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Theorem lspsneq 17568
Description: Equal spans of singletons must have proportional vectors. See lspsnss2 17451 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 17552 . . . . . . . . . 10  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 16 . . . . . . . . 9  |-  ( ph  ->  W  e.  LMod )
4 lspsneq.s . . . . . . . . . 10  |-  S  =  (Scalar `  W )
54lmodrng 17320 . . . . . . . . 9  |-  ( W  e.  LMod  ->  S  e. 
Ring )
6 lspsneq.k . . . . . . . . . 10  |-  K  =  ( Base `  S
)
7 eqid 2467 . . . . . . . . . 10  |-  ( 1r
`  S )  =  ( 1r `  S
)
86, 7rngidcl 17020 . . . . . . . . 9  |-  ( S  e.  Ring  ->  ( 1r
`  S )  e.  K )
93, 5, 83syl 20 . . . . . . . 8  |-  ( ph  ->  ( 1r `  S
)  e.  K )
104lvecdrng 17551 . . . . . . . . 9  |-  ( W  e.  LVec  ->  S  e.  DivRing )
11 lspsneq.o . . . . . . . . . 10  |-  .0.  =  ( 0g `  S )
1211, 7drngunz 17211 . . . . . . . . 9  |-  ( S  e.  DivRing  ->  ( 1r `  S )  =/=  .0.  )
131, 10, 123syl 20 . . . . . . . 8  |-  ( ph  ->  ( 1r `  S
)  =/=  .0.  )
14 eldifsn 4152 . . . . . . . 8  |-  ( ( 1r `  S )  e.  ( K  \  {  .0.  } )  <->  ( ( 1r `  S )  e.  K  /\  ( 1r
`  S )  =/= 
.0.  ) )
159, 13, 14sylanbrc 664 . . . . . . 7  |-  ( ph  ->  ( 1r `  S
)  e.  ( K 
\  {  .0.  }
) )
1615ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =  ( 0g `  W ) )  -> 
( 1r `  S
)  e.  ( K 
\  {  .0.  }
) )
17 lspsneq.v . . . . . . . . . . 11  |-  V  =  ( Base `  W
)
18 eqid 2467 . . . . . . . . . . 11  |-  ( 0g
`  W )  =  ( 0g `  W
)
1917, 18lmod0vcl 17341 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  ( 0g
`  W )  e.  V )
201, 2, 193syl 20 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  W
)  e.  V )
21 lspsneq.t . . . . . . . . . 10  |-  .x.  =  ( .s `  W )
2217, 4, 21, 7lmodvs1 17340 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  ( 0g `  W )  e.  V )  ->  (
( 1r `  S
)  .x.  ( 0g `  W ) )  =  ( 0g `  W
) )
233, 20, 22syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( ( 1r `  S )  .x.  ( 0g `  W ) )  =  ( 0g `  W ) )
2423ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =  ( 0g `  W ) )  -> 
( ( 1r `  S )  .x.  ( 0g `  W ) )  =  ( 0g `  W ) )
25 oveq2 6292 . . . . . . . 8  |-  ( Y  =  ( 0g `  W )  ->  (
( 1r `  S
)  .x.  Y )  =  ( ( 1r
`  S )  .x.  ( 0g `  W ) ) )
2625adantl 466 . . . . . . 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 465 . . . . . . . . 9  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  W  e.  LMod )
29 lspsneq.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  V )
3029adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  X  e.  V
)
31 lspsneq.y . . . . . . . . . 10  |-  ( ph  ->  Y  e.  V )
3231adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  Y  e.  V
)
33 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  ( N `  { X } )  =  ( N `  { Y } ) )
3417, 18, 27, 28, 30, 32, 33lspsneq0b 17459 . . . . . . . 8  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  ( X  =  ( 0g `  W
)  <->  Y  =  ( 0g `  W ) ) )
3534biimpar 485 . . . . . . 7  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =  ( 0g `  W ) )  ->  X  =  ( 0g `  W ) )
3624, 26, 353eqtr4rd 2519 . . . . . 6  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =  ( 0g `  W ) )  ->  X  =  ( ( 1r `  S )  .x.  Y ) )
37 oveq1 6291 . . . . . . . 8  |-  ( j  =  ( 1r `  S )  ->  (
j  .x.  Y )  =  ( ( 1r
`  S )  .x.  Y ) )
3837eqeq2d 2481 . . . . . . 7  |-  ( j  =  ( 1r `  S )  ->  ( X  =  ( j  .x.  Y )  <->  X  =  ( ( 1r `  S )  .x.  Y
) ) )
3938rspcev 3214 . . . . . 6  |-  ( ( ( 1r `  S
)  e.  ( K 
\  {  .0.  }
)  /\  X  =  ( ( 1r `  S )  .x.  Y
) )  ->  E. j  e.  ( K  \  {  .0.  } ) X  =  ( j  .x.  Y
) )
4016, 36, 39syl2anc 661 . . . . 5  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =  ( 0g `  W ) )  ->  E. j  e.  ( K  \  {  .0.  }
) X  =  ( j  .x.  Y ) )
41 eqimss 3556 . . . . . . . . . 10  |-  ( ( N `  { X } )  =  ( N `  { Y } )  ->  ( N `  { X } )  C_  ( N `  { Y } ) )
4241adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  ( N `  { X } )  C_  ( N `  { Y } ) )
43 eqid 2467 . . . . . . . . . 10  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
4417, 43, 27lspsncl 17423 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
453, 31, 44syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
4645adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  ( N `  { Y } )  e.  ( LSubSp `  W )
)
4717, 43, 27, 28, 46, 30lspsnel5 17441 . . . . . . . . 9  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  ( X  e.  ( N `  { Y } )  <->  ( N `  { X } ) 
C_  ( N `  { Y } ) ) )
4842, 47mpbird 232 . . . . . . . 8  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  X  e.  ( N `  { Y } ) )
494, 6, 17, 21, 27lspsnel 17449 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  ( X  e.  ( N `  { Y } )  <->  E. j  e.  K  X  =  ( j  .x.  Y ) ) )
5028, 32, 49syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  ( X  e.  ( N `  { Y } )  <->  E. j  e.  K  X  =  ( j  .x.  Y
) ) )
5148, 50mpbid 210 . . . . . . 7  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  E. j  e.  K  X  =  ( j  .x.  Y ) )
5251adantr 465 . . . . . 6  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  ->  E. j  e.  K  X  =  ( j  .x.  Y ) )
53 simprl 755 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  j  e.  K )
54 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( j  e.  K  /\  X  =  ( j  .x.  Y ) )  ->  X  =  ( j  .x.  Y ) )
5554adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  X  =  ( j  .x.  Y ) )
5634biimpd 207 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  ( X  =  ( 0g `  W
)  ->  Y  =  ( 0g `  W ) ) )
5756necon3d 2691 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  ( Y  =/=  ( 0g `  W
)  ->  X  =/=  ( 0g `  W ) ) )
5857imp 429 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  ->  X  =/=  ( 0g `  W ) )
5958adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  X  =/=  ( 0g `  W
) )
6055, 59eqnetrrd 2761 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  (
j  .x.  Y )  =/=  ( 0g `  W
) )
611adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  W  e.  LVec )
6261ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  W  e.  LVec )
6332ad2antrr 725 . . . . . . . . . . . . 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 17555 . . . . . . . . . . . 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 210 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  (
j  =/=  .0.  /\  Y  =/=  ( 0g `  W ) ) )
6665simpld 459 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  j  =/=  .0.  )
67 eldifsn 4152 . . . . . . . . . 10  |-  ( j  e.  ( K  \  {  .0.  } )  <->  ( j  e.  K  /\  j  =/=  .0.  ) )
6853, 66, 67sylanbrc 664 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =/=  ( 0g `  W ) )  /\  ( j  e.  K  /\  X  =  (
j  .x.  Y )
) )  ->  j  e.  ( K  \  {  .0.  } ) )
6968, 55jca 532 . . . . . . . 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 434 . . . . . . 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 2934 . . . . . 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 2785 . . . 4  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  E. j  e.  ( K  \  {  .0.  } ) X  =  ( j  .x.  Y ) )
7473ex 434 . . 3  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  ->  E. j  e.  ( K  \  {  .0.  } ) X  =  ( j  .x.  Y
) ) )
751adantr 465 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( K  \  {  .0.  } ) )  ->  W  e.  LVec )
76 eldifi 3626 . . . . . . . 8  |-  ( j  e.  ( K  \  {  .0.  } )  -> 
j  e.  K )
7776adantl 466 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( K  \  {  .0.  } ) )  ->  j  e.  K )
78 eldifsni 4153 . . . . . . . 8  |-  ( j  e.  ( K  \  {  .0.  } )  -> 
j  =/=  .0.  )
7978adantl 466 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( K  \  {  .0.  } ) )  ->  j  =/=  .0.  )
8031adantr 465 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( K  \  {  .0.  } ) )  ->  Y  e.  V )
8117, 4, 21, 6, 11, 27lspsnvs 17560 . . . . . . 7  |-  ( ( W  e.  LVec  /\  (
j  e.  K  /\  j  =/=  .0.  )  /\  Y  e.  V )  ->  ( N `  {
( j  .x.  Y
) } )  =  ( N `  { Y } ) )
8275, 77, 79, 80, 81syl121anc 1233 . . . . . 6  |-  ( (
ph  /\  j  e.  ( K  \  {  .0.  } ) )  ->  ( N `  { (
j  .x.  Y ) } )  =  ( N `  { Y } ) )
8382ex 434 . . . . 5  |-  ( ph  ->  ( j  e.  ( K  \  {  .0.  } )  ->  ( N `  { ( j  .x.  Y ) } )  =  ( N `  { Y } ) ) )
84 sneq 4037 . . . . . . . 8  |-  ( X  =  ( j  .x.  Y )  ->  { X }  =  { (
j  .x.  Y ) } )
8584fveq2d 5870 . . . . . . 7  |-  ( X  =  ( j  .x.  Y )  ->  ( N `  { X } )  =  ( N `  { ( j  .x.  Y ) } ) )
8685eqeq1d 2469 . . . . . 6  |-  ( X  =  ( j  .x.  Y )  ->  (
( N `  { X } )  =  ( N `  { Y } )  <->  ( N `  { ( j  .x.  Y ) } )  =  ( N `  { Y } ) ) )
8786biimprcd 225 . . . . 5  |-  ( ( N `  { ( j  .x.  Y ) } )  =  ( N `  { Y } )  ->  ( X  =  ( j  .x.  Y )  ->  ( N `  { X } )  =  ( N `  { Y } ) ) )
8883, 87syl6 33 . . . 4  |-  ( ph  ->  ( j  e.  ( K  \  {  .0.  } )  ->  ( X  =  ( j  .x.  Y )  ->  ( N `  { X } )  =  ( N `  { Y } ) ) ) )
8988rexlimdv 2953 . . 3  |-  ( ph  ->  ( E. j  e.  ( K  \  {  .0.  } ) X  =  ( j  .x.  Y
)  ->  ( N `  { X } )  =  ( N `  { Y } ) ) )
9074, 89impbid 191 . 2  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  <->  E. j  e.  ( K  \  {  .0.  } ) X  =  ( j  .x.  Y
) ) )
91 oveq1 6291 . . . 4  |-  ( j  =  k  ->  (
j  .x.  Y )  =  ( k  .x.  Y ) )
9291eqeq2d 2481 . . 3  |-  ( j  =  k  ->  ( X  =  ( j  .x.  Y )  <->  X  =  ( k  .x.  Y
) ) )
9392cbvrexv 3089 . 2  |-  ( E. j  e.  ( K 
\  {  .0.  }
) X  =  ( j  .x.  Y )  <->  E. k  e.  ( K  \  {  .0.  }
) X  =  ( k  .x.  Y ) )
9490, 93syl6bb 261 1  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  <->  E. k  e.  ( K  \  {  .0.  } ) X  =  ( k  .x.  Y
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    \ cdif 3473    C_ wss 3476   {csn 4027   ` cfv 5588  (class class class)co 6284   Basecbs 14490  Scalarcsca 14558   .scvsca 14559   0gc0g 14695   1rcur 16955   Ringcrg 17000   DivRingcdr 17196   LModclmod 17312   LSubSpclss 17378   LSpanclspn 17417   LVecclvec 17548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-tpos 6955  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-0g 14697  df-mnd 15732  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mgp 16944  df-ur 16956  df-rng 17002  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-invr 17122  df-drng 17198  df-lmod 17314  df-lss 17379  df-lsp 17418  df-lvec 17549
This theorem is referenced by:  lspsneu  17569  mapdpglem26  36513  mapdpglem27  36514  hdmap14lem2a  36685  hdmap14lem2N  36687
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