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Theorem mapdheq4lem 32368
Description: Lemma for mapdheq4 32369. Part (4) in [Baer] p. 46. (Contributed by NM, 12-Apr-2015.)
Hypotheses
Ref Expression
mapdh.q  |-  Q  =  ( 0g `  C
)
mapdh.i  |-  I  =  ( x  e.  _V  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) )
mapdh.h  |-  H  =  ( LHyp `  K
)
mapdh.m  |-  M  =  ( (mapd `  K
) `  W )
mapdh.u  |-  U  =  ( ( DVecH `  K
) `  W )
mapdh.v  |-  V  =  ( Base `  U
)
mapdh.s  |-  .-  =  ( -g `  U )
mapdhc.o  |-  .0.  =  ( 0g `  U )
mapdh.n  |-  N  =  ( LSpan `  U )
mapdh.c  |-  C  =  ( (LCDual `  K
) `  W )
mapdh.d  |-  D  =  ( Base `  C
)
mapdh.r  |-  R  =  ( -g `  C
)
mapdh.j  |-  J  =  ( LSpan `  C )
mapdh.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
mapdhc.f  |-  ( ph  ->  F  e.  D )
mapdh.mn  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )
mapdhcl.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
mapdhe4.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
mapdhe.z  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
mapdh.xn  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
mapdh.yz  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
mapdh.eg  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
mapdh.ee  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )
Assertion
Ref Expression
mapdheq4lem  |-  ( ph  ->  ( M `  ( N `  { ( Y  .-  Z ) } ) )  =  ( J `  { ( G R E ) } ) )
Distinct variable groups:    x, D, h    h, F, x    x, J    x, M    x, N    x,  .0.    x, Q    x, R    x, 
.-    h, X, x    h, Y, x    ph, h    .0. , h    C, h    D, h   
h, J    h, M    h, N    R, h    U, h    .- , h    h, G, x   
h, E    h, Z, x
Allowed substitution hints:    ph( x)    C( x)    Q( h)    U( x)    E( x)    H( x, h)    I( x, h)    K( x, h)    V( x, h)    W( x, h)

Proof of Theorem mapdheq4lem
StepHypRef Expression
1 mapdh.h . . . 4  |-  H  =  ( LHyp `  K
)
2 mapdh.m . . . 4  |-  M  =  ( (mapd `  K
) `  W )
3 mapdh.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
4 eqid 2435 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 mapdh.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 31747 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 mapdhe4.y . . . . . . 7  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
87eldifad 3324 . . . . . 6  |-  ( ph  ->  Y  e.  V )
9 mapdh.v . . . . . . 7  |-  V  =  ( Base `  U
)
10 mapdh.n . . . . . . 7  |-  N  =  ( LSpan `  U )
119, 4, 10lspsncl 16041 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
126, 8, 11syl2anc 643 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
13 mapdhe.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
1413eldifad 3324 . . . . . 6  |-  ( ph  ->  Z  e.  V )
159, 4, 10lspsncl 16041 . . . . . 6  |-  ( ( U  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
166, 14, 15syl2anc 643 . . . . 5  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
17 eqid 2435 . . . . . 6  |-  ( LSSum `  U )  =  (
LSSum `  U )
184, 17lsmcl 16143 . . . . 5  |-  ( ( U  e.  LMod  /\  ( N `  { Y } )  e.  (
LSubSp `  U )  /\  ( N `  { Z } )  e.  (
LSubSp `  U ) )  ->  ( ( N `
 { Y }
) ( LSSum `  U
) ( N `  { Z } ) )  e.  ( LSubSp `  U
) )
196, 12, 16, 18syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  e.  ( LSubSp `  U )
)
20 mapdhcl.x . . . . . . . 8  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
2120eldifad 3324 . . . . . . 7  |-  ( ph  ->  X  e.  V )
22 mapdh.s . . . . . . . 8  |-  .-  =  ( -g `  U )
239, 22lmodvsubcl 15977 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .-  Y )  e.  V )
246, 21, 8, 23syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( X  .-  Y
)  e.  V )
259, 4, 10lspsncl 16041 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Y )  e.  V )  ->  ( N `  { ( X  .-  Y ) } )  e.  ( LSubSp `  U ) )
266, 24, 25syl2anc 643 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Y
) } )  e.  ( LSubSp `  U )
)
279, 22lmodvsubcl 15977 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Z  e.  V )  ->  ( X  .-  Z )  e.  V )
286, 21, 14, 27syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( X  .-  Z
)  e.  V )
299, 4, 10lspsncl 16041 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Z )  e.  V )  ->  ( N `  { ( X  .-  Z ) } )  e.  ( LSubSp `  U ) )
306, 28, 29syl2anc 643 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Z
) } )  e.  ( LSubSp `  U )
)
314, 17lsmcl 16143 . . . . 5  |-  ( ( U  e.  LMod  /\  ( N `  { ( X  .-  Y ) } )  e.  ( LSubSp `  U )  /\  ( N `  { ( X  .-  Z ) } )  e.  ( LSubSp `  U ) )  -> 
( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  {
( X  .-  Z
) } ) )  e.  ( LSubSp `  U
) )
326, 26, 30, 31syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  {
( X  .-  Z
) } ) )  e.  ( LSubSp `  U
) )
331, 2, 3, 4, 5, 19, 32mapdin 32299 . . 3  |-  ( ph  ->  ( M `  (
( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  {
( X  .-  Z
) } ) ) ) )  =  ( ( M `  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  i^i  ( M `  (
( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { ( X  .-  Z ) } ) ) ) ) )
34 mapdh.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
35 eqid 2435 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
361, 2, 3, 4, 17, 34, 35, 5, 12, 16mapdlsm 32301 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  =  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) ) )
37 mapdh.eg . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
38 mapdh.q . . . . . . . . 9  |-  Q  =  ( 0g `  C
)
39 mapdh.i . . . . . . . . 9  |-  I  =  ( x  e.  _V  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) )
40 mapdhc.o . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
41 mapdh.d . . . . . . . . 9  |-  D  =  ( Base `  C
)
42 mapdh.r . . . . . . . . 9  |-  R  =  ( -g `  C
)
43 mapdh.j . . . . . . . . 9  |-  J  =  ( LSpan `  C )
44 mapdhc.f . . . . . . . . 9  |-  ( ph  ->  F  e.  D )
45 mapdh.mn . . . . . . . . 9  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )
461, 3, 5dvhlvec 31746 . . . . . . . . . . . . 13  |-  ( ph  ->  U  e.  LVec )
47 mapdh.yz . . . . . . . . . . . . 13  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
48 mapdh.xn . . . . . . . . . . . . 13  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
499, 40, 10, 46, 8, 13, 21, 47, 48lspindp2 16195 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  -.  Z  e.  ( N `  { X ,  Y } ) ) )
5049simpld 446 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
5138, 39, 1, 2, 3, 9, 22, 40, 10, 34, 41, 42, 43, 5, 44, 45, 20, 8, 50mapdhcl 32364 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5237, 51eqeltrrd 2510 . . . . . . . . 9  |-  ( ph  ->  G  e.  D )
5338, 39, 1, 2, 3, 9, 22, 40, 10, 34, 41, 42, 43, 5, 44, 45, 20, 7, 52, 50mapdheq 32365 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) ) )
5437, 53mpbid 202 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) )
5554simpld 446 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( J `  { G } ) )
56 mapdh.ee . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )
579, 40, 10, 46, 7, 14, 21, 47, 48lspindp1 16193 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Z } )  /\  -.  Y  e.  ( N `  { X ,  Z } ) ) )
5857simpld 446 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z } ) )
5938, 39, 1, 2, 3, 9, 22, 40, 10, 34, 41, 42, 43, 5, 44, 45, 20, 14, 58mapdhcl 32364 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6056, 59eqeltrrd 2510 . . . . . . . . 9  |-  ( ph  ->  E  e.  D )
6138, 39, 1, 2, 3, 9, 22, 40, 10, 34, 41, 42, 43, 5, 44, 45, 20, 13, 60, 58mapdheq 32365 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Z >. )  =  E  <-> 
( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) ) )
6256, 61mpbid 202 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) )
6362simpld 446 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Z } ) )  =  ( J `  { E } ) )
6455, 63oveq12d 6090 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( J `  { G } ) (
LSSum `  C ) ( J `  { E } ) ) )
6536, 64eqtrd 2467 . . . 4  |-  ( ph  ->  ( M `  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  =  ( ( J `  { G } ) (
LSSum `  C ) ( J `  { E } ) ) )
661, 2, 3, 4, 17, 34, 35, 5, 26, 30mapdlsm 32301 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { ( X  .-  Z ) } ) ) )  =  ( ( M `
 ( N `  { ( X  .-  Y ) } ) ) ( LSSum `  C
) ( M `  ( N `  { ( X  .-  Z ) } ) ) ) )
6754simprd 450 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  { ( F R G ) } ) )
6862simprd 450 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `  { ( F R E ) } ) )
6967, 68oveq12d 6090 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  Y ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { ( X 
.-  Z ) } ) ) )  =  ( ( J `  { ( F R G ) } ) ( LSSum `  C )
( J `  {
( F R E ) } ) ) )
7066, 69eqtrd 2467 . . . 4  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { ( X  .-  Z ) } ) ) )  =  ( ( J `
 { ( F R G ) } ) ( LSSum `  C
) ( J `  { ( F R E ) } ) ) )
7165, 70ineq12d 3535 . . 3  |-  ( ph  ->  ( ( M `  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) ) )  i^i  ( M `  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  {
( X  .-  Z
) } ) ) ) )  =  ( ( ( J `  { G } ) (
LSSum `  C ) ( J `  { E } ) )  i^i  ( ( J `  { ( F R G ) } ) ( LSSum `  C )
( J `  {
( F R E ) } ) ) ) )
7233, 71eqtrd 2467 . 2  |-  ( ph  ->  ( M `  (
( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  {
( X  .-  Z
) } ) ) ) )  =  ( ( ( J `  { G } ) (
LSSum `  C ) ( J `  { E } ) )  i^i  ( ( J `  { ( F R G ) } ) ( LSSum `  C )
( J `  {
( F R E ) } ) ) ) )
739, 22, 40, 17, 10, 46, 21, 48, 47, 7, 13baerlem3 32350 . . 3  |-  ( ph  ->  ( N `  {
( Y  .-  Z
) } )  =  ( ( ( N `
 { Y }
) ( LSSum `  U
) ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  Y ) } ) ( LSSum `  U
) ( N `  { ( X  .-  Z ) } ) ) ) )
7473fveq2d 5723 . 2  |-  ( ph  ->  ( M `  ( N `  { ( Y  .-  Z ) } ) )  =  ( M `  ( ( ( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) )  i^i  (
( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { ( X  .-  Z ) } ) ) ) ) )
75 eqid 2435 . . 3  |-  ( 0g
`  C )  =  ( 0g `  C
)
761, 34, 5lcdlvec 32228 . . 3  |-  ( ph  ->  C  e.  LVec )
771, 2, 3, 9, 10, 34, 41, 43, 5, 44, 45, 21, 8, 52, 55, 14, 60, 63, 48mapdindp 32308 . . 3  |-  ( ph  ->  -.  F  e.  ( J `  { G ,  E } ) )
781, 2, 3, 9, 10, 34, 41, 43, 5, 52, 55, 8, 14, 60, 63, 47mapdncol 32307 . . 3  |-  ( ph  ->  ( J `  { G } )  =/=  ( J `  { E } ) )
791, 2, 3, 9, 10, 34, 41, 43, 5, 52, 55, 40, 75, 7mapdn0 32306 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { ( 0g
`  C ) } ) )
801, 2, 3, 9, 10, 34, 41, 43, 5, 60, 63, 40, 75, 13mapdn0 32306 . . 3  |-  ( ph  ->  E  e.  ( D 
\  { ( 0g
`  C ) } ) )
8141, 42, 75, 35, 43, 76, 44, 77, 78, 79, 80baerlem3 32350 . 2  |-  ( ph  ->  ( J `  {
( G R E ) } )  =  ( ( ( J `
 { G }
) ( LSSum `  C
) ( J `  { E } ) )  i^i  ( ( J `
 { ( F R G ) } ) ( LSSum `  C
) ( J `  { ( F R E ) } ) ) ) )
8272, 74, 813eqtr4d 2477 1  |-  ( ph  ->  ( M `  ( N `  { ( Y  .-  Z ) } ) )  =  ( J `  { ( G R E ) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    \ cdif 3309    i^i cin 3311   ifcif 3731   {csn 3806   {cpr 3807   <.cotp 3810    e. cmpt 4258   ` cfv 5445  (class class class)co 6072   1stc1st 6338   2ndc2nd 6339   iota_crio 6533   Basecbs 13457   0gc0g 13711   -gcsg 14676   LSSumclsm 15256   LModclmod 15938   LSubSpclss 15996   LSpanclspn 16035   HLchlt 29987   LHypclh 30620   DVecHcdvh 31715  LCDualclcd 32223  mapdcmpd 32261
This theorem is referenced by:  mapdheq4  32369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-ot 3816  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-tpos 6470  df-undef 6534  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-er 6896  df-map 7011  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-n0 10211  df-z 10272  df-uz 10478  df-fz 11033  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-sca 13533  df-vsca 13534  df-0g 13715  df-mre 13799  df-mrc 13800  df-acs 13802  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-mnd 14678  df-submnd 14727  df-grp 14800  df-minusg 14801  df-sbg 14802  df-subg 14929  df-cntz 15104  df-oppg 15130  df-lsm 15258  df-cmn 15402  df-abl 15403  df-mgp 15637  df-rng 15651  df-ur 15653  df-oppr 15716  df-dvdsr 15734  df-unit 15735  df-invr 15765  df-dvr 15776  df-drng 15825  df-lmod 15940  df-lss 15997  df-lsp 16036  df-lvec 16163  df-lsatoms 29613  df-lshyp 29614  df-lcv 29656  df-lfl 29695  df-lkr 29723  df-ldual 29761  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135  df-lvols 30136  df-lines 30137  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624  df-laut 30625  df-ldil 30740  df-ltrn 30741  df-trl 30795  df-tgrp 31379  df-tendo 31391  df-edring 31393  df-dveca 31639  df-disoa 31666  df-dvech 31716  df-dib 31776  df-dic 31810  df-dih 31866  df-doch 31985  df-djh 32032  df-lcdual 32224  df-mapd 32262
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