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Theorem mapdpglem24 36378
Description: Lemma for mapdpg 36380. Existence part - consolidate hypotheses in mapdpglem23 36368. (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 3483 . . 3  |-  ( ph  ->  X  e.  V )
11 mapdpg.y . . . 4  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
1211eldifad 3483 . . 3  |-  ( ph  ->  Y  e.  V )
13 eqid 2462 . . 3  |-  ( LSSum `  C )  =  (
LSSum `  C )
14 mapdpg.j . . 3  |-  J  =  ( LSpan `  C )
151, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 14mapdpglem2 36347 . 2  |-  ( ph  ->  E. t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) ) ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )
1683ad2ant1 1012 . . . . 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 1012 . . . . 5  |-  ( (
ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  ->  X  e.  V )
18123ad2ant1 1012 . . . . 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 992 . . . . 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 2462 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
22 eqid 2462 . . . . 5  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
23 eqid 2462 . . . . 5  |-  ( .s
`  C )  =  ( .s `  C
)
24 mapdpg.r . . . . 5  |-  R  =  ( -g `  C
)
25 mapdpg.g . . . . . 6  |-  ( ph  ->  G  e.  F )
26253ad2ant1 1012 . . . . 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 1012 . . . . 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 36349 . . . 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 1012 . . . . . . 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 1012 . . . . . . 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 1012 . . . . . . 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 1022 . . . . . . 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 1012 . . . . . . 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 1012 . . . . . . 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 1012 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  -> 
( N `  { X } )  =/=  ( N `  { Y } ) )
39383ad2ant1 1012 . . . . . . 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 1023 . . . . . . 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 2462 . . . . . . 7  |-  ( 0g
`  (Scalar `  U )
)  =  ( 0g
`  (Scalar `  U )
)
42 simp2l 1017 . . . . . . 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 1018 . . . . . . 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 993 . . . . . . 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 4148 . . . . . . . . . 10  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  =/=  .0.  )
469, 45syl 16 . . . . . . . . 9  |-  ( ph  ->  X  =/=  .0.  )
47463ad2ant1 1012 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  ->  X  =/=  .0.  )
48473ad2ant1 1012 . . . . . . 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 4148 . . . . . . . . . 10  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  =/=  .0.  )
5011, 49syl 16 . . . . . . . . 9  |-  ( ph  ->  Y  =/=  .0.  )
51503ad2ant1 1012 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( M `  ( N `  { X } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { t } ) )  ->  Y  =/=  .0.  )
52513ad2ant1 1012 . . . . . . 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 2462 . . . . . . 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 36368 . . . . . 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 1190 . . . . 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 2956 . . . 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 2952 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657   E.wrex 2810    \ cdif 3468   {csn 4022   ` cfv 5581  (class class class)co 6277   Basecbs 14481  Scalarcsca 14549   .scvsca 14550   0gc0g 14686   -gcsg 15721   LSSumclsm 16445   invrcinvr 17099   LSpanclspn 17395   HLchlt 34024   LHypclh 34657   DVecHcdvh 35752  LCDualclcd 36260  mapdcmpd 36298
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 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-riotaBAD 33633
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 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-tpos 6947  df-undef 6994  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-sca 14562  df-vsca 14563  df-0g 14688  df-mre 14832  df-mrc 14833  df-acs 14835  df-poset 15424  df-plt 15436  df-lub 15452  df-glb 15453  df-join 15454  df-meet 15455  df-p0 15517  df-p1 15518  df-lat 15524  df-clat 15586  df-mnd 15723  df-submnd 15773  df-grp 15853  df-minusg 15854  df-sbg 15855  df-subg 15988  df-cntz 16145  df-oppg 16171  df-lsm 16447  df-cmn 16591  df-abl 16592  df-mgp 16927  df-ur 16939  df-rng 16983  df-oppr 17051  df-dvdsr 17069  df-unit 17070  df-invr 17100  df-dvr 17111  df-drng 17176  df-lmod 17292  df-lss 17357  df-lsp 17396  df-lvec 17527  df-lsatoms 33650  df-lshyp 33651  df-lcv 33693  df-lfl 33732  df-lkr 33760  df-ldual 33798  df-oposet 33850  df-ol 33852  df-oml 33853  df-covers 33940  df-ats 33941  df-atl 33972  df-cvlat 33996  df-hlat 34025  df-llines 34171  df-lplanes 34172  df-lvols 34173  df-lines 34174  df-psubsp 34176  df-pmap 34177  df-padd 34469  df-lhyp 34661  df-laut 34662  df-ldil 34777  df-ltrn 34778  df-trl 34832  df-tgrp 35416  df-tendo 35428  df-edring 35430  df-dveca 35676  df-disoa 35703  df-dvech 35753  df-dib 35813  df-dic 35847  df-dih 35903  df-doch 36022  df-djh 36069  df-lcdual 36261  df-mapd 36299
This theorem is referenced by:  mapdpg  36380
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