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Theorem lspsneq 17325
Description: Equal spans of singletons must have proportional vectors. See lspsnss2 17208 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 17309 . . . . . . . . . 10  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 16 . . . . . . . . 9  |-  ( ph  ->  W  e.  LMod )
4 lspsneq.s . . . . . . . . . 10  |-  S  =  (Scalar `  W )
54lmodrng 17078 . . . . . . . . 9  |-  ( W  e.  LMod  ->  S  e. 
Ring )
6 lspsneq.k . . . . . . . . . 10  |-  K  =  ( Base `  S
)
7 eqid 2454 . . . . . . . . . 10  |-  ( 1r
`  S )  =  ( 1r `  S
)
86, 7rngidcl 16787 . . . . . . . . 9  |-  ( S  e.  Ring  ->  ( 1r
`  S )  e.  K )
93, 5, 83syl 20 . . . . . . . 8  |-  ( ph  ->  ( 1r `  S
)  e.  K )
104lvecdrng 17308 . . . . . . . . 9  |-  ( W  e.  LVec  ->  S  e.  DivRing )
11 lspsneq.o . . . . . . . . . 10  |-  .0.  =  ( 0g `  S )
1211, 7drngunz 16969 . . . . . . . . 9  |-  ( S  e.  DivRing  ->  ( 1r `  S )  =/=  .0.  )
131, 10, 123syl 20 . . . . . . . 8  |-  ( ph  ->  ( 1r `  S
)  =/=  .0.  )
14 eldifsn 4107 . . . . . . . 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 2454 . . . . . . . . . . 11  |-  ( 0g
`  W )  =  ( 0g `  W
)
1917, 18lmod0vcl 17099 . . . . . . . . . 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 17098 . . . . . . . . 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 6207 . . . . . . . 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 17216 . . . . . . . 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 2506 . . . . . 6  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  Y  =  ( 0g `  W ) )  ->  X  =  ( ( 1r `  S )  .x.  Y ) )
37 oveq1 6206 . . . . . . . 8  |-  ( j  =  ( 1r `  S )  ->  (
j  .x.  Y )  =  ( ( 1r
`  S )  .x.  Y ) )
3837eqeq2d 2468 . . . . . . 7  |-  ( j  =  ( 1r `  S )  ->  ( X  =  ( j  .x.  Y )  <->  X  =  ( ( 1r `  S )  .x.  Y
) ) )
3938rspcev 3177 . . . . . 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 3515 . . . . . . . . . 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 2454 . . . . . . . . . 10  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
4417, 43, 27lspsncl 17180 . . . . . . . . . . . 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 17198 . . . . . . . . 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 17206 . . . . . . . . 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 2675 . . . . . . . . . . . . . . 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 2745 . . . . . . . . . . . 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 17312 . . . . . . . . . . . 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 4107 . . . . . . . . . 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 2929 . . . . . 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 2769 . . . 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 3585 . . . . . . . 8  |-  ( j  e.  ( K  \  {  .0.  } )  -> 
j  e.  K )
7776adantl 466 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( K  \  {  .0.  } ) )  ->  j  e.  K )
78 eldifsni 4108 . . . . . . . 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 17317 . . . . . . 7  |-  ( ( W  e.  LVec  /\  (
j  e.  K  /\  j  =/=  .0.  )  /\  Y  e.  V )  ->  ( N `  {
( j  .x.  Y
) } )  =  ( N `  { Y } ) )
8275, 77, 79, 80, 81syl121anc 1224 . . . . . 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 3994 . . . . . . . 8  |-  ( X  =  ( j  .x.  Y )  ->  { X }  =  { (
j  .x.  Y ) } )
8584fveq2d 5802 . . . . . . 7  |-  ( X  =  ( j  .x.  Y )  ->  ( N `  { X } )  =  ( N `  { ( j  .x.  Y ) } ) )
8685eqeq1d 2456 . . . . . 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 2944 . . 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 6206 . . . 4  |-  ( j  =  k  ->  (
j  .x.  Y )  =  ( k  .x.  Y ) )
9291eqeq2d 2468 . . 3  |-  ( j  =  k  ->  ( X  =  ( j  .x.  Y )  <->  X  =  ( k  .x.  Y
) ) )
9392cbvrexv 3052 . 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 1370    e. wcel 1758    =/= wne 2647   E.wrex 2799    \ cdif 3432    C_ wss 3435   {csn 3984   ` cfv 5525  (class class class)co 6199   Basecbs 14291  Scalarcsca 14359   .scvsca 14360   0gc0g 14496   1rcur 16724   Ringcrg 16767   DivRingcdr 16954   LModclmod 17070   LSubSpclss 17135   LSpanclspn 17174   LVecclvec 17305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-tpos 6854  df-recs 6941  df-rdg 6975  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-3 10491  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-0g 14498  df-mnd 15533  df-grp 15663  df-minusg 15664  df-sbg 15665  df-mgp 16713  df-ur 16725  df-rng 16769  df-oppr 16837  df-dvdsr 16855  df-unit 16856  df-invr 16886  df-drng 16956  df-lmod 17072  df-lss 17136  df-lsp 17175  df-lvec 17306
This theorem is referenced by:  lspsneu  17326  mapdpglem26  35666  mapdpglem27  35667  hdmap14lem2a  35838  hdmap14lem2N  35840
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