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Theorem mapdh6lem2N 36406
Description: Lemma for mapdh6N 36419. Part (6) in [Baer] p. 47, lines 20-22. (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
mapdh6lem2N  |-  ( ph  ->  ( M `  ( N `  { ( Y  .+  Z ) } ) )  =  ( J `  { ( 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 mapdh6lem2N
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 2460 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 mapdh.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 35782 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 mapdhe6.y . . . . . . 7  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
87eldifad 3481 . . . . . 6  |-  ( ph  ->  Y  e.  V )
9 mapdh.v . . . . . . 7  |-  V  =  ( Base `  U
)
10 mapdh.n . . . . . . 7  |-  N  =  ( LSpan `  U )
119, 4, 10lspsncl 17399 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
126, 8, 11syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
13 mapdhe6.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
1413eldifad 3481 . . . . . 6  |-  ( ph  ->  Z  e.  V )
159, 4, 10lspsncl 17399 . . . . . 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 2460 . . . . . 6  |-  ( LSSum `  U )  =  (
LSSum `  U )
184, 17lsmcl 17505 . . . . 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 1223 . . . 4  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  e.  ( LSubSp `  U )
)
20 mapdhcl.x . . . . . . . 8  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
2120eldifad 3481 . . . . . . 7  |-  ( ph  ->  X  e.  V )
22 mapdh.p . . . . . . . . 9  |-  .+  =  ( +g  `  U )
239, 22lmodvacl 17302 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  Y  e.  V  /\  Z  e.  V )  ->  ( Y  .+  Z )  e.  V )
246, 8, 14, 23syl3anc 1223 . . . . . . 7  |-  ( ph  ->  ( Y  .+  Z
)  e.  V )
25 mapdh.s . . . . . . . 8  |-  .-  =  ( -g `  U )
269, 25lmodvsubcl 17331 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  ( Y  .+  Z )  e.  V )  ->  ( X  .-  ( Y  .+  Z ) )  e.  V )
276, 21, 24, 26syl3anc 1223 . . . . . 6  |-  ( ph  ->  ( X  .-  ( Y  .+  Z ) )  e.  V )
289, 4, 10lspsncl 17399 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  ( Y  .+  Z ) )  e.  V )  ->  ( N `  { ( X  .-  ( Y  .+  Z ) ) } )  e.  ( LSubSp `  U ) )
296, 27, 28syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  Z ) ) } )  e.  (
LSubSp `  U ) )
309, 4, 10lspsncl 17399 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
316, 21, 30syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
324, 17lsmcl 17505 . . . . 5  |-  ( ( U  e.  LMod  /\  ( N `  { ( X  .-  ( Y  .+  Z ) ) } )  e.  ( LSubSp `  U )  /\  ( N `  { X } )  e.  (
LSubSp `  U ) )  ->  ( ( N `
 { ( X 
.-  ( Y  .+  Z ) ) } ) ( LSSum `  U
) ( N `  { X } ) )  e.  ( LSubSp `  U
) )
336, 29, 31, 32syl3anc 1223 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) (
LSSum `  U ) ( N `  { X } ) )  e.  ( LSubSp `  U )
)
341, 2, 3, 4, 5, 19, 33mapdin 36334 . . 3  |-  ( ph  ->  ( M `  (
( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) (
LSSum `  U ) ( N `  { X } ) ) ) )  =  ( ( M `  ( ( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  i^i  ( M `  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) ) ) )
35 mapdh.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
36 eqid 2460 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
371, 2, 3, 4, 17, 35, 36, 5, 12, 16mapdlsm 36336 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  =  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) ) )
38 mapdh6.fg . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
39 mapdh.q . . . . . . . . 9  |-  Q  =  ( 0g `  C
)
40 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 ) } ) ) ) ) )
41 mapdhc.o . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
42 mapdh.d . . . . . . . . 9  |-  D  =  ( Base `  C
)
43 mapdh.r . . . . . . . . 9  |-  R  =  ( -g `  C
)
44 mapdh.j . . . . . . . . 9  |-  J  =  ( LSpan `  C )
45 mapdhc.f . . . . . . . . 9  |-  ( ph  ->  F  e.  D )
46 mapdh.mn . . . . . . . . 9  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )
471, 3, 5dvhlvec 35781 . . . . . . . . . . . . 13  |-  ( ph  ->  U  e.  LVec )
48 mapdh6.yz . . . . . . . . . . . . 13  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
49 mapdhe6.xn . . . . . . . . . . . . 13  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
509, 41, 10, 47, 8, 13, 21, 48, 49lspindp2 17557 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  -.  Z  e.  ( N `  { X ,  Y } ) ) )
5150simpld 459 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
5239, 40, 1, 2, 3, 9, 25, 41, 10, 35, 42, 43, 44, 5, 45, 46, 20, 8, 51mapdhcl 36399 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5338, 52eqeltrrd 2549 . . . . . . . . 9  |-  ( ph  ->  G  e.  D )
5439, 40, 1, 2, 3, 9, 25, 41, 10, 35, 42, 43, 44, 5, 45, 46, 20, 7, 53, 51mapdheq 36400 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) ) )
5538, 54mpbid 210 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) )
5655simpld 459 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( J `  { G } ) )
57 mapdh6.fe . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )
589, 41, 10, 47, 7, 14, 21, 48, 49lspindp1 17555 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Z } )  /\  -.  Y  e.  ( N `  { X ,  Z } ) ) )
5958simpld 459 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z } ) )
6039, 40, 1, 2, 3, 9, 25, 41, 10, 35, 42, 43, 44, 5, 45, 46, 20, 14, 59mapdhcl 36399 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6157, 60eqeltrrd 2549 . . . . . . . . 9  |-  ( ph  ->  E  e.  D )
6239, 40, 1, 2, 3, 9, 25, 41, 10, 35, 42, 43, 44, 5, 45, 46, 20, 13, 61, 59mapdheq 36400 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Z >. )  =  E  <-> 
( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) ) )
6357, 62mpbid 210 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) )
6463simpld 459 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Z } ) )  =  ( J `  { E } ) )
6556, 64oveq12d 6293 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( J `  { G } ) (
LSSum `  C ) ( J `  { E } ) ) )
6637, 65eqtrd 2501 . . . 4  |-  ( ph  ->  ( M `  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  =  ( ( J `  { G } ) (
LSSum `  C ) ( J `  { E } ) ) )
671, 2, 3, 4, 17, 35, 36, 5, 29, 31mapdlsm 36336 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) )  =  ( ( M `  ( N `  { ( X  .-  ( Y 
.+  Z ) ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { X }
) ) ) )
68 mapdh.a . . . . . . 7  |-  .+b  =  ( +g  `  C )
6939, 40, 1, 2, 3, 9, 25, 41, 10, 35, 42, 43, 44, 5, 45, 46, 20, 22, 68, 7, 13, 49, 48, 38, 57mapdh6lem1N 36405 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( J `  { ( F R ( G 
.+b  E ) ) } ) )
7069, 46oveq12d 6293 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  ( Y 
.+  Z ) ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { X }
) ) )  =  ( ( J `  { ( F R ( G  .+b  E
) ) } ) ( LSSum `  C )
( J `  { F } ) ) )
7167, 70eqtrd 2501 . . . 4  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) )  =  ( ( J `  { ( F R ( G  .+b  E
) ) } ) ( LSSum `  C )
( J `  { F } ) ) )
7266, 71ineq12d 3694 . . 3  |-  ( ph  ->  ( ( M `  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) ) )  i^i  ( M `  ( ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) (
LSSum `  U ) ( N `  { X } ) ) ) )  =  ( ( ( J `  { G } ) ( LSSum `  C ) ( J `
 { E }
) )  i^i  (
( J `  {
( F R ( G  .+b  E )
) } ) (
LSSum `  C ) ( J `  { F } ) ) ) )
7334, 72eqtrd 2501 . 2  |-  ( ph  ->  ( M `  (
( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) (
LSSum `  U ) ( N `  { X } ) ) ) )  =  ( ( ( J `  { G } ) ( LSSum `  C ) ( J `
 { E }
) )  i^i  (
( J `  {
( F R ( G  .+b  E )
) } ) (
LSSum `  C ) ( J `  { F } ) ) ) )
749, 25, 41, 17, 10, 47, 21, 49, 48, 7, 13, 22baerlem5b 36387 . . 3  |-  ( ph  ->  ( N `  {
( Y  .+  Z
) } )  =  ( ( ( N `
 { Y }
) ( LSSum `  U
) ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  ( Y  .+  Z ) ) } ) ( LSSum `  U
) ( N `  { X } ) ) ) )
7574fveq2d 5861 . 2  |-  ( ph  ->  ( M `  ( N `  { ( Y  .+  Z ) } ) )  =  ( M `  ( ( ( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) )  i^i  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) ) ) )
761, 35, 5lcdlvec 36263 . . 3  |-  ( ph  ->  C  e.  LVec )
771, 2, 3, 9, 10, 35, 42, 44, 5, 45, 46, 21, 8, 53, 56, 14, 61, 64, 49mapdindp 36343 . . 3  |-  ( ph  ->  -.  F  e.  ( J `  { G ,  E } ) )
781, 2, 3, 9, 10, 35, 42, 44, 5, 53, 56, 8, 14, 61, 64, 48mapdncol 36342 . . 3  |-  ( ph  ->  ( J `  { G } )  =/=  ( J `  { E } ) )
791, 2, 3, 9, 10, 35, 42, 44, 5, 53, 56, 41, 39, 7mapdn0 36341 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { Q }
) )
801, 2, 3, 9, 10, 35, 42, 44, 5, 61, 64, 41, 39, 13mapdn0 36341 . . 3  |-  ( ph  ->  E  e.  ( D 
\  { Q }
) )
8142, 43, 39, 36, 44, 76, 45, 77, 78, 79, 80, 68baerlem5b 36387 . 2  |-  ( ph  ->  ( J `  {
( G  .+b  E
) } )  =  ( ( ( J `
 { G }
) ( LSSum `  C
) ( J `  { E } ) )  i^i  ( ( J `
 { ( F R ( G  .+b  E ) ) } ) ( LSSum `  C )
( J `  { F } ) ) ) )
8273, 75, 813eqtr4d 2511 1  |-  ( ph  ->  ( M `  ( N `  { ( Y  .+  Z ) } ) )  =  ( J `  { ( G  .+b  E ) } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   _Vcvv 3106    \ cdif 3466    i^i cin 3468   ifcif 3932   {csn 4020   {cpr 4022   <.cotp 4028    |-> cmpt 4498   ` cfv 5579   iota_crio 6235  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   Basecbs 14479   +g cplusg 14544   0gc0g 14684   -gcsg 15719   LSSumclsm 16443   LModclmod 17288   LSubSpclss 17354   LSpanclspn 17393   HLchlt 34022   LHypclh 34655   DVecHcdvh 35750  LCDualclcd 36258  mapdcmpd 36296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-riotaBAD 33631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-ot 4029  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-tpos 6945  df-undef 6992  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-sca 14560  df-vsca 14561  df-0g 14686  df-mre 14830  df-mrc 14831  df-acs 14833  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-mnd 15721  df-submnd 15771  df-grp 15851  df-minusg 15852  df-sbg 15853  df-subg 15986  df-cntz 16143  df-oppg 16169  df-lsm 16445  df-cmn 16589  df-abl 16590  df-mgp 16925  df-ur 16937  df-rng 16981  df-oppr 17049  df-dvdsr 17067  df-unit 17068  df-invr 17098  df-dvr 17109  df-drng 17174  df-lmod 17290  df-lss 17355  df-lsp 17394  df-lvec 17525  df-lsatoms 33648  df-lshyp 33649  df-lcv 33691  df-lfl 33730  df-lkr 33758  df-ldual 33796  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169  df-lplanes 34170  df-lvols 34171  df-lines 34172  df-psubsp 34174  df-pmap 34175  df-padd 34467  df-lhyp 34659  df-laut 34660  df-ldil 34775  df-ltrn 34776  df-trl 34830  df-tgrp 35414  df-tendo 35426  df-edring 35428  df-dveca 35674  df-disoa 35701  df-dvech 35751  df-dib 35811  df-dic 35845  df-dih 35901  df-doch 36020  df-djh 36067  df-lcdual 36259  df-mapd 36297
This theorem is referenced by:  mapdh6aN  36407
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