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Theorem lspsneq 16122
Description: Equal spans of singletons must have proportional vectors. See lspsnss2 16009 for comparable span version. TODO: can proof be shortened? (Contributed by NM, 21-Mar-2015.)
Hypotheses
Ref Expression
lspsneq.v  |-  V  =  ( Base `  W
)
lspsneq.s  |-  S  =  (Scalar `  W )
lspsneq.k  |-  K  =  ( Base `  S
)
lspsneq.o  |-  .0.  =  ( 0g `  S )
lspsneq.t  |-  .x.  =  ( .s `  W )
lspsneq.n  |-  N  =  ( LSpan `  W )
lspsneq.w  |-  ( ph  ->  W  e.  LVec )
lspsneq.x  |-  ( ph  ->  X  e.  V )
lspsneq.y  |-  ( ph  ->  Y  e.  V )
Assertion
Ref Expression
lspsneq  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  <->  E. k  e.  ( K  \  {  .0.  } ) X  =  ( k  .x.  Y
) ) )
Distinct variable groups:    k, K    .0. , k    .x. , k    k, X    k, Y
Allowed substitution hints:    ph( k)    S( k)    N( k)    V( k)    W( k)

Proof of Theorem lspsneq
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 lspsneq.w . . . . . . . . . 10  |-  ( ph  ->  W  e.  LVec )
2 lveclmod 16106 . . . . . . . . . 10  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 16 . . . . . . . . 9  |-  ( ph  ->  W  e.  LMod )
4 lspsneq.s . . . . . . . . . 10  |-  S  =  (Scalar `  W )
54lmodrng 15886 . . . . . . . . 9  |-  ( W  e.  LMod  ->  S  e. 
Ring )
6 lspsneq.k . . . . . . . . . 10  |-  K  =  ( Base `  S
)
7 eqid 2388 . . . . . . . . . 10  |-  ( 1r
`  S )  =  ( 1r `  S
)
86, 7rngidcl 15612 . . . . . . . . 9  |-  ( S  e.  Ring  ->  ( 1r
`  S )  e.  K )
93, 5, 83syl 19 . . . . . . . 8  |-  ( ph  ->  ( 1r `  S
)  e.  K )
104lvecdrng 16105 . . . . . . . . 9  |-  ( W  e.  LVec  ->  S  e.  DivRing )
11 lspsneq.o . . . . . . . . . 10  |-  .0.  =  ( 0g `  S )
1211, 7drngunz 15778 . . . . . . . . 9  |-  ( S  e.  DivRing  ->  ( 1r `  S )  =/=  .0.  )
131, 10, 123syl 19 . . . . . . . 8  |-  ( ph  ->  ( 1r `  S
)  =/=  .0.  )
14 eldifsn 3871 . . . . . . . 8  |-  ( ( 1r `  S )  e.  ( K  \  {  .0.  } )  <->  ( ( 1r `  S )  e.  K  /\  ( 1r
`  S )  =/= 
.0.  ) )
159, 13, 14sylanbrc 646 . . . . . . 7  |-  ( ph  ->  ( 1r `  S
)  e.  ( K 
\  {  .0.  }
) )
1615ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =  ( 0g `  W ) )  -> 
( 1r `  S
)  e.  ( K 
\  {  .0.  }
) )
17 lspsneq.v . . . . . . . . . . 11  |-  V  =  ( Base `  W
)
18 eqid 2388 . . . . . . . . . . 11  |-  ( 0g
`  W )  =  ( 0g `  W
)
1917, 18lmod0vcl 15907 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  ( 0g
`  W )  e.  V )
201, 2, 193syl 19 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  W
)  e.  V )
21 lspsneq.t . . . . . . . . . 10  |-  .x.  =  ( .s `  W )
2217, 4, 21, 7lmodvs1 15906 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  ( 0g `  W )  e.  V )  ->  (
( 1r `  S
)  .x.  ( 0g `  W ) )  =  ( 0g `  W
) )
233, 20, 22syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( ( 1r `  S )  .x.  ( 0g `  W ) )  =  ( 0g `  W ) )
2423ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =  ( 0g `  W ) )  -> 
( ( 1r `  S )  .x.  ( 0g `  W ) )  =  ( 0g `  W ) )
25 oveq2 6029 . . . . . . . 8  |-  ( Y  =  ( 0g `  W )  ->  (
( 1r `  S
)  .x.  Y )  =  ( ( 1r
`  S )  .x.  ( 0g `  W ) ) )
2625adantl 453 . . . . . . 7  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =  ( 0g `  W ) )  -> 
( ( 1r `  S )  .x.  Y
)  =  ( ( 1r `  S ) 
.x.  ( 0g `  W ) ) )
27 lspsneq.n . . . . . . . . 9  |-  N  =  ( LSpan `  W )
283adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  W  e.  LMod )
29 lspsneq.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  V )
3029adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  X  e.  V
)
31 lspsneq.y . . . . . . . . . 10  |-  ( ph  ->  Y  e.  V )
3231adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  Y  e.  V
)
33 simpr 448 . . . . . . . . 9  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  ( N `  { X } )  =  ( N `  { Y } ) )
3417, 18, 27, 28, 30, 32, 33lspsneq0b 16017 . . . . . . . 8  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  ( X  =  ( 0g `  W
)  <->  Y  =  ( 0g `  W ) ) )
3534biimpar 472 . . . . . . 7  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =  ( 0g `  W ) )  ->  X  =  ( 0g `  W ) )
3624, 26, 353eqtr4rd 2431 . . . . . 6  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =  ( 0g `  W ) )  ->  X  =  ( ( 1r `  S )  .x.  Y ) )
37 oveq1 6028 . . . . . . . 8  |-  ( j  =  ( 1r `  S )  ->  (
j  .x.  Y )  =  ( ( 1r
`  S )  .x.  Y ) )
3837eqeq2d 2399 . . . . . . 7  |-  ( j  =  ( 1r `  S )  ->  ( X  =  ( j  .x.  Y )  <->  X  =  ( ( 1r `  S )  .x.  Y
) ) )
3938rspcev 2996 . . . . . 6  |-  ( ( ( 1r `  S
)  e.  ( K 
\  {  .0.  }
)  /\  X  =  ( ( 1r `  S )  .x.  Y
) )  ->  E. j  e.  ( K  \  {  .0.  } ) X  =  ( j  .x.  Y
) )
4016, 36, 39syl2anc 643 . . . . 5  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =  ( 0g `  W ) )  ->  E. j  e.  ( K  \  {  .0.  }
) X  =  ( j  .x.  Y ) )
41 eqimss 3344 . . . . . . . . . 10  |-  ( ( N `  { X } )  =  ( N `  { Y } )  ->  ( N `  { X } )  C_  ( N `  { Y } ) )
4241adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  ( N `  { X } )  C_  ( N `  { Y } ) )
43 eqid 2388 . . . . . . . . . 10  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
4417, 43, 27lspsncl 15981 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
453, 31, 44syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
4645adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  ( N `  { Y } )  e.  ( LSubSp `  W )
)
4717, 43, 27, 28, 46, 30lspsnel5 15999 . . . . . . . . 9  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  ( X  e.  ( N `  { Y } )  <->  ( N `  { X } ) 
C_  ( N `  { Y } ) ) )
4842, 47mpbird 224 . . . . . . . 8  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  X  e.  ( N `  { Y } ) )
494, 6, 17, 21, 27lspsnel 16007 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  ( X  e.  ( N `  { Y } )  <->  E. j  e.  K  X  =  ( j  .x.  Y ) ) )
5028, 32, 49syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  ( X  e.  ( N `  { Y } )  <->  E. j  e.  K  X  =  ( j  .x.  Y
) ) )
5148, 50mpbid 202 . . . . . . 7  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  E. j  e.  K  X  =  ( j  .x.  Y ) )
5251adantr 452 . . . . . 6  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  ->  E. j  e.  K  X  =  ( j  .x.  Y ) )
53 simprl 733 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  j  e.  K )
54 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( j  e.  K  /\  X  =  ( j  .x.  Y ) )  ->  X  =  ( j  .x.  Y ) )
5554adantl 453 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  X  =  ( j  .x.  Y ) )
5634biimpd 199 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  ( X  =  ( 0g `  W
)  ->  Y  =  ( 0g `  W ) ) )
5756necon3d 2589 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  ( Y  =/=  ( 0g `  W
)  ->  X  =/=  ( 0g `  W ) ) )
5857imp 419 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  ->  X  =/=  ( 0g `  W ) )
5958adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  X  =/=  ( 0g `  W
) )
6055, 59eqnetrrd 2571 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  (
j  .x.  Y )  =/=  ( 0g `  W
) )
611adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  W  e.  LVec )
6261ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  W  e.  LVec )
6332ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  Y  e.  V )
6417, 21, 4, 6, 11, 18, 62, 53, 63lvecvsn0 16109 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  (
( j  .x.  Y
)  =/=  ( 0g
`  W )  <->  ( j  =/=  .0.  /\  Y  =/=  ( 0g `  W
) ) ) )
6560, 64mpbid 202 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  (
j  =/=  .0.  /\  Y  =/=  ( 0g `  W ) ) )
6665simpld 446 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  j  =/=  .0.  )
67 eldifsn 3871 . . . . . . . . . 10  |-  ( j  e.  ( K  \  {  .0.  } )  <->  ( j  e.  K  /\  j  =/=  .0.  ) )
6853, 66, 67sylanbrc 646 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  j  e.  ( K  \  {  .0.  } ) )
6968, 55jca 519 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  (
j  e.  ( K 
\  {  .0.  }
)  /\  X  =  ( j  .x.  Y
) ) )
7069ex 424 . . . . . . 7  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  -> 
( ( j  e.  K  /\  X  =  ( j  .x.  Y
) )  ->  (
j  e.  ( K 
\  {  .0.  }
)  /\  X  =  ( j  .x.  Y
) ) ) )
7170reximdv2 2759 . . . . . 6  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  -> 
( E. j  e.  K  X  =  ( j  .x.  Y )  ->  E. j  e.  ( K  \  {  .0.  } ) X  =  ( j  .x.  Y ) ) )
7252, 71mpd 15 . . . . 5  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  ->  E. j  e.  ( K  \  {  .0.  }
) X  =  ( j  .x.  Y ) )
7340, 72pm2.61dane 2629 . . . 4  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  E. j  e.  ( K  \  {  .0.  } ) X  =  ( j  .x.  Y ) )
7473ex 424 . . 3  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  ->  E. j  e.  ( K  \  {  .0.  } ) X  =  ( j  .x.  Y
) ) )
751adantr 452 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( K  \  {  .0.  } ) )  ->  W  e.  LVec )
76 eldifi 3413 . . . . . . . 8  |-  ( j  e.  ( K  \  {  .0.  } )  -> 
j  e.  K )
7776adantl 453 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( K  \  {  .0.  } ) )  ->  j  e.  K )
78 eldifsni 3872 . . . . . . . 8  |-  ( j  e.  ( K  \  {  .0.  } )  -> 
j  =/=  .0.  )
7978adantl 453 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( K  \  {  .0.  } ) )  ->  j  =/=  .0.  )
8031adantr 452 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( K  \  {  .0.  } ) )  ->  Y  e.  V )
8117, 4, 21, 6, 11, 27lspsnvs 16114 . . . . . . 7  |-  ( ( W  e.  LVec  /\  (
j  e.  K  /\  j  =/=  .0.  )  /\  Y  e.  V )  ->  ( N `  {
( j  .x.  Y
) } )  =  ( N `  { Y } ) )
8275, 77, 79, 80, 81syl121anc 1189 . . . . . 6  |-  ( (
ph  /\  j  e.  ( K  \  {  .0.  } ) )  ->  ( N `  { (
j  .x.  Y ) } )  =  ( N `  { Y } ) )
8382ex 424 . . . . 5  |-  ( ph  ->  ( j  e.  ( K  \  {  .0.  } )  ->  ( N `  { ( j  .x.  Y ) } )  =  ( N `  { Y } ) ) )
84 sneq 3769 . . . . . . . 8  |-  ( X  =  ( j  .x.  Y )  ->  { X }  =  { (
j  .x.  Y ) } )
8584fveq2d 5673 . . . . . . 7  |-  ( X  =  ( j  .x.  Y )  ->  ( N `  { X } )  =  ( N `  { ( j  .x.  Y ) } ) )
8685eqeq1d 2396 . . . . . 6  |-  ( X  =  ( j  .x.  Y )  ->  (
( N `  { X } )  =  ( N `  { Y } )  <->  ( N `  { ( j  .x.  Y ) } )  =  ( N `  { Y } ) ) )
8786biimprcd 217 . . . . 5  |-  ( ( N `  { ( j  .x.  Y ) } )  =  ( N `  { Y } )  ->  ( X  =  ( j  .x.  Y )  ->  ( N `  { X } )  =  ( N `  { Y } ) ) )
8883, 87syl6 31 . . . 4  |-  ( ph  ->  ( j  e.  ( K  \  {  .0.  } )  ->  ( X  =  ( j  .x.  Y )  ->  ( N `  { X } )  =  ( N `  { Y } ) ) ) )
8988rexlimdv 2773 . . 3  |-  ( ph  ->  ( E. j  e.  ( K  \  {  .0.  } ) X  =  ( j  .x.  Y
)  ->  ( N `  { X } )  =  ( N `  { Y } ) ) )
9074, 89impbid 184 . 2  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  <->  E. j  e.  ( K  \  {  .0.  } ) X  =  ( j  .x.  Y
) ) )
91 oveq1 6028 . . . 4  |-  ( j  =  k  ->  (
j  .x.  Y )  =  ( k  .x.  Y ) )
9291eqeq2d 2399 . . 3  |-  ( j  =  k  ->  ( X  =  ( j  .x.  Y )  <->  X  =  ( k  .x.  Y
) ) )
9392cbvrexv 2877 . 2  |-  ( E. j  e.  ( K 
\  {  .0.  }
) X  =  ( j  .x.  Y )  <->  E. k  e.  ( K  \  {  .0.  }
) X  =  ( k  .x.  Y ) )
9490, 93syl6bb 253 1  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  <->  E. k  e.  ( K  \  {  .0.  } ) X  =  ( k  .x.  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   E.wrex 2651    \ cdif 3261    C_ wss 3264   {csn 3758   ` cfv 5395  (class class class)co 6021   Basecbs 13397  Scalarcsca 13460   .scvsca 13461   0gc0g 13651   Ringcrg 15588   1rcur 15590   DivRingcdr 15763   LModclmod 15878   LSubSpclss 15936   LSpanclspn 15975   LVecclvec 16102
This theorem is referenced by:  lspsneu  16123  mapdpglem26  31814  mapdpglem27  31815  hdmap14lem2a  31986  hdmap14lem2N  31988
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-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-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-1st 6289  df-2nd 6290  df-tpos 6416  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  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-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-0g 13655  df-mnd 14618  df-grp 14740  df-minusg 14741  df-sbg 14742  df-mgp 15577  df-rng 15591  df-ur 15593  df-oppr 15656  df-dvdsr 15674  df-unit 15675  df-invr 15705  df-drng 15765  df-lmod 15880  df-lss 15937  df-lsp 15976  df-lvec 16103
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