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Theorem mapdheq4lem 36745
Description: Lemma for mapdheq4 36746. 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 2467 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 mapdh.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 36124 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 mapdhe4.y . . . . . . 7  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
87eldifad 3488 . . . . . 6  |-  ( ph  ->  Y  e.  V )
9 mapdh.v . . . . . . 7  |-  V  =  ( Base `  U
)
10 mapdh.n . . . . . . 7  |-  N  =  ( LSpan `  U )
119, 4, 10lspsncl 17435 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
126, 8, 11syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
13 mapdhe.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
1413eldifad 3488 . . . . . 6  |-  ( ph  ->  Z  e.  V )
159, 4, 10lspsncl 17435 . . . . . 6  |-  ( ( U  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
166, 14, 15syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
17 eqid 2467 . . . . . 6  |-  ( LSSum `  U )  =  (
LSSum `  U )
184, 17lsmcl 17541 . . . . 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 1228 . . . 4  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  e.  ( LSubSp `  U )
)
20 mapdhcl.x . . . . . . . 8  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
2120eldifad 3488 . . . . . . 7  |-  ( ph  ->  X  e.  V )
22 mapdh.s . . . . . . . 8  |-  .-  =  ( -g `  U )
239, 22lmodvsubcl 17367 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .-  Y )  e.  V )
246, 21, 8, 23syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( X  .-  Y
)  e.  V )
259, 4, 10lspsncl 17435 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Y )  e.  V )  ->  ( N `  { ( X  .-  Y ) } )  e.  ( LSubSp `  U ) )
266, 24, 25syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Y
) } )  e.  ( LSubSp `  U )
)
279, 22lmodvsubcl 17367 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Z  e.  V )  ->  ( X  .-  Z )  e.  V )
286, 21, 14, 27syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( X  .-  Z
)  e.  V )
299, 4, 10lspsncl 17435 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Z )  e.  V )  ->  ( N `  { ( X  .-  Z ) } )  e.  ( LSubSp `  U ) )
306, 28, 29syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Z
) } )  e.  ( LSubSp `  U )
)
314, 17lsmcl 17541 . . . . 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 1228 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  {
( X  .-  Z
) } ) )  e.  ( LSubSp `  U
) )
331, 2, 3, 4, 5, 19, 32mapdin 36676 . . 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 2467 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
361, 2, 3, 4, 17, 34, 35, 5, 12, 16mapdlsm 36678 . . . . 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 36123 . . . . . . . . . . . . 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 17593 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  -.  Z  e.  ( N `  { X ,  Y } ) ) )
5049simpld 459 . . . . . . . . . . 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 36741 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5237, 51eqeltrrd 2556 . . . . . . . . 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 36742 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) ) )
5437, 53mpbid 210 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) )
5554simpld 459 . . . . . 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 17591 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Z } )  /\  -.  Y  e.  ( N `  { X ,  Z } ) ) )
5857simpld 459 . . . . . . . . . . 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 36741 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6056, 59eqeltrrd 2556 . . . . . . . . 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 36742 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Z >. )  =  E  <-> 
( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) ) )
6256, 61mpbid 210 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) )
6362simpld 459 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Z } ) )  =  ( J `  { E } ) )
6455, 63oveq12d 6303 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( J `  { G } ) (
LSSum `  C ) ( J `  { E } ) ) )
6536, 64eqtrd 2508 . . . 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 36678 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { ( X  .-  Z ) } ) ) )  =  ( ( M `
 ( N `  { ( X  .-  Y ) } ) ) ( LSSum `  C
) ( M `  ( N `  { ( X  .-  Z ) } ) ) ) )
6754simprd 463 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  { ( F R G ) } ) )
6862simprd 463 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `  { ( F R E ) } ) )
6967, 68oveq12d 6303 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  Y ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { ( X 
.-  Z ) } ) ) )  =  ( ( J `  { ( F R G ) } ) ( LSSum `  C )
( J `  {
( F R E ) } ) ) )
7066, 69eqtrd 2508 . . . 4  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { ( X  .-  Z ) } ) ) )  =  ( ( J `
 { ( F R G ) } ) ( LSSum `  C
) ( J `  { ( F R E ) } ) ) )
7165, 70ineq12d 3701 . . 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 2508 . 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 36727 . . 3  |-  ( ph  ->  ( N `  {
( Y  .-  Z
) } )  =  ( ( ( N `
 { Y }
) ( LSSum `  U
) ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  Y ) } ) ( LSSum `  U
) ( N `  { ( X  .-  Z ) } ) ) ) )
7473fveq2d 5870 . 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 2467 . . 3  |-  ( 0g
`  C )  =  ( 0g `  C
)
761, 34, 5lcdlvec 36605 . . 3  |-  ( ph  ->  C  e.  LVec )
771, 2, 3, 9, 10, 34, 41, 43, 5, 44, 45, 21, 8, 52, 55, 14, 60, 63, 48mapdindp 36685 . . 3  |-  ( ph  ->  -.  F  e.  ( J `  { G ,  E } ) )
781, 2, 3, 9, 10, 34, 41, 43, 5, 52, 55, 8, 14, 60, 63, 47mapdncol 36684 . . 3  |-  ( ph  ->  ( J `  { G } )  =/=  ( J `  { E } ) )
791, 2, 3, 9, 10, 34, 41, 43, 5, 52, 55, 40, 75, 7mapdn0 36683 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { ( 0g
`  C ) } ) )
801, 2, 3, 9, 10, 34, 41, 43, 5, 60, 63, 40, 75, 13mapdn0 36683 . . 3  |-  ( ph  ->  E  e.  ( D 
\  { ( 0g
`  C ) } ) )
8141, 42, 75, 35, 43, 76, 44, 77, 78, 79, 80baerlem3 36727 . 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 2518 1  |-  ( ph  ->  ( M `  ( N `  { ( Y  .-  Z ) } ) )  =  ( J `  { ( G R E ) } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    \ cdif 3473    i^i cin 3475   ifcif 3939   {csn 4027   {cpr 4029   <.cotp 4035    |-> cmpt 4505   ` cfv 5588   iota_crio 6245  (class class class)co 6285   1stc1st 6783   2ndc2nd 6784   Basecbs 14493   0gc0g 14698   -gcsg 15733   LSSumclsm 16469   LModclmod 17324   LSubSpclss 17390   LSpanclspn 17429   HLchlt 34364   LHypclh 34997   DVecHcdvh 36092  LCDualclcd 36600  mapdcmpd 36638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-riotaBAD 33973
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6956  df-undef 7003  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-sca 14574  df-vsca 14575  df-0g 14700  df-mre 14844  df-mrc 14845  df-acs 14847  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-p1 15530  df-lat 15536  df-clat 15598  df-mnd 15735  df-submnd 15790  df-grp 15871  df-minusg 15872  df-sbg 15873  df-subg 16012  df-cntz 16169  df-oppg 16195  df-lsm 16471  df-cmn 16615  df-abl 16616  df-mgp 16956  df-ur 16968  df-rng 17014  df-oppr 17085  df-dvdsr 17103  df-unit 17104  df-invr 17134  df-dvr 17145  df-drng 17210  df-lmod 17326  df-lss 17391  df-lsp 17430  df-lvec 17561  df-lsatoms 33990  df-lshyp 33991  df-lcv 34033  df-lfl 34072  df-lkr 34100  df-ldual 34138  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-llines 34511  df-lplanes 34512  df-lvols 34513  df-lines 34514  df-psubsp 34516  df-pmap 34517  df-padd 34809  df-lhyp 35001  df-laut 35002  df-ldil 35117  df-ltrn 35118  df-trl 35172  df-tgrp 35756  df-tendo 35768  df-edring 35770  df-dveca 36016  df-disoa 36043  df-dvech 36093  df-dib 36153  df-dic 36187  df-dih 36243  df-doch 36362  df-djh 36409  df-lcdual 36601  df-mapd 36639
This theorem is referenced by:  mapdheq4  36746
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