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Theorem mapdpglem24 32439
Description: Lemma for mapdpg 32441. Existence part - consolidate hypotheses in mapdpglem23 32429. (Contributed by NM, 21-Mar-2015.)
Hypotheses
Ref Expression
mapdpg.h  |-  H  =  ( LHyp `  K
)
mapdpg.m  |-  M  =  ( (mapd `  K
) `  W )
mapdpg.u  |-  U  =  ( ( DVecH `  K
) `  W )
mapdpg.v  |-  V  =  ( Base `  U
)
mapdpg.s  |-  .-  =  ( -g `  U )
mapdpg.z  |-  .0.  =  ( 0g `  U )
mapdpg.n  |-  N  =  ( LSpan `  U )
mapdpg.c  |-  C  =  ( (LCDual `  K
) `  W )
mapdpg.f  |-  F  =  ( Base `  C
)
mapdpg.r  |-  R  =  ( -g `  C
)
mapdpg.j  |-  J  =  ( LSpan `  C )
mapdpg.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
mapdpg.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
mapdpg.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
mapdpg.g  |-  ( ph  ->  G  e.  F )
mapdpg.ne  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
mapdpg.e  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )
Assertion
Ref Expression
mapdpglem24  |-  ( ph  ->  E. h  e.  F  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) ) )
Distinct variable groups:    C, h    h, F    h, G    h, J    h, M    h, N    R, h    .- , h    U, h    h, X    h, Y
Allowed substitution hints:    ph( h)    H( h)    K( h)    V( h)    W( h)    .0. ( h)

Proof of Theorem mapdpglem24
Dummy variables  g 
t  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mapdpg.h . . 3  |-  H  =  ( LHyp `  K
)
2 mapdpg.m . . 3  |-  M  =  ( (mapd `  K
) `  W )
3 mapdpg.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
4 mapdpg.v . . 3  |-  V  =  ( Base `  U
)
5 mapdpg.s . . 3  |-  .-  =  ( -g `  U )
6 mapdpg.n . . 3  |-  N  =  ( LSpan `  U )
7 mapdpg.c . . 3  |-  C  =  ( (LCDual `  K
) `  W )
8 mapdpg.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
9 mapdpg.x . . . 4  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
109eldifad 3324 . . 3  |-  ( ph  ->  X  e.  V )
11 mapdpg.y . . . 4  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
1211eldifad 3324 . . 3  |-  ( ph  ->  Y  e.  V )
13 eqid 2435 . . 3  |-  ( LSSum `  C )  =  (
LSSum `  C )
14 mapdpg.j . . 3  |-  J  =  ( LSpan `  C )
151, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 14mapdpglem2 32408 . 2  |-  ( ph  ->  E. t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) ) ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )
1683ad2ant1 978 . . . . 5  |-  ( (
ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
17103ad2ant1 978 . . . . 5  |-  ( (
ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  ->  X  e.  V )
18123ad2ant1 978 . . . . 5  |-  ( (
ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  ->  Y  e.  V )
19 mapdpg.f . . . . 5  |-  F  =  ( Base `  C
)
20 simp2 958 . . . . 5  |-  ( (
ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  -> 
t  e.  ( ( M `  ( N `
 { X }
) ) ( LSSum `  C ) ( M `
 ( N `  { Y } ) ) ) )
21 eqid 2435 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
22 eqid 2435 . . . . 5  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
23 eqid 2435 . . . . 5  |-  ( .s
`  C )  =  ( .s `  C
)
24 mapdpg.r . . . . 5  |-  R  =  ( -g `  C
)
25 mapdpg.g . . . . . 6  |-  ( ph  ->  G  e.  F )
26253ad2ant1 978 . . . . 5  |-  ( (
ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  ->  G  e.  F )
27 mapdpg.e . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )
28273ad2ant1 978 . . . . 5  |-  ( (
ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  -> 
( M `  ( N `  { X } ) )  =  ( J `  { G } ) )
291, 2, 3, 4, 5, 6, 7, 16, 17, 18, 13, 14, 19, 20, 21, 22, 23, 24, 26, 28mapdpglem3 32410 . . . 4  |-  ( (
ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  ->  E. g  e.  ( Base `  (Scalar `  U
) ) E. z  e.  ( M `  ( N `  { Y } ) ) t  =  ( ( g ( .s `  C
) G ) R z ) )
30163ad2ant1 978 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  /\  ( g  e.  (
Base `  (Scalar `  U
) )  /\  z  e.  ( M `  ( N `  { Y } ) ) )  /\  t  =  ( ( g ( .s
`  C ) G ) R z ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
31173ad2ant1 978 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  /\  ( g  e.  (
Base `  (Scalar `  U
) )  /\  z  e.  ( M `  ( N `  { Y } ) ) )  /\  t  =  ( ( g ( .s
`  C ) G ) R z ) )  ->  X  e.  V )
32183ad2ant1 978 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  /\  ( g  e.  (
Base `  (Scalar `  U
) )  /\  z  e.  ( M `  ( N `  { Y } ) ) )  /\  t  =  ( ( g ( .s
`  C ) G ) R z ) )  ->  Y  e.  V )
33 simp12 988 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  /\  ( g  e.  (
Base `  (Scalar `  U
) )  /\  z  e.  ( M `  ( N `  { Y } ) ) )  /\  t  =  ( ( g ( .s
`  C ) G ) R z ) )  ->  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) ) )
34263ad2ant1 978 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  /\  ( g  e.  (
Base `  (Scalar `  U
) )  /\  z  e.  ( M `  ( N `  { Y } ) ) )  /\  t  =  ( ( g ( .s
`  C ) G ) R z ) )  ->  G  e.  F )
35283ad2ant1 978 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  /\  ( g  e.  (
Base `  (Scalar `  U
) )  /\  z  e.  ( M `  ( N `  { Y } ) ) )  /\  t  =  ( ( g ( .s
`  C ) G ) R z ) )  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )
36 mapdpg.z . . . . . . 7  |-  .0.  =  ( 0g `  U )
37 mapdpg.ne . . . . . . . . 9  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
38373ad2ant1 978 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  -> 
( N `  { X } )  =/=  ( N `  { Y } ) )
39383ad2ant1 978 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  /\  ( g  e.  (
Base `  (Scalar `  U
) )  /\  z  e.  ( M `  ( N `  { Y } ) ) )  /\  t  =  ( ( g ( .s
`  C ) G ) R z ) )  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
40 simp13 989 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  /\  ( g  e.  (
Base `  (Scalar `  U
) )  /\  z  e.  ( M `  ( N `  { Y } ) ) )  /\  t  =  ( ( g ( .s
`  C ) G ) R z ) )  ->  ( M `  ( N `  {
( X  .-  Y
) } ) )  =  ( J `  { t } ) )
41 eqid 2435 . . . . . . 7  |-  ( 0g
`  (Scalar `  U )
)  =  ( 0g
`  (Scalar `  U )
)
42 simp2l 983 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  /\  ( g  e.  (
Base `  (Scalar `  U
) )  /\  z  e.  ( M `  ( N `  { Y } ) ) )  /\  t  =  ( ( g ( .s
`  C ) G ) R z ) )  ->  g  e.  ( Base `  (Scalar `  U
) ) )
43 simp2r 984 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  /\  ( g  e.  (
Base `  (Scalar `  U
) )  /\  z  e.  ( M `  ( N `  { Y } ) ) )  /\  t  =  ( ( g ( .s
`  C ) G ) R z ) )  ->  z  e.  ( M `  ( N `
 { Y }
) ) )
44 simp3 959 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  /\  ( g  e.  (
Base `  (Scalar `  U
) )  /\  z  e.  ( M `  ( N `  { Y } ) ) )  /\  t  =  ( ( g ( .s
`  C ) G ) R z ) )  ->  t  =  ( ( g ( .s `  C ) G ) R z ) )
45 eldifsni 3920 . . . . . . . . . 10  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  =/=  .0.  )
469, 45syl 16 . . . . . . . . 9  |-  ( ph  ->  X  =/=  .0.  )
47463ad2ant1 978 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  ->  X  =/=  .0.  )
48473ad2ant1 978 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  /\  ( g  e.  (
Base `  (Scalar `  U
) )  /\  z  e.  ( M `  ( N `  { Y } ) ) )  /\  t  =  ( ( g ( .s
`  C ) G ) R z ) )  ->  X  =/=  .0.  )
49 eldifsni 3920 . . . . . . . . . 10  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  =/=  .0.  )
5011, 49syl 16 . . . . . . . . 9  |-  ( ph  ->  Y  =/=  .0.  )
51503ad2ant1 978 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  ->  Y  =/=  .0.  )
52513ad2ant1 978 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  /\  ( g  e.  (
Base `  (Scalar `  U
) )  /\  z  e.  ( M `  ( N `  { Y } ) ) )  /\  t  =  ( ( g ( .s
`  C ) G ) R z ) )  ->  Y  =/=  .0.  )
53 eqid 2435 . . . . . . 7  |-  ( ( ( invr `  (Scalar `  U ) ) `  g ) ( .s
`  C ) z )  =  ( ( ( invr `  (Scalar `  U ) ) `  g ) ( .s
`  C ) z )
541, 2, 3, 4, 5, 6, 7, 30, 31, 32, 13, 14, 19, 33, 21, 22, 23, 24, 34, 35, 36, 39, 40, 41, 42, 43, 44, 48, 52, 53mapdpglem23 32429 . . . . . 6  |-  ( ( ( ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  /\  ( g  e.  (
Base `  (Scalar `  U
) )  /\  z  e.  ( M `  ( N `  { Y } ) ) )  /\  t  =  ( ( g ( .s
`  C ) G ) R z ) )  ->  E. h  e.  F  ( ( M `  ( N `  { Y } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( X  .-  Y
) } ) )  =  ( J `  { ( G R h ) } ) ) )
55543exp 1152 . . . . 5  |-  ( (
ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  -> 
( ( g  e.  ( Base `  (Scalar `  U ) )  /\  z  e.  ( M `  ( N `  { Y } ) ) )  ->  ( t  =  ( ( g ( .s `  C ) G ) R z )  ->  E. h  e.  F  ( ( M `  ( N `  { Y } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( X  .-  Y
) } ) )  =  ( J `  { ( G R h ) } ) ) ) ) )
5655rexlimdvv 2828 . . . 4  |-  ( (
ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  -> 
( E. g  e.  ( Base `  (Scalar `  U ) ) E. z  e.  ( M `
 ( N `  { Y } ) ) t  =  ( ( g ( .s `  C ) G ) R z )  ->  E. h  e.  F  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) ) ) )
5729, 56mpd 15 . . 3  |-  ( (
ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  ->  E. h  e.  F  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) ) )
5857rexlimdv3a 2824 . 2  |-  ( ph  ->  ( E. t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) ) ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } )  ->  E. h  e.  F  ( ( M `  ( N `  { Y } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( X  .-  Y
) } ) )  =  ( J `  { ( G R h ) } ) ) ) )
5915, 58mpd 15 1  |-  ( ph  ->  E. h  e.  F  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    \ cdif 3309   {csn 3806   ` cfv 5446  (class class class)co 6073   Basecbs 13461  Scalarcsca 13524   .scvsca 13525   0gc0g 13715   -gcsg 14680   LSSumclsm 15260   invrcinvr 15768   LSpanclspn 16039   HLchlt 30085   LHypclh 30718   DVecHcdvh 31813  LCDualclcd 32321  mapdcmpd 32359
This theorem is referenced by:  mapdpg  32441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-undef 6535  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-0g 13719  df-mre 13803  df-mrc 13804  df-acs 13806  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-cntz 15108  df-oppg 15134  df-lsm 15262  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-dvr 15780  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167  df-lsatoms 29711  df-lshyp 29712  df-lcv 29754  df-lfl 29793  df-lkr 29821  df-ldual 29859  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893  df-tgrp 31477  df-tendo 31489  df-edring 31491  df-dveca 31737  df-disoa 31764  df-dvech 31814  df-dib 31874  df-dic 31908  df-dih 31964  df-doch 32083  df-djh 32130  df-lcdual 32322  df-mapd 32360
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