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Theorem mapdheq4lem 35052
Description: Lemma for mapdheq4 35053. 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 2420 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 mapdh.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 34431 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 mapdhe4.y . . . . . . 7  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
87eldifad 3445 . . . . . 6  |-  ( ph  ->  Y  e.  V )
9 mapdh.v . . . . . . 7  |-  V  =  ( Base `  U
)
10 mapdh.n . . . . . . 7  |-  N  =  ( LSpan `  U )
119, 4, 10lspsncl 18141 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
126, 8, 11syl2anc 665 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
13 mapdhe.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
1413eldifad 3445 . . . . . 6  |-  ( ph  ->  Z  e.  V )
159, 4, 10lspsncl 18141 . . . . . 6  |-  ( ( U  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
166, 14, 15syl2anc 665 . . . . 5  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
17 eqid 2420 . . . . . 6  |-  ( LSSum `  U )  =  (
LSSum `  U )
184, 17lsmcl 18247 . . . . 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 1264 . . . 4  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  e.  ( LSubSp `  U )
)
20 mapdhcl.x . . . . . . . 8  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
2120eldifad 3445 . . . . . . 7  |-  ( ph  ->  X  e.  V )
22 mapdh.s . . . . . . . 8  |-  .-  =  ( -g `  U )
239, 22lmodvsubcl 18074 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .-  Y )  e.  V )
246, 21, 8, 23syl3anc 1264 . . . . . 6  |-  ( ph  ->  ( X  .-  Y
)  e.  V )
259, 4, 10lspsncl 18141 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Y )  e.  V )  ->  ( N `  { ( X  .-  Y ) } )  e.  ( LSubSp `  U ) )
266, 24, 25syl2anc 665 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Y
) } )  e.  ( LSubSp `  U )
)
279, 22lmodvsubcl 18074 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Z  e.  V )  ->  ( X  .-  Z )  e.  V )
286, 21, 14, 27syl3anc 1264 . . . . . 6  |-  ( ph  ->  ( X  .-  Z
)  e.  V )
299, 4, 10lspsncl 18141 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Z )  e.  V )  ->  ( N `  { ( X  .-  Z ) } )  e.  ( LSubSp `  U ) )
306, 28, 29syl2anc 665 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Z
) } )  e.  ( LSubSp `  U )
)
314, 17lsmcl 18247 . . . . 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 1264 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  {
( X  .-  Z
) } ) )  e.  ( LSubSp `  U
) )
331, 2, 3, 4, 5, 19, 32mapdin 34983 . . 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 2420 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
361, 2, 3, 4, 17, 34, 35, 5, 12, 16mapdlsm 34985 . . . . 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 34430 . . . . . . . . . . . . 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 18299 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  -.  Z  e.  ( N `  { X ,  Y } ) ) )
5049simpld 460 . . . . . . . . . . 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 35048 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5237, 51eqeltrrd 2509 . . . . . . . . 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 35049 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) ) )
5437, 53mpbid 213 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) )
5554simpld 460 . . . . . 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 18297 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Z } )  /\  -.  Y  e.  ( N `  { X ,  Z } ) ) )
5857simpld 460 . . . . . . . . . . 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 35048 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6056, 59eqeltrrd 2509 . . . . . . . . 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 35049 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Z >. )  =  E  <-> 
( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) ) )
6256, 61mpbid 213 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) )
6362simpld 460 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Z } ) )  =  ( J `  { E } ) )
6455, 63oveq12d 6314 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( J `  { G } ) (
LSSum `  C ) ( J `  { E } ) ) )
6536, 64eqtrd 2461 . . . 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 34985 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { ( X  .-  Z ) } ) ) )  =  ( ( M `
 ( N `  { ( X  .-  Y ) } ) ) ( LSSum `  C
) ( M `  ( N `  { ( X  .-  Z ) } ) ) ) )
6754simprd 464 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  { ( F R G ) } ) )
6862simprd 464 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `  { ( F R E ) } ) )
6967, 68oveq12d 6314 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  Y ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { ( X 
.-  Z ) } ) ) )  =  ( ( J `  { ( F R G ) } ) ( LSSum `  C )
( J `  {
( F R E ) } ) ) )
7066, 69eqtrd 2461 . . . 4  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { ( X  .-  Z ) } ) ) )  =  ( ( J `
 { ( F R G ) } ) ( LSSum `  C
) ( J `  { ( F R E ) } ) ) )
7165, 70ineq12d 3662 . . 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 2461 . 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 35034 . . 3  |-  ( ph  ->  ( N `  {
( Y  .-  Z
) } )  =  ( ( ( N `
 { Y }
) ( LSSum `  U
) ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  Y ) } ) ( LSSum `  U
) ( N `  { ( X  .-  Z ) } ) ) ) )
7473fveq2d 5876 . 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 2420 . . 3  |-  ( 0g
`  C )  =  ( 0g `  C
)
761, 34, 5lcdlvec 34912 . . 3  |-  ( ph  ->  C  e.  LVec )
771, 2, 3, 9, 10, 34, 41, 43, 5, 44, 45, 21, 8, 52, 55, 14, 60, 63, 48mapdindp 34992 . . 3  |-  ( ph  ->  -.  F  e.  ( J `  { G ,  E } ) )
781, 2, 3, 9, 10, 34, 41, 43, 5, 52, 55, 8, 14, 60, 63, 47mapdncol 34991 . . 3  |-  ( ph  ->  ( J `  { G } )  =/=  ( J `  { E } ) )
791, 2, 3, 9, 10, 34, 41, 43, 5, 52, 55, 40, 75, 7mapdn0 34990 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { ( 0g
`  C ) } ) )
801, 2, 3, 9, 10, 34, 41, 43, 5, 60, 63, 40, 75, 13mapdn0 34990 . . 3  |-  ( ph  ->  E  e.  ( D 
\  { ( 0g
`  C ) } ) )
8141, 42, 75, 35, 43, 76, 44, 77, 78, 79, 80baerlem3 35034 . 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 2471 1  |-  ( ph  ->  ( M `  ( N `  { ( Y  .-  Z ) } ) )  =  ( J `  { ( G R E ) } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616   _Vcvv 3078    \ cdif 3430    i^i cin 3432   ifcif 3906   {csn 3993   {cpr 3995   <.cotp 4001    |-> cmpt 4475   ` cfv 5592   iota_crio 6257  (class class class)co 6296   1stc1st 6796   2ndc2nd 6797   Basecbs 15081   0gc0g 15298   -gcsg 16623   LSSumclsm 17227   LModclmod 18032   LSubSpclss 18096   LSpanclspn 18135   HLchlt 32669   LHypclh 33302   DVecHcdvh 34399  LCDualclcd 34907  mapdcmpd 34945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-riotaBAD 32278
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-ot 4002  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-tpos 6972  df-undef 7019  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-n0 10859  df-z 10927  df-uz 11149  df-fz 11772  df-struct 15083  df-ndx 15084  df-slot 15085  df-base 15086  df-sets 15087  df-ress 15088  df-plusg 15163  df-mulr 15164  df-sca 15166  df-vsca 15167  df-0g 15300  df-mre 15444  df-mrc 15445  df-acs 15447  df-preset 16125  df-poset 16143  df-plt 16156  df-lub 16172  df-glb 16173  df-join 16174  df-meet 16175  df-p0 16237  df-p1 16238  df-lat 16244  df-clat 16306  df-mgm 16440  df-sgrp 16479  df-mnd 16489  df-submnd 16535  df-grp 16625  df-minusg 16626  df-sbg 16627  df-subg 16766  df-cntz 16923  df-oppg 16949  df-lsm 17229  df-cmn 17373  df-abl 17374  df-mgp 17665  df-ur 17677  df-ring 17723  df-oppr 17792  df-dvdsr 17810  df-unit 17811  df-invr 17841  df-dvr 17852  df-drng 17918  df-lmod 18034  df-lss 18097  df-lsp 18136  df-lvec 18267  df-lsatoms 32295  df-lshyp 32296  df-lcv 32338  df-lfl 32377  df-lkr 32405  df-ldual 32443  df-oposet 32495  df-ol 32497  df-oml 32498  df-covers 32585  df-ats 32586  df-atl 32617  df-cvlat 32641  df-hlat 32670  df-llines 32816  df-lplanes 32817  df-lvols 32818  df-lines 32819  df-psubsp 32821  df-pmap 32822  df-padd 33114  df-lhyp 33306  df-laut 33307  df-ldil 33422  df-ltrn 33423  df-trl 33478  df-tgrp 34063  df-tendo 34075  df-edring 34077  df-dveca 34323  df-disoa 34350  df-dvech 34400  df-dib 34460  df-dic 34494  df-dih 34550  df-doch 34669  df-djh 34716  df-lcdual 34908  df-mapd 34946
This theorem is referenced by:  mapdheq4  35053
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