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Theorem lfl1dim2N 34275
Description: Equivalent expressions for a 1-dim subspace (ray) of functionals. TODO: delete this if not useful; lfl1dim 34274 may be more compatible with lspsn 17519. (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  oF  .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 17623 . . . . . . . . 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 2467 . . . . . . . . 9  |-  ( 0g
`  D )  =  ( 0g `  D
)
74, 5, 6lmod0cl 17409 . . . . . . . 8  |-  ( W  e.  LMod  ->  ( 0g
`  D )  e.  K )
83, 7syl 16 . . . . . . 7  |-  ( ph  ->  ( 0g `  D
)  e.  K )
98ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( 0g `  D )  e.  K
)
10 simpr 461 . . . . . . 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 725 . . . . . . . 8  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  W  e.  LMod )
15 lfl1dim.g . . . . . . . . 9  |-  ( ph  ->  G  e.  F )
1615ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  G  e.  F )
1711, 4, 12, 5, 13, 6, 14, 16lfl0sc 34235 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( G  oF  .x.  ( V  X.  { ( 0g
`  D ) } ) )  =  ( V  X.  { ( 0g `  D ) } ) )
1810, 17eqtr4d 2511 . . . . . 6  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  g  =  ( G  oF  .x.  ( V  X.  {
( 0g `  D
) } ) ) )
19 sneq 4043 . . . . . . . . . 10  |-  ( k  =  ( 0g `  D )  ->  { k }  =  { ( 0g `  D ) } )
2019xpeq2d 5029 . . . . . . . . 9  |-  ( k  =  ( 0g `  D )  ->  ( V  X.  { k } )  =  ( V  X.  { ( 0g
`  D ) } ) )
2120oveq2d 6311 . . . . . . . 8  |-  ( k  =  ( 0g `  D )  ->  ( G  oF  .x.  ( V  X.  { k } ) )  =  ( G  oF  .x.  ( V  X.  { ( 0g `  D ) } ) ) )
2221eqeq2d 2481 . . . . . . 7  |-  ( k  =  ( 0g `  D )  ->  (
g  =  ( G  oF  .x.  ( V  X.  { k } ) )  <->  g  =  ( G  oF  .x.  ( V  X.  {
( 0g `  D
) } ) ) ) )
2322rspcev 3219 . . . . . 6  |-  ( ( ( 0g `  D
)  e.  K  /\  g  =  ( G  oF  .x.  ( V  X.  { ( 0g
`  D ) } ) ) )  ->  E. k  e.  K  g  =  ( G  oF  .x.  ( V  X.  { k } ) ) )
249, 18, 23syl2anc 661 . . . . 5  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  E. k  e.  K  g  =  ( G  oF  .x.  ( V  X.  {
k } ) ) )
2524a1d 25 . . . 4  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( ( L `  G )  C_  ( L `  g
)  ->  E. k  e.  K  g  =  ( G  oF  .x.  ( V  X.  {
k } ) ) ) )
268ad3antrrr 729 . . . . . 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 729 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  W  e.  LMod )
29 simpllr 758 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  g  e.  F )
3011, 12, 27, 28, 29lkrssv 34249 . . . . . . . . 9  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  ( L `  g )  C_  V )
313adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  g  e.  F )  ->  W  e.  LMod )
3215adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  g  e.  F )  ->  G  e.  F )
334, 6, 11, 12, 27lkr0f 34247 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( L `  G
)  =  V  <->  G  =  ( V  X.  { ( 0g `  D ) } ) ) )
3431, 32, 33syl2anc 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  g  e.  F )  ->  (
( L `  G
)  =  V  <->  G  =  ( V  X.  { ( 0g `  D ) } ) ) )
3534biimpar 485 . . . . . . . . . . 11  |-  ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( L `  G )  =  V )
3635sseq1d 3536 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( ( L `  G )  C_  ( L `  g
)  <->  V  C_  ( L `
 g ) ) )
3736biimpa 484 . . . . . . . . 9  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  V  C_  ( L `  g
) )
3830, 37eqssd 3526 . . . . . . . 8  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  ( L `  g )  =  V )
394, 6, 11, 12, 27lkr0f 34247 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  g  e.  F )  ->  (
( L `  g
)  =  V  <->  g  =  ( V  X.  { ( 0g `  D ) } ) ) )
4028, 29, 39syl2anc 661 . . . . . . . 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 210 . . . . . . 7  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  g  =  ( V  X.  { ( 0g `  D ) } ) )
4215ad3antrrr 729 . . . . . . . 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 34235 . . . . . . 7  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  ( G  oF  .x.  ( V  X.  { ( 0g
`  D ) } ) )  =  ( V  X.  { ( 0g `  D ) } ) )
4441, 43eqtr4d 2511 . . . . . 6  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  g  =  ( G  oF  .x.  ( V  X.  { ( 0g `  D ) } ) ) )
4526, 44, 23syl2anc 661 . . . . 5  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  E. k  e.  K  g  =  ( G  oF  .x.  ( V  X.  {
k } ) ) )
4645ex 434 . . . 4  |-  ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( ( L `  G )  C_  ( L `  g
)  ->  E. k  e.  K  g  =  ( G  oF  .x.  ( V  X.  {
k } ) ) ) )
47 eqid 2467 . . . . . 6  |-  (LSHyp `  W )  =  (LSHyp `  W )
481ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  W  e.  LVec )
4915ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  G  e.  F )
50 simprr 756 . . . . . . 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 34258 . . . . . . 7  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) )  ->  ( L `  G )  e.  (LSHyp `  W ) )
5248, 49, 50, 51syl3anc 1228 . . . . . 6  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  ( L `  G )  e.  (LSHyp `  W ) )
53 simplr 754 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  g  e.  F )
54 simprl 755 . . . . . . 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 34258 . . . . . . 7  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  g  =/=  ( V  X.  {
( 0g `  D
) } ) )  ->  ( L `  g )  e.  (LSHyp `  W ) )
5648, 53, 54, 55syl3anc 1228 . . . . . 6  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  ( L `  g )  e.  (LSHyp `  W ) )
5747, 48, 52, 56lshpcmp 34141 . . . . 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 729 . . . . . . 7  |-  ( ( ( ( ph  /\  g  e.  F )  /\  ( g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  /\  ( L `
 G )  =  ( L `  g
) )  ->  W  e.  LVec )
5915ad3antrrr 729 . . . . . . 7  |-  ( ( ( ( ph  /\  g  e.  F )  /\  ( g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  /\  ( L `
 G )  =  ( L `  g
) )  ->  G  e.  F )
60 simpllr 758 . . . . . . 7  |-  ( ( ( ( ph  /\  g  e.  F )  /\  ( g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  /\  ( L `
 G )  =  ( L `  g
) )  ->  g  e.  F )
61 simpr 461 . . . . . . 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 34253 . . . . . . 7  |-  ( ( W  e.  LVec  /\  ( G  e.  F  /\  g  e.  F )  /\  ( L `  G
)  =  ( L `
 g ) )  ->  E. k  e.  K  g  =  ( G  oF  .x.  ( V  X.  { k } ) ) )
6358, 59, 60, 61, 62syl121anc 1233 . . . . . 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  oF  .x.  ( V  X.  {
k } ) ) )
6463ex 434 . . . . 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  oF  .x.  ( V  X.  {
k } ) ) ) )
6557, 64sylbid 215 . . . 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  oF  .x.  ( V  X.  {
k } ) ) ) )
6625, 46, 65pm2.61da2ne 2786 . . 3  |-  ( (
ph  /\  g  e.  F )  ->  (
( L `  G
)  C_  ( L `  g )  ->  E. k  e.  K  g  =  ( G  oF  .x.  ( V  X.  {
k } ) ) ) )
671ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  k  e.  K )  ->  W  e.  LVec )
6815ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  k  e.  K )  ->  G  e.  F )
69 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  k  e.  K )  ->  k  e.  K )
7011, 4, 5, 13, 12, 27, 67, 68, 69lkrscss 34251 . . . . . 6  |-  ( ( ( ph  /\  g  e.  F )  /\  k  e.  K )  ->  ( L `  G )  C_  ( L `  ( G  oF  .x.  ( V  X.  { k } ) ) ) )
7170ex 434 . . . . 5  |-  ( (
ph  /\  g  e.  F )  ->  (
k  e.  K  -> 
( L `  G
)  C_  ( L `  ( G  oF  .x.  ( V  X.  { k } ) ) ) ) )
72 fveq2 5872 . . . . . . 7  |-  ( g  =  ( G  oF  .x.  ( V  X.  { k } ) )  ->  ( L `  g )  =  ( L `  ( G  oF  .x.  ( V  X.  { k } ) ) ) )
7372sseq2d 3537 . . . . . 6  |-  ( g  =  ( G  oF  .x.  ( V  X.  { k } ) )  ->  ( ( L `  G )  C_  ( L `  g
)  <->  ( L `  G )  C_  ( L `  ( G  oF  .x.  ( V  X.  { k } ) ) ) ) )
7473biimprcd 225 . . . . 5  |-  ( ( L `  G ) 
C_  ( L `  ( G  oF  .x.  ( V  X.  {
k } ) ) )  ->  ( g  =  ( G  oF  .x.  ( V  X.  { k } ) )  ->  ( L `  G )  C_  ( L `  g )
) )
7571, 74syl6 33 . . . 4  |-  ( (
ph  /\  g  e.  F )  ->  (
k  e.  K  -> 
( g  =  ( G  oF  .x.  ( V  X.  { k } ) )  -> 
( L `  G
)  C_  ( L `  g ) ) ) )
7675rexlimdv 2957 . . 3  |-  ( (
ph  /\  g  e.  F )  ->  ( E. k  e.  K  g  =  ( G  oF  .x.  ( V  X.  { k } ) )  ->  ( L `  G )  C_  ( L `  g
) ) )
7766, 76impbid 191 . 2  |-  ( (
ph  /\  g  e.  F )  ->  (
( L `  G
)  C_  ( L `  g )  <->  E. k  e.  K  g  =  ( G  oF  .x.  ( V  X.  {
k } ) ) ) )
7877rabbidva 3109 1  |-  ( ph  ->  { g  e.  F  |  ( L `  G )  C_  ( L `  g ) }  =  { g  e.  F  |  E. k  e.  K  g  =  ( G  oF  .x.  ( V  X.  { k } ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   {crab 2821    C_ wss 3481   {csn 4033    X. cxp 5003   ` cfv 5594  (class class class)co 6295    oFcof 6533   Basecbs 14507   .rcmulr 14573  Scalarcsca 14575   0gc0g 14712   LModclmod 17383   LVecclvec 17619  LSHypclsh 34128  LFnlclfn 34210  LKerclk 34238
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-cntz 16227  df-lsm 16529  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-drng 17269  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lvec 17620  df-lshyp 34130  df-lfl 34211  df-lkr 34239
This theorem is referenced by: (None)
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