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Theorem mapdpglem24 31820
Description: Lemma for mapdpg 31822. Existence part - consolidate hypotheses in mapdpglem23 31810. (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 3276 . . 3  |-  ( ph  ->  X  e.  V )
11 mapdpg.y . . . 4  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
1211eldifad 3276 . . 3  |-  ( ph  ->  Y  e.  V )
13 eqid 2388 . . 3  |-  ( LSSum `  C )  =  (
LSSum `  C )
14 mapdpg.j . . 3  |-  J  =  ( LSpan `  C )
151, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 14mapdpglem2 31789 . 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 2388 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
22 eqid 2388 . . . . 5  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
23 eqid 2388 . . . . 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 31791 . . . 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 2388 . . . . . . 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 3872 . . . . . . . . . 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 3872 . . . . . . . . . 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 2388 . . . . . . 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 31810 . . . . . 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 2780 . . . 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 2776 . 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 1649    e. wcel 1717    =/= wne 2551   E.wrex 2651    \ cdif 3261   {csn 3758   ` cfv 5395  (class class class)co 6021   Basecbs 13397  Scalarcsca 13460   .scvsca 13461   0gc0g 13651   -gcsg 14616   LSSumclsm 15196   invrcinvr 15704   LSpanclspn 15975   HLchlt 29466   LHypclh 30099   DVecHcdvh 31194  LCDualclcd 31702  mapdcmpd 31740
This theorem is referenced by:  mapdpg  31822
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-tpos 6416  df-undef 6480  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-sca 13473  df-vsca 13474  df-0g 13655  df-mre 13739  df-mrc 13740  df-acs 13742  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-p1 14397  df-lat 14403  df-clat 14465  df-mnd 14618  df-submnd 14667  df-grp 14740  df-minusg 14741  df-sbg 14742  df-subg 14869  df-cntz 15044  df-oppg 15070  df-lsm 15198  df-cmn 15342  df-abl 15343  df-mgp 15577  df-rng 15591  df-ur 15593  df-oppr 15656  df-dvdsr 15674  df-unit 15675  df-invr 15705  df-dvr 15716  df-drng 15765  df-lmod 15880  df-lss 15937  df-lsp 15976  df-lvec 16103  df-lsatoms 29092  df-lshyp 29093  df-lcv 29135  df-lfl 29174  df-lkr 29202  df-ldual 29240  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-llines 29613  df-lplanes 29614  df-lvols 29615  df-lines 29616  df-psubsp 29618  df-pmap 29619  df-padd 29911  df-lhyp 30103  df-laut 30104  df-ldil 30219  df-ltrn 30220  df-trl 30274  df-tgrp 30858  df-tendo 30870  df-edring 30872  df-dveca 31118  df-disoa 31145  df-dvech 31195  df-dib 31255  df-dic 31289  df-dih 31345  df-doch 31464  df-djh 31511  df-lcdual 31703  df-mapd 31741
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