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Theorem lfl1dim2N 29934
Description: Equivalent expressions for a 1-dim subspace (ray) of functionals. TODO: delete this if not useful; lfl1dim 29933 may be more compatible with lspsn 15775. (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 15875 . . . . . . . . 9  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 15 . . . . . . . 8  |-  ( ph  ->  W  e.  LMod )
4 lfl1dim.d . . . . . . . . 9  |-  D  =  (Scalar `  W )
5 lfl1dim.k . . . . . . . . 9  |-  K  =  ( Base `  D
)
6 eqid 2296 . . . . . . . . 9  |-  ( 0g
`  D )  =  ( 0g `  D
)
74, 5, 6lmod0cl 15672 . . . . . . . 8  |-  ( W  e.  LMod  ->  ( 0g
`  D )  e.  K )
83, 7syl 15 . . . . . . 7  |-  ( ph  ->  ( 0g `  D
)  e.  K )
98ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( 0g `  D )  e.  K
)
10 simpr 447 . . . . . . 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 706 . . . . . . . 8  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  W  e.  LMod )
15 lfl1dim.g . . . . . . . . 9  |-  ( ph  ->  G  e.  F )
1615ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  G  e.  F )
1711, 4, 12, 5, 13, 6, 14, 16lfl0sc 29894 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( G  o F  .x.  ( V  X.  { ( 0g
`  D ) } ) )  =  ( V  X.  { ( 0g `  D ) } ) )
1810, 17eqtr4d 2331 . . . . . 6  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  g  =  ( G  o F  .x.  ( V  X.  {
( 0g `  D
) } ) ) )
19 sneq 3664 . . . . . . . . . 10  |-  ( k  =  ( 0g `  D )  ->  { k }  =  { ( 0g `  D ) } )
2019xpeq2d 4729 . . . . . . . . 9  |-  ( k  =  ( 0g `  D )  ->  ( V  X.  { k } )  =  ( V  X.  { ( 0g
`  D ) } ) )
2120oveq2d 5890 . . . . . . . 8  |-  ( k  =  ( 0g `  D )  ->  ( G  o F  .x.  ( V  X.  { k } ) )  =  ( G  o F  .x.  ( V  X.  { ( 0g `  D ) } ) ) )
2221eqeq2d 2307 . . . . . . 7  |-  ( k  =  ( 0g `  D )  ->  (
g  =  ( G  o F  .x.  ( V  X.  { k } ) )  <->  g  =  ( G  o F  .x.  ( V  X.  {
( 0g `  D
) } ) ) ) )
2322rspcev 2897 . . . . . 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 642 . . . . 5  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) )
2524a1d 22 . . . 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 710 . . . . . 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 710 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  W  e.  LMod )
29 simpllr 735 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  g  e.  F )
3011, 12, 27, 28, 29lkrssv 29908 . . . . . . . . 9  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  ( L `  g )  C_  V )
313adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  g  e.  F )  ->  W  e.  LMod )
3215adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  g  e.  F )  ->  G  e.  F )
334, 6, 11, 12, 27lkr0f 29906 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( L `  G
)  =  V  <->  G  =  ( V  X.  { ( 0g `  D ) } ) ) )
3431, 32, 33syl2anc 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  g  e.  F )  ->  (
( L `  G
)  =  V  <->  G  =  ( V  X.  { ( 0g `  D ) } ) ) )
3534biimpar 471 . . . . . . . . . . 11  |-  ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( L `  G )  =  V )
3635sseq1d 3218 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( ( L `  G )  C_  ( L `  g
)  <->  V  C_  ( L `
 g ) ) )
3736biimpa 470 . . . . . . . . 9  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  V  C_  ( L `  g
) )
3830, 37eqssd 3209 . . . . . . . 8  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  ( L `  g )  =  V )
394, 6, 11, 12, 27lkr0f 29906 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  g  e.  F )  ->  (
( L `  g
)  =  V  <->  g  =  ( V  X.  { ( 0g `  D ) } ) ) )
4028, 29, 39syl2anc 642 . . . . . . . 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 201 . . . . . . 7  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  g  =  ( V  X.  { ( 0g `  D ) } ) )
4215ad3antrrr 710 . . . . . . . 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 29894 . . . . . . 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 2331 . . . . . 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 642 . . . . 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 423 . . . 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 2296 . . . . . 6  |-  (LSHyp `  W )  =  (LSHyp `  W )
481ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  W  e.  LVec )
4915ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  G  e.  F )
50 simprr 733 . . . . . . 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 29917 . . . . . . 7  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) )  ->  ( L `  G )  e.  (LSHyp `  W ) )
5248, 49, 50, 51syl3anc 1182 . . . . . 6  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  ( L `  G )  e.  (LSHyp `  W ) )
53 simplr 731 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  g  e.  F )
54 simprl 732 . . . . . . 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 29917 . . . . . . 7  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  g  =/=  ( V  X.  {
( 0g `  D
) } ) )  ->  ( L `  g )  e.  (LSHyp `  W ) )
5648, 53, 54, 55syl3anc 1182 . . . . . 6  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  ( L `  g )  e.  (LSHyp `  W ) )
5747, 48, 52, 56lshpcmp 29800 . . . . 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 710 . . . . . . 7  |-  ( ( ( ( ph  /\  g  e.  F )  /\  ( g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  /\  ( L `
 G )  =  ( L `  g
) )  ->  W  e.  LVec )
5915ad3antrrr 710 . . . . . . 7  |-  ( ( ( ( ph  /\  g  e.  F )  /\  ( g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  /\  ( L `
 G )  =  ( L `  g
) )  ->  G  e.  F )
60 simpllr 735 . . . . . . 7  |-  ( ( ( ( ph  /\  g  e.  F )  /\  ( g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  /\  ( L `
 G )  =  ( L `  g
) )  ->  g  e.  F )
61 simpr 447 . . . . . . 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 29912 . . . . . . 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 1187 . . . . . 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 423 . . . . 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 206 . . . 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 2538 . . 3  |-  ( (
ph  /\  g  e.  F )  ->  (
( L `  G
)  C_  ( L `  g )  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) ) )
671ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  k  e.  K )  ->  W  e.  LVec )
6815ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  k  e.  K )  ->  G  e.  F )
69 simpr 447 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  k  e.  K )  ->  k  e.  K )
7011, 4, 5, 13, 12, 27, 67, 68, 69lkrscss 29910 . . . . . 6  |-  ( ( ( ph  /\  g  e.  F )  /\  k  e.  K )  ->  ( L `  G )  C_  ( L `  ( G  o F  .x.  ( V  X.  { k } ) ) ) )
7170ex 423 . . . . 5  |-  ( (
ph  /\  g  e.  F )  ->  (
k  e.  K  -> 
( L `  G
)  C_  ( L `  ( G  o F 
.x.  ( V  X.  { k } ) ) ) ) )
72 fveq2 5541 . . . . . . 7  |-  ( g  =  ( G  o F  .x.  ( V  X.  { k } ) )  ->  ( L `  g )  =  ( L `  ( G  o F  .x.  ( V  X.  { k } ) ) ) )
7372sseq2d 3219 . . . . . 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 216 . . . . 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 29 . . . 4  |-  ( (
ph  /\  g  e.  F )  ->  (
k  e.  K  -> 
( g  =  ( G  o F  .x.  ( V  X.  { k } ) )  -> 
( L `  G
)  C_  ( L `  g ) ) ) )
7675rexlimdv 2679 . . 3  |-  ( (
ph  /\  g  e.  F )  ->  ( E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  { k } ) )  ->  ( L `  G )  C_  ( L `  g
) ) )
7766, 76impbid 183 . 2  |-  ( (
ph  /\  g  e.  F )  ->  (
( L `  G
)  C_  ( L `  g )  <->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) ) )
7877rabbidva 2792 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560    C_ wss 3165   {csn 3653    X. cxp 4703   ` cfv 5271  (class class class)co 5874    o Fcof 6092   Basecbs 13164   .rcmulr 13225  Scalarcsca 13227   0gc0g 13416   LModclmod 15643   LVecclvec 15871  LSHypclsh 29787  LFnlclfn 29869  LKerclk 29897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872  df-lshyp 29789  df-lfl 29870  df-lkr 29898
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