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Theorem mapdh6lem1N 35372
Description: Lemma for mapdh6N 35386. Part (6) in [Baer] p. 47, lines 16-18. (Contributed by NM, 13-Apr-2015.) (New usage is discouraged.)
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.  }
) )
mapdh.p  |-  .+  =  ( +g  `  U )
mapdh.a  |-  .+b  =  ( +g  `  C )
mapdhe6.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
mapdhe6.z  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
mapdhe6.xn  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
mapdh6.yz  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
mapdh6.fg  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
mapdh6.fe  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )
Assertion
Ref Expression
mapdh6lem1N  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( J `  { ( F R ( G 
.+b  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   
.+b , h    h, I    .+ , h, x
Allowed substitution hints:    ph( x)    C( x)   
.+b ( x)    Q( h)    U( x)    E( x)    H( x, h)    I( x)    K( x, h)    V( x, h)    W( x, h)

Proof of Theorem mapdh6lem1N
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 2471 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 mapdh.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 34749 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 mapdhcl.x . . . . . . . 8  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
87eldifad 3402 . . . . . . 7  |-  ( ph  ->  X  e.  V )
9 mapdhe6.y . . . . . . . 8  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
109eldifad 3402 . . . . . . 7  |-  ( ph  ->  Y  e.  V )
11 mapdh.v . . . . . . . 8  |-  V  =  ( Base `  U
)
12 mapdh.s . . . . . . . 8  |-  .-  =  ( -g `  U )
1311, 12lmodvsubcl 18211 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .-  Y )  e.  V )
146, 8, 10, 13syl3anc 1292 . . . . . 6  |-  ( ph  ->  ( X  .-  Y
)  e.  V )
15 mapdh.n . . . . . . 7  |-  N  =  ( LSpan `  U )
1611, 4, 15lspsncl 18278 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Y )  e.  V )  ->  ( N `  { ( X  .-  Y ) } )  e.  ( LSubSp `  U ) )
176, 14, 16syl2anc 673 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Y
) } )  e.  ( LSubSp `  U )
)
18 mapdhe6.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
1918eldifad 3402 . . . . . 6  |-  ( ph  ->  Z  e.  V )
2011, 4, 15lspsncl 18278 . . . . . 6  |-  ( ( U  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
216, 19, 20syl2anc 673 . . . . 5  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
22 eqid 2471 . . . . . 6  |-  ( LSSum `  U )  =  (
LSSum `  U )
234, 22lsmcl 18384 . . . . 5  |-  ( ( U  e.  LMod  /\  ( N `  { ( X  .-  Y ) } )  e.  ( LSubSp `  U )  /\  ( N `  { Z } )  e.  (
LSubSp `  U ) )  ->  ( ( N `
 { ( X 
.-  Y ) } ) ( LSSum `  U
) ( N `  { Z } ) )  e.  ( LSubSp `  U
) )
246, 17, 21, 23syl3anc 1292 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  e.  ( LSubSp `  U )
)
2511, 12lmodvsubcl 18211 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Z  e.  V )  ->  ( X  .-  Z )  e.  V )
266, 8, 19, 25syl3anc 1292 . . . . . 6  |-  ( ph  ->  ( X  .-  Z
)  e.  V )
2711, 4, 15lspsncl 18278 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Z )  e.  V )  ->  ( N `  { ( X  .-  Z ) } )  e.  ( LSubSp `  U ) )
286, 26, 27syl2anc 673 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Z
) } )  e.  ( LSubSp `  U )
)
2911, 4, 15lspsncl 18278 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
306, 10, 29syl2anc 673 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
314, 22lsmcl 18384 . . . . 5  |-  ( ( U  e.  LMod  /\  ( N `  { ( X  .-  Z ) } )  e.  ( LSubSp `  U )  /\  ( N `  { Y } )  e.  (
LSubSp `  U ) )  ->  ( ( N `
 { ( X 
.-  Z ) } ) ( LSSum `  U
) ( N `  { Y } ) )  e.  ( LSubSp `  U
) )
326, 28, 30, 31syl3anc 1292 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) )  e.  ( LSubSp `  U )
)
331, 2, 3, 4, 5, 24, 32mapdin 35301 . . 3  |-  ( ph  ->  ( M `  (
( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )  =  ( ( M `  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  i^i  ( M `  (
( N `  {
( X  .-  Z
) } ) (
LSSum `  U ) ( N `  { Y } ) ) ) ) )
34 mapdh.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
35 eqid 2471 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
361, 2, 3, 4, 22, 34, 35, 5, 17, 21mapdlsm 35303 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { Z } ) ) )  =  ( ( M `
 ( N `  { ( X  .-  Y ) } ) ) ( LSSum `  C
) ( M `  ( N `  { Z } ) ) ) )
371, 2, 3, 4, 22, 34, 35, 5, 28, 30mapdlsm 35303 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  Z
) } ) (
LSSum `  U ) ( N `  { Y } ) ) )  =  ( ( M `
 ( N `  { ( X  .-  Z ) } ) ) ( LSSum `  C
) ( M `  ( N `  { Y } ) ) ) )
3836, 37ineq12d 3626 . . . 4  |-  ( ph  ->  ( ( M `  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) ) )  i^i  ( M `  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )  =  ( ( ( M `  ( N `  { ( X  .-  Y ) } ) ) ( LSSum `  C ) ( M `
 ( N `  { Z } ) ) )  i^i  ( ( M `  ( N `
 { ( X 
.-  Z ) } ) ) ( LSSum `  C ) ( M `
 ( N `  { Y } ) ) ) ) )
39 mapdh6.fg . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
40 mapdh.q . . . . . . . . 9  |-  Q  =  ( 0g `  C
)
41 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 ) } ) ) ) ) )
42 mapdhc.o . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
43 mapdh.d . . . . . . . . 9  |-  D  =  ( Base `  C
)
44 mapdh.r . . . . . . . . 9  |-  R  =  ( -g `  C
)
45 mapdh.j . . . . . . . . 9  |-  J  =  ( LSpan `  C )
46 mapdhc.f . . . . . . . . 9  |-  ( ph  ->  F  e.  D )
47 mapdh.mn . . . . . . . . 9  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )
481, 3, 5dvhlvec 34748 . . . . . . . . . . . . 13  |-  ( ph  ->  U  e.  LVec )
49 mapdh6.yz . . . . . . . . . . . . 13  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
50 mapdhe6.xn . . . . . . . . . . . . 13  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
5111, 42, 15, 48, 10, 18, 8, 49, 50lspindp2 18436 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  -.  Z  e.  ( N `  { X ,  Y } ) ) )
5251simpld 466 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
5340, 41, 1, 2, 3, 11, 12, 42, 15, 34, 43, 44, 45, 5, 46, 47, 7, 10, 52mapdhcl 35366 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5439, 53eqeltrrd 2550 . . . . . . . . 9  |-  ( ph  ->  G  e.  D )
5540, 41, 1, 2, 3, 11, 12, 42, 15, 34, 43, 44, 45, 5, 46, 47, 7, 9, 54, 52mapdheq 35367 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) ) )
5639, 55mpbid 215 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) )
5756simprd 470 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  { ( F R G ) } ) )
58 mapdh6.fe . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )
5911, 42, 15, 48, 9, 19, 8, 49, 50lspindp1 18434 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Z } )  /\  -.  Y  e.  ( N `  { X ,  Z } ) ) )
6059simpld 466 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z } ) )
6140, 41, 1, 2, 3, 11, 12, 42, 15, 34, 43, 44, 45, 5, 46, 47, 7, 19, 60mapdhcl 35366 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6258, 61eqeltrrd 2550 . . . . . . . . 9  |-  ( ph  ->  E  e.  D )
6340, 41, 1, 2, 3, 11, 12, 42, 15, 34, 43, 44, 45, 5, 46, 47, 7, 18, 62, 60mapdheq 35367 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Z >. )  =  E  <-> 
( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) ) )
6458, 63mpbid 215 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) )
6564simpld 466 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Z } ) )  =  ( J `  { E } ) )
6657, 65oveq12d 6326 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  Y ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( J `  { ( F R G ) } ) ( LSSum `  C )
( J `  { E } ) ) )
6764simprd 470 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `  { ( F R E ) } ) )
6856simpld 466 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( J `  { G } ) )
6967, 68oveq12d 6326 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  Z ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  =  ( ( J `  { ( F R E ) } ) ( LSSum `  C )
( J `  { G } ) ) )
7066, 69ineq12d 3626 . . . 4  |-  ( ph  ->  ( ( ( M `
 ( N `  { ( X  .-  Y ) } ) ) ( LSSum `  C
) ( M `  ( N `  { Z } ) ) )  i^i  ( ( M `
 ( N `  { ( X  .-  Z ) } ) ) ( LSSum `  C
) ( M `  ( N `  { Y } ) ) ) )  =  ( ( ( J `  {
( F R G ) } ) (
LSSum `  C ) ( J `  { E } ) )  i^i  ( ( J `  { ( F R E ) } ) ( LSSum `  C )
( J `  { G } ) ) ) )
7138, 70eqtrd 2505 . . 3  |-  ( ph  ->  ( ( M `  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) ) )  i^i  ( M `  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )  =  ( ( ( J `  {
( F R G ) } ) (
LSSum `  C ) ( J `  { E } ) )  i^i  ( ( J `  { ( F R E ) } ) ( LSSum `  C )
( J `  { G } ) ) ) )
7233, 71eqtrd 2505 . 2  |-  ( ph  ->  ( M `  (
( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )  =  ( ( ( J `  {
( F R G ) } ) (
LSSum `  C ) ( J `  { E } ) )  i^i  ( ( J `  { ( F R E ) } ) ( LSSum `  C )
( J `  { G } ) ) ) )
73 mapdh.p . . . 4  |-  .+  =  ( +g  `  U )
7411, 12, 42, 22, 15, 48, 8, 50, 49, 9, 18, 73baerlem5a 35353 . . 3  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  Z ) ) } )  =  ( ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )
7574fveq2d 5883 . 2  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( M `  ( ( ( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) ) )
761, 34, 5lcdlvec 35230 . . 3  |-  ( ph  ->  C  e.  LVec )
771, 2, 3, 11, 15, 34, 43, 45, 5, 46, 47, 8, 10, 54, 68, 19, 62, 65, 50mapdindp 35310 . . 3  |-  ( ph  ->  -.  F  e.  ( J `  { G ,  E } ) )
781, 2, 3, 11, 15, 34, 43, 45, 5, 54, 68, 10, 19, 62, 65, 49mapdncol 35309 . . 3  |-  ( ph  ->  ( J `  { G } )  =/=  ( J `  { E } ) )
791, 2, 3, 11, 15, 34, 43, 45, 5, 54, 68, 42, 40, 9mapdn0 35308 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { Q }
) )
801, 2, 3, 11, 15, 34, 43, 45, 5, 62, 65, 42, 40, 18mapdn0 35308 . . 3  |-  ( ph  ->  E  e.  ( D 
\  { Q }
) )
81 mapdh.a . . 3  |-  .+b  =  ( +g  `  C )
8243, 44, 40, 35, 45, 76, 46, 77, 78, 79, 80, 81baerlem5a 35353 . 2  |-  ( ph  ->  ( J `  {
( F R ( G  .+b  E )
) } )  =  ( ( ( J `
 { ( F R G ) } ) ( LSSum `  C
) ( J `  { E } ) )  i^i  ( ( J `
 { ( F R E ) } ) ( LSSum `  C
) ( J `  { G } ) ) ) )
8372, 75, 823eqtr4d 2515 1  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( J `  { ( F R ( G 
.+b  E ) ) } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031    \ cdif 3387    i^i cin 3389   ifcif 3872   {csn 3959   {cpr 3961   <.cotp 3967    |-> cmpt 4454   ` cfv 5589   iota_crio 6269  (class class class)co 6308   1stc1st 6810   2ndc2nd 6811   Basecbs 15199   +g cplusg 15268   0gc0g 15416   -gcsg 16749   LSSumclsm 17364   LModclmod 18169   LSubSpclss 18233   LSpanclspn 18272   HLchlt 32987   LHypclh 33620   DVecHcdvh 34717  LCDualclcd 35225  mapdcmpd 35263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-riotaBAD 32589
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-ot 3968  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-tpos 6991  df-undef 7038  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-sca 15284  df-vsca 15285  df-0g 15418  df-mre 15570  df-mrc 15571  df-acs 15573  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-p1 16364  df-lat 16370  df-clat 16432  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-subg 16892  df-cntz 17049  df-oppg 17075  df-lsm 17366  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-dvr 17989  df-drng 18055  df-lmod 18171  df-lss 18234  df-lsp 18273  df-lvec 18404  df-lsatoms 32613  df-lshyp 32614  df-lcv 32656  df-lfl 32695  df-lkr 32723  df-ldual 32761  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-llines 33134  df-lplanes 33135  df-lvols 33136  df-lines 33137  df-psubsp 33139  df-pmap 33140  df-padd 33432  df-lhyp 33624  df-laut 33625  df-ldil 33740  df-ltrn 33741  df-trl 33796  df-tgrp 34381  df-tendo 34393  df-edring 34395  df-dveca 34641  df-disoa 34668  df-dvech 34718  df-dib 34778  df-dic 34812  df-dih 34868  df-doch 34987  df-djh 35034  df-lcdual 35226  df-mapd 35264
This theorem is referenced by:  mapdh6lem2N  35373  mapdh6aN  35374
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