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Theorem islmib 24829
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 24817 . . . . . . . 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 5865 . . . . . . . . . . 11  |-  ( g  =  G  ->  (LineG `  g )  =  (LineG `  G ) )
5 lmif.l . . . . . . . . . . 11  |-  L  =  (LineG `  G )
64, 5syl6eqr 2503 . . . . . . . . . 10  |-  ( g  =  G  ->  (LineG `  g )  =  L )
76rneqd 5062 . . . . . . . . 9  |-  ( g  =  G  ->  ran  (LineG `  g )  =  ran  L )
8 fveq2 5865 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
9 ismid.p . . . . . . . . . . 11  |-  P  =  ( Base `  G
)
108, 9syl6eqr 2503 . . . . . . . . . 10  |-  ( g  =  G  ->  ( Base `  g )  =  P )
11 fveq2 5865 . . . . . . . . . . . . . 14  |-  ( g  =  G  ->  (midG `  g )  =  (midG `  G ) )
1211oveqd 6307 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  (
a (midG `  g
) b )  =  ( a (midG `  G ) b ) )
1312eleq1d 2513 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (
( a (midG `  g ) b )  e.  d  <->  ( a
(midG `  G )
b )  e.  d ) )
14 eqidd 2452 . . . . . . . . . . . . . 14  |-  ( g  =  G  ->  d  =  d )
15 fveq2 5865 . . . . . . . . . . . . . 14  |-  ( g  =  G  ->  (⟂G `  g )  =  (⟂G `  G ) )
166oveqd 6307 . . . . . . . . . . . . . 14  |-  ( g  =  G  ->  (
a (LineG `  g
) b )  =  ( a L b ) )
1714, 15, 16breq123d 4416 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  (
d (⟂G `  g )
( a (LineG `  g ) b )  <-> 
d (⟂G `  G )
( a L b ) ) )
1817orbi1d 709 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (
( d (⟂G `  g
) ( a (LineG `  g ) b )  \/  a  =  b )  <->  ( d (⟂G `  G ) ( a L b )  \/  a  =  b ) ) )
1913, 18anbi12d 717 . . . . . . . . . . 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 6255 . . . . . . . . . 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 4481 . . . . . . . . 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 4481 . . . . . . . 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 468 . . . . . . 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 3054 . . . . . . . 8  |-  ( G  e. TarskiG  ->  G  e.  _V )
2624, 25syl 17 . . . . . . 7  |-  ( ph  ->  G  e.  _V )
27 fvex 5875 . . . . . . . . . 10  |-  (LineG `  G )  e.  _V
285, 27eqeltri 2525 . . . . . . . . 9  |-  L  e. 
_V
29 rnexg 6725 . . . . . . . . 9  |-  ( L  e.  _V  ->  ran  L  e.  _V )
30 mptexg 6135 . . . . . . . . 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 5954 . . . . . 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 2518 . . . . . . . . . 10  |-  ( d  =  D  ->  (
( a (midG `  G ) b )  e.  d  <->  ( a
(midG `  G )
b )  e.  D
) )
35 breq1 4405 . . . . . . . . . . 11  |-  ( d  =  D  ->  (
d (⟂G `  G )
( a L b )  <->  D (⟂G `  G
) ( a L b ) ) )
3635orbi1d 709 . . . . . . . . . 10  |-  ( d  =  D  ->  (
( d (⟂G `  G
) ( a L b )  \/  a  =  b )  <->  ( D
(⟂G `  G ) ( a L b )  \/  a  =  b ) ) )
3734, 36anbi12d 717 . . . . . . . . 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 6254 . . . . . . . 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 4490 . . . . . . 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 468 . . . . . 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 5875 . . . . . . . . 9  |-  ( Base `  G )  e.  _V
439, 42eqeltri 2525 . . . . . . . 8  |-  P  e. 
_V
4443mptex 6136 . . . . . . 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 5954 . . . . 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 2497 . . . 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 6297 . . . . . . . 8  |-  ( a  =  A  ->  (
a (midG `  G
) b )  =  ( A (midG `  G ) b ) )
4948eleq1d 2513 . . . . . . 7  |-  ( a  =  A  ->  (
( a (midG `  G ) b )  e.  D  <->  ( A
(midG `  G )
b )  e.  D
) )
50 oveq1 6297 . . . . . . . . 9  |-  ( a  =  A  ->  (
a L b )  =  ( A L b ) )
5150breq2d 4414 . . . . . . . 8  |-  ( a  =  A  ->  ( D (⟂G `  G )
( a L b )  <->  D (⟂G `  G
) ( A L b ) ) )
52 eqeq1 2455 . . . . . . . 8  |-  ( a  =  A  ->  (
a  =  b  <->  A  =  b ) )
5351, 52orbi12d 716 . . . . . . 7  |-  ( a  =  A  ->  (
( D (⟂G `  G
) ( a L b )  \/  a  =  b )  <->  ( D
(⟂G `  G ) ( A L b )  \/  A  =  b ) ) )
5449, 53anbi12d 717 . . . . . 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 6254 . . . . 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 468 . . . 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 24826 . . . . 5  |-  ( ph  ->  E! b  e.  P  ( ( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) )
62 riotacl 6266 . . . . 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 17 . . . 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 5954 . . 3  |-  ( ph  ->  ( M `  A
)  =  ( iota_ b  e.  P  ( ( A (midG `  G
) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) ) )
6564eqeq2d 2461 . 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 6298 . . . . . . 7  |-  ( b  =  B  ->  ( A (midG `  G )
b )  =  ( A (midG `  G
) B ) )
6867eleq1d 2513 . . . . . 6  |-  ( b  =  B  ->  (
( A (midG `  G ) b )  e.  D  <->  ( A
(midG `  G ) B )  e.  D
) )
69 oveq2 6298 . . . . . . . 8  |-  ( b  =  B  ->  ( A L b )  =  ( A L B ) )
7069breq2d 4414 . . . . . . 7  |-  ( b  =  B  ->  ( D (⟂G `  G )
( A L b )  <->  D (⟂G `  G
) ( A L B ) ) )
71 eqeq2 2462 . . . . . . 7  |-  ( b  =  B  ->  ( A  =  b  <->  A  =  B ) )
7270, 71orbi12d 716 . . . . . 6  |-  ( b  =  B  ->  (
( D (⟂G `  G
) ( A L b )  \/  A  =  b )  <->  ( D
(⟂G `  G ) ( A L B )  \/  A  =  B ) ) )
7368, 72anbi12d 717 . . . . 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 6274 . . . 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 667 . . 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 2458 . . 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 267 . 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 260 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 188    \/ wo 370    /\ wa 371    = wceq 1444    e. wcel 1887   E!wreu 2739   _Vcvv 3045   class class class wbr 4402    |-> cmpt 4461   ran crn 4835   ` cfv 5582   iota_crio 6251  (class class class)co 6290   2c2 10659   Basecbs 15121   distcds 15199  TarskiGcstrkg 24478  DimTarskiGcstrkgld 24482  Itvcitv 24484  LineGclng 24485  ⟂Gcperpg 24740  midGcmid 24814  lInvGclmi 24815
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  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  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-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  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-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  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-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  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-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12516  df-word 12664  df-concat 12666  df-s1 12667  df-s2 12944  df-s3 12945  df-trkgc 24496  df-trkgb 24497  df-trkgcb 24498  df-trkgld 24500  df-trkg 24501  df-cgrg 24556  df-leg 24628  df-mir 24698  df-rag 24739  df-perpg 24741  df-mid 24816  df-lmi 24817
This theorem is referenced by:  lmicom  24830  lmiinv  24834  lmimid  24836  lmiisolem  24838  hypcgrlem1  24841  hypcgrlem2  24842  lmiopp  24844  trgcopyeulem  24847
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