2 6278 . . . . . . 7  |-  ( b  =  B  ->  ( A (midG `  G )
b )  =  ( A (midG `  G
) B ) )
6867eleq1d 2523 . . . . . 6  |-  ( b  =  B  ->  (
( A (midG `  G ) b )  e.  D  <->  ( A
(midG `  G ) B )  e.  D
) )
69 oveq2 6278 . . . . . . . 8  |-  ( b  =  B  ->  ( A L b )  =  ( A L B ) )
7069breq2d 4451 . . . . . . 7  |-  ( b  =  B  ->  ( D (⟂G `  G )
( A L b )  <->  D (⟂G `  G
) ( A L B ) ) )
71 eqeq2 2469 . . . . . . 7  |-  ( b  =  B  ->  ( A  =  b  <->  A  =  B ) )
7270, 71orbi12d 707 . . . . . 6  |-  ( b  =  B  ->  (
( D (⟂G `  G
) ( A L b )  \/  A  =  b )  <->  ( D
(⟂G `  G ) ( A L B )  \/  A  =  B ) ) )
7368, 72anbi12d 708 . . . . 5  |-  ( b  =  B  ->  (
( ( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) )  <->  ( ( A (midG `  G ) B )  e.  D  /\  ( D (⟂G `  G
) ( A L B )  \/  A  =  B ) ) ) )
7473riota2 6254 . . . 4  |-  ( ( B  e.  P  /\  E! b  e.  P  ( ( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) )  ->  ( ( ( A (midG `  G
) B )  e.  D  /\  ( D (⟂G `  G )
( A L B )  \/  A  =  B ) )  <->  ( iota_ b  e.  P  ( ( A (midG `  G
) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) )  =  B ) )
7566, 61, 74syl2anc 659 . . 3  |-  ( ph  ->  ( ( ( A (midG `  G ) B )  e.  D  /\  ( D (⟂G `  G
) ( A L B )  \/  A  =  B ) )  <->  ( iota_ b  e.  P  ( ( A (midG `  G
) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) )  =  B ) )
76 eqcom 2463 . . 3  |-  ( B  =  ( iota_ b  e.  P  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  <->  ( iota_ b  e.  P  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  =  B )
7775, 76syl6bbr 263 . 2  |-  ( ph  ->  ( ( ( A (midG `  G ) B )  e.  D  /\  ( D (⟂G `  G
) ( A L B )  \/  A  =  B ) )  <->  B  =  ( iota_ b  e.  P  ( ( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) ) ) )
7865, 77bitr4d 256 1  |-  ( ph  ->  ( B  =  ( M `  A )  <-> 
( ( A (midG `  G ) B )  e.  D  /\  ( D (⟂G `  G )
( A L B )  \/  A  =  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823   E!wreu 2806   _Vcvv 3106   class class class wbr 4439    |-> cmpt 4497   ran crn 4989   ` cfv 5570   iota_crio 6231  (class class class)co 6270   2c2 10581   Basecbs 14716   distcds 14793  TarskiGcstrkg 24023  DimTarskiGcstrkgld 24027  Itvcitv 24030  LineGclng 24031  ⟂Gcperpg 24273  midGcmid 24339  lInvGclmi 24340
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-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
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-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-concat 12528  df-s1 12529  df-s2 12804  df-s3 12805  df-trkgc 24042  df-trkgb 24043  df-trkgcb 24044  df-trkgld 24046  df-trkg 24048  df-cgrg 24104  df-leg 24171  df-mir 24235  df-rag 24272  df-perpg 24274  df-mid 24341  df-lmi 24342
This theorem is referenced by:  lmicom  24355  lmiinv  24359  lmimid  24360  lmiisolem  24362  hypcgrlem1  24365  hypcgrlem2  24366
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