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Theorem mapdh6lem2N 32221
Description: Lemma for mapdh6N 32234. 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 2408 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 mapdh.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 31597 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 mapdhe6.y . . . . . . 7  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
87eldifad 3296 . . . . . 6  |-  ( ph  ->  Y  e.  V )
9 mapdh.v . . . . . . 7  |-  V  =  ( Base `  U
)
10 mapdh.n . . . . . . 7  |-  N  =  ( LSpan `  U )
119, 4, 10lspsncl 16012 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
126, 8, 11syl2anc 643 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
13 mapdhe6.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
1413eldifad 3296 . . . . . 6  |-  ( ph  ->  Z  e.  V )
159, 4, 10lspsncl 16012 . . . . . 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 2408 . . . . . 6  |-  ( LSSum `  U )  =  (
LSSum `  U )
184, 17lsmcl 16114 . . . . 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 3296 . . . . . . 7  |-  ( ph  ->  X  e.  V )
22 mapdh.p . . . . . . . . 9  |-  .+  =  ( +g  `  U )
239, 22lmodvacl 15923 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  Y  e.  V  /\  Z  e.  V )  ->  ( Y  .+  Z )  e.  V )
246, 8, 14, 23syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( Y  .+  Z
)  e.  V )
25 mapdh.s . . . . . . . 8  |-  .-  =  ( -g `  U )
269, 25lmodvsubcl 15948 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  ( Y  .+  Z )  e.  V )  ->  ( X  .-  ( Y  .+  Z ) )  e.  V )
276, 21, 24, 26syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( X  .-  ( Y  .+  Z ) )  e.  V )
289, 4, 10lspsncl 16012 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  ( Y  .+  Z ) )  e.  V )  ->  ( N `  { ( X  .-  ( Y  .+  Z ) ) } )  e.  ( LSubSp `  U ) )
296, 27, 28syl2anc 643 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  Z ) ) } )  e.  (
LSubSp `  U ) )
309, 4, 10lspsncl 16012 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
316, 21, 30syl2anc 643 . . . . 5  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
324, 17lsmcl 16114 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) (
LSSum `  U ) ( N `  { X } ) )  e.  ( LSubSp `  U )
)
341, 2, 3, 4, 5, 19, 33mapdin 32149 . . 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 2408 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
371, 2, 3, 4, 17, 35, 36, 5, 12, 16mapdlsm 32151 . . . . 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 31596 . . . . . . . . . . . . 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 16166 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  -.  Z  e.  ( N `  { X ,  Y } ) ) )
5150simpld 446 . . . . . . . . . . 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 32214 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5338, 52eqeltrrd 2483 . . . . . . . . 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 32215 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) ) )
5538, 54mpbid 202 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) )
5655simpld 446 . . . . . 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 16164 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Z } )  /\  -.  Y  e.  ( N `  { X ,  Z } ) ) )
5958simpld 446 . . . . . . . . . . 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 32214 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6157, 60eqeltrrd 2483 . . . . . . . . 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 32215 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Z >. )  =  E  <-> 
( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) ) )
6357, 62mpbid 202 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) )
6463simpld 446 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Z } ) )  =  ( J `  { E } ) )
6556, 64oveq12d 6062 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( J `  { G } ) (
LSSum `  C ) ( J `  { E } ) ) )
6637, 65eqtrd 2440 . . . 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 32151 . . . . 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 32220 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( J `  { ( F R ( G 
.+b  E ) ) } ) )
7069, 46oveq12d 6062 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  ( Y 
.+  Z ) ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { X }
) ) )  =  ( ( J `  { ( F R ( G  .+b  E
) ) } ) ( LSSum `  C )
( J `  { F } ) ) )
7167, 70eqtrd 2440 . . . 4  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) )  =  ( ( J `  { ( F R ( G  .+b  E
) ) } ) ( LSSum `  C )
( J `  { F } ) ) )
7266, 71ineq12d 3507 . . 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 2440 . 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 32202 . . 3  |-  ( ph  ->  ( N `  {
( Y  .+  Z
) } )  =  ( ( ( N `
 { Y }
) ( LSSum `  U
) ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  ( Y  .+  Z ) ) } ) ( LSSum `  U
) ( N `  { X } ) ) ) )
7574fveq2d 5695 . 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 32078 . . 3  |-  ( ph  ->  C  e.  LVec )
771, 2, 3, 9, 10, 35, 42, 44, 5, 45, 46, 21, 8, 53, 56, 14, 61, 64, 49mapdindp 32158 . . 3  |-  ( ph  ->  -.  F  e.  ( J `  { G ,  E } ) )
781, 2, 3, 9, 10, 35, 42, 44, 5, 53, 56, 8, 14, 61, 64, 48mapdncol 32157 . . 3  |-  ( ph  ->  ( J `  { G } )  =/=  ( J `  { E } ) )
791, 2, 3, 9, 10, 35, 42, 44, 5, 53, 56, 41, 39, 7mapdn0 32156 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { Q }
) )
801, 2, 3, 9, 10, 35, 42, 44, 5, 61, 64, 41, 39, 13mapdn0 32156 . . 3  |-  ( ph  ->  E  e.  ( D 
\  { Q }
) )
8142, 43, 39, 36, 44, 76, 45, 77, 78, 79, 80, 68baerlem5b 32202 . 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 2450 1  |-  ( ph  ->  ( M `  ( N `  { ( Y  .+  Z ) } ) )  =  ( J `  { ( G  .+b  E ) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2571   _Vcvv 2920    \ cdif 3281    i^i cin 3283   ifcif 3703   {csn 3778   {cpr 3779   <.cotp 3782    e. cmpt 4230   ` cfv 5417  (class class class)co 6044   1stc1st 6310   2ndc2nd 6311   iota_crio 6505   Basecbs 13428   +g cplusg 13488   0gc0g 13682   -gcsg 14647   LSSumclsm 15227   LModclmod 15909   LSubSpclss 15967   LSpanclspn 16006   HLchlt 29837   LHypclh 30470   DVecHcdvh 31565  LCDualclcd 32073  mapdcmpd 32111
This theorem is referenced by:  mapdh6aN  32222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-ot 3788  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-1st 6312  df-2nd 6313  df-tpos 6442  df-undef 6506  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-n0 10182  df-z 10243  df-uz 10449  df-fz 11004  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-sca 13504  df-vsca 13505  df-0g 13686  df-mre 13770  df-mrc 13771  df-acs 13773  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-mnd 14649  df-submnd 14698  df-grp 14771  df-minusg 14772  df-sbg 14773  df-subg 14900  df-cntz 15075  df-oppg 15101  df-lsm 15229  df-cmn 15373  df-abl 15374  df-mgp 15608  df-rng 15622  df-ur 15624  df-oppr 15687  df-dvdsr 15705  df-unit 15706  df-invr 15736  df-dvr 15747  df-drng 15796  df-lmod 15911  df-lss 15968  df-lsp 16007  df-lvec 16134  df-lsatoms 29463  df-lshyp 29464  df-lcv 29506  df-lfl 29545  df-lkr 29573  df-ldual 29611  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984  df-lplanes 29985  df-lvols 29986  df-lines 29987  df-psubsp 29989  df-pmap 29990  df-padd 30282  df-lhyp 30474  df-laut 30475  df-ldil 30590  df-ltrn 30591  df-trl 30645  df-tgrp 31229  df-tendo 31241  df-edring 31243  df-dveca 31489  df-disoa 31516  df-dvech 31566  df-dib 31626  df-dic 31660  df-dih 31716  df-doch 31835  df-djh 31882  df-lcdual 32074  df-mapd 32112
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