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Theorem lfl1dim2N 29920
Description: Equivalent expressions for a 1-dim subspace (ray) of functionals. TODO: delete this if not useful; lfl1dim 29919 may be more compatible with lspsn 16078. (Contributed by NM, 24-Oct-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lfl1dim.v  |-  V  =  ( Base `  W
)
lfl1dim.d  |-  D  =  (Scalar `  W )
lfl1dim.f  |-  F  =  (LFnl `  W )
lfl1dim.l  |-  L  =  (LKer `  W )
lfl1dim.k  |-  K  =  ( Base `  D
)
lfl1dim.t  |-  .x.  =  ( .r `  D )
lfl1dim.w  |-  ( ph  ->  W  e.  LVec )
lfl1dim.g  |-  ( ph  ->  G  e.  F )
Assertion
Ref Expression
lfl1dim2N  |-  ( ph  ->  { g  e.  F  |  ( L `  G )  C_  ( L `  g ) }  =  { g  e.  F  |  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  { k } ) ) } )
Distinct variable groups:    D, k    k, F    k, G    k, K    k, L    k, V    k, W    g, k, ph    .x. , k
Allowed substitution hints:    D( g)    .x. ( g)    F( g)    G( g)    K( g)    L( g)    V( g)    W( g)

Proof of Theorem lfl1dim2N
StepHypRef Expression
1 lfl1dim.w . . . . . . . . 9  |-  ( ph  ->  W  e.  LVec )
2 lveclmod 16178 . . . . . . . . 9  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 16 . . . . . . . 8  |-  ( ph  ->  W  e.  LMod )
4 lfl1dim.d . . . . . . . . 9  |-  D  =  (Scalar `  W )
5 lfl1dim.k . . . . . . . . 9  |-  K  =  ( Base `  D
)
6 eqid 2436 . . . . . . . . 9  |-  ( 0g
`  D )  =  ( 0g `  D
)
74, 5, 6lmod0cl 15976 . . . . . . . 8  |-  ( W  e.  LMod  ->  ( 0g
`  D )  e.  K )
83, 7syl 16 . . . . . . 7  |-  ( ph  ->  ( 0g `  D
)  e.  K )
98ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( 0g `  D )  e.  K
)
10 simpr 448 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  g  =  ( V  X.  { ( 0g `  D ) } ) )
11 lfl1dim.v . . . . . . . 8  |-  V  =  ( Base `  W
)
12 lfl1dim.f . . . . . . . 8  |-  F  =  (LFnl `  W )
13 lfl1dim.t . . . . . . . 8  |-  .x.  =  ( .r `  D )
143ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  W  e.  LMod )
15 lfl1dim.g . . . . . . . . 9  |-  ( ph  ->  G  e.  F )
1615ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  G  e.  F )
1711, 4, 12, 5, 13, 6, 14, 16lfl0sc 29880 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( G  o F  .x.  ( V  X.  { ( 0g
`  D ) } ) )  =  ( V  X.  { ( 0g `  D ) } ) )
1810, 17eqtr4d 2471 . . . . . 6  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  g  =  ( G  o F  .x.  ( V  X.  {
( 0g `  D
) } ) ) )
19 sneq 3825 . . . . . . . . . 10  |-  ( k  =  ( 0g `  D )  ->  { k }  =  { ( 0g `  D ) } )
2019xpeq2d 4902 . . . . . . . . 9  |-  ( k  =  ( 0g `  D )  ->  ( V  X.  { k } )  =  ( V  X.  { ( 0g
`  D ) } ) )
2120oveq2d 6097 . . . . . . . 8  |-  ( k  =  ( 0g `  D )  ->  ( G  o F  .x.  ( V  X.  { k } ) )  =  ( G  o F  .x.  ( V  X.  { ( 0g `  D ) } ) ) )
2221eqeq2d 2447 . . . . . . 7  |-  ( k  =  ( 0g `  D )  ->  (
g  =  ( G  o F  .x.  ( V  X.  { k } ) )  <->  g  =  ( G  o F  .x.  ( V  X.  {
( 0g `  D
) } ) ) ) )
2322rspcev 3052 . . . . . 6  |-  ( ( ( 0g `  D
)  e.  K  /\  g  =  ( G  o F  .x.  ( V  X.  { ( 0g
`  D ) } ) ) )  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  { k } ) ) )
249, 18, 23syl2anc 643 . . . . 5  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) )
2524a1d 23 . . . 4  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( ( L `  G )  C_  ( L `  g
)  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) ) )
268ad3antrrr 711 . . . . . 6  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  ( 0g `  D )  e.  K )
27 lfl1dim.l . . . . . . . . . 10  |-  L  =  (LKer `  W )
283ad3antrrr 711 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  W  e.  LMod )
29 simpllr 736 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  g  e.  F )
3011, 12, 27, 28, 29lkrssv 29894 . . . . . . . . 9  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  ( L `  g )  C_  V )
313adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  g  e.  F )  ->  W  e.  LMod )
3215adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  g  e.  F )  ->  G  e.  F )
334, 6, 11, 12, 27lkr0f 29892 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( L `  G
)  =  V  <->  G  =  ( V  X.  { ( 0g `  D ) } ) ) )
3431, 32, 33syl2anc 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  g  e.  F )  ->  (
( L `  G
)  =  V  <->  G  =  ( V  X.  { ( 0g `  D ) } ) ) )
3534biimpar 472 . . . . . . . . . . 11  |-  ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( L `  G )  =  V )
3635sseq1d 3375 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( ( L `  G )  C_  ( L `  g
)  <->  V  C_  ( L `
 g ) ) )
3736biimpa 471 . . . . . . . . 9  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  V  C_  ( L `  g
) )
3830, 37eqssd 3365 . . . . . . . 8  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  ( L `  g )  =  V )
394, 6, 11, 12, 27lkr0f 29892 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  g  e.  F )  ->  (
( L `  g
)  =  V  <->  g  =  ( V  X.  { ( 0g `  D ) } ) ) )
4028, 29, 39syl2anc 643 . . . . . . . 8  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  (
( L `  g
)  =  V  <->  g  =  ( V  X.  { ( 0g `  D ) } ) ) )
4138, 40mpbid 202 . . . . . . 7  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  g  =  ( V  X.  { ( 0g `  D ) } ) )
4215ad3antrrr 711 . . . . . . . 8  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  G  e.  F )
4311, 4, 12, 5, 13, 6, 28, 42lfl0sc 29880 . . . . . . 7  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  ( G  o F  .x.  ( V  X.  { ( 0g
`  D ) } ) )  =  ( V  X.  { ( 0g `  D ) } ) )
4441, 43eqtr4d 2471 . . . . . 6  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  g  =  ( G  o F  .x.  ( V  X.  { ( 0g `  D ) } ) ) )
4526, 44, 23syl2anc 643 . . . . 5  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) )
4645ex 424 . . . 4  |-  ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( ( L `  G )  C_  ( L `  g
)  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) ) )
47 eqid 2436 . . . . . 6  |-  (LSHyp `  W )  =  (LSHyp `  W )
481ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  W  e.  LVec )
4915ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  G  e.  F )
50 simprr 734 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  G  =/=  ( V  X.  { ( 0g `  D ) } ) )
5111, 4, 6, 47, 12, 27lkrshp 29903 . . . . . . 7  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) )  ->  ( L `  G )  e.  (LSHyp `  W ) )
5248, 49, 50, 51syl3anc 1184 . . . . . 6  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  ( L `  G )  e.  (LSHyp `  W ) )
53 simplr 732 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  g  e.  F )
54 simprl 733 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  g  =/=  ( V  X.  { ( 0g `  D ) } ) )
5511, 4, 6, 47, 12, 27lkrshp 29903 . . . . . . 7  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  g  =/=  ( V  X.  {
( 0g `  D
) } ) )  ->  ( L `  g )  e.  (LSHyp `  W ) )
5648, 53, 54, 55syl3anc 1184 . . . . . 6  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  ( L `  g )  e.  (LSHyp `  W ) )
5747, 48, 52, 56lshpcmp 29786 . . . . 5  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  ( ( L `  G )  C_  ( L `  g
)  <->  ( L `  G )  =  ( L `  g ) ) )
581ad3antrrr 711 . . . . . . 7  |-  ( ( ( ( ph  /\  g  e.  F )  /\  ( g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  /\  ( L `
 G )  =  ( L `  g
) )  ->  W  e.  LVec )
5915ad3antrrr 711 . . . . . . 7  |-  ( ( ( ( ph  /\  g  e.  F )  /\  ( g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  /\  ( L `
 G )  =  ( L `  g
) )  ->  G  e.  F )
60 simpllr 736 . . . . . . 7  |-  ( ( ( ( ph  /\  g  e.  F )  /\  ( g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  /\  ( L `
 G )  =  ( L `  g
) )  ->  g  e.  F )
61 simpr 448 . . . . . . 7  |-  ( ( ( ( ph  /\  g  e.  F )  /\  ( g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  /\  ( L `
 G )  =  ( L `  g
) )  ->  ( L `  G )  =  ( L `  g ) )
624, 5, 13, 11, 12, 27eqlkr2 29898 . . . . . . 7  |-  ( ( W  e.  LVec  /\  ( G  e.  F  /\  g  e.  F )  /\  ( L `  G
)  =  ( L `
 g ) )  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  { k } ) ) )
6358, 59, 60, 61, 62syl121anc 1189 . . . . . 6  |-  ( ( ( ( ph  /\  g  e.  F )  /\  ( g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  /\  ( L `
 G )  =  ( L `  g
) )  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) )
6463ex 424 . . . . 5  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  ( ( L `  G )  =  ( L `  g )  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) ) )
6557, 64sylbid 207 . . . 4  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  ( ( L `  G )  C_  ( L `  g
)  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) ) )
6625, 46, 65pm2.61da2ne 2683 . . 3  |-  ( (
ph  /\  g  e.  F )  ->  (
( L `  G
)  C_  ( L `  g )  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) ) )
671ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  k  e.  K )  ->  W  e.  LVec )
6815ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  k  e.  K )  ->  G  e.  F )
69 simpr 448 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  k  e.  K )  ->  k  e.  K )
7011, 4, 5, 13, 12, 27, 67, 68, 69lkrscss 29896 . . . . . 6  |-  ( ( ( ph  /\  g  e.  F )  /\  k  e.  K )  ->  ( L `  G )  C_  ( L `  ( G  o F  .x.  ( V  X.  { k } ) ) ) )
7170ex 424 . . . . 5  |-  ( (
ph  /\  g  e.  F )  ->  (
k  e.  K  -> 
( L `  G
)  C_  ( L `  ( G  o F 
.x.  ( V  X.  { k } ) ) ) ) )
72 fveq2 5728 . . . . . . 7  |-  ( g  =  ( G  o F  .x.  ( V  X.  { k } ) )  ->  ( L `  g )  =  ( L `  ( G  o F  .x.  ( V  X.  { k } ) ) ) )
7372sseq2d 3376 . . . . . 6  |-  ( g  =  ( G  o F  .x.  ( V  X.  { k } ) )  ->  ( ( L `  G )  C_  ( L `  g
)  <->  ( L `  G )  C_  ( L `  ( G  o F  .x.  ( V  X.  { k } ) ) ) ) )
7473biimprcd 217 . . . . 5  |-  ( ( L `  G ) 
C_  ( L `  ( G  o F  .x.  ( V  X.  {
k } ) ) )  ->  ( g  =  ( G  o F  .x.  ( V  X.  { k } ) )  ->  ( L `  G )  C_  ( L `  g )
) )
7571, 74syl6 31 . . . 4  |-  ( (
ph  /\  g  e.  F )  ->  (
k  e.  K  -> 
( g  =  ( G  o F  .x.  ( V  X.  { k } ) )  -> 
( L `  G
)  C_  ( L `  g ) ) ) )
7675rexlimdv 2829 . . 3  |-  ( (
ph  /\  g  e.  F )  ->  ( E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  { k } ) )  ->  ( L `  G )  C_  ( L `  g
) ) )
7766, 76impbid 184 . 2  |-  ( (
ph  /\  g  e.  F )  ->  (
( L `  G
)  C_  ( L `  g )  <->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) ) )
7877rabbidva 2947 1  |-  ( ph  ->  { g  e.  F  |  ( L `  G )  C_  ( L `  g ) }  =  { g  e.  F  |  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  { k } ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   {crab 2709    C_ wss 3320   {csn 3814    X. cxp 4876   ` cfv 5454  (class class class)co 6081    o Fcof 6303   Basecbs 13469   .rcmulr 13530  Scalarcsca 13532   0gc0g 13723   LModclmod 15950   LVecclvec 16174  LSHypclsh 29773  LFnlclfn 29855  LKerclk 29883
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-cntz 15116  df-lsm 15270  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-drng 15837  df-lmod 15952  df-lss 16009  df-lsp 16048  df-lvec 16175  df-lshyp 29775  df-lfl 29856  df-lkr 29884
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