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Theorem dochkr1 31486
Description: A non-zero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 29078. (Contributed by NM, 2-Jan-2015.)
Hypotheses
Ref Expression
dochkr1.h  |-  H  =  ( LHyp `  K
)
dochkr1.o  |-  ._|_  =  ( ( ocH `  K
) `  W )
dochkr1.u  |-  U  =  ( ( DVecH `  K
) `  W )
dochkr1.v  |-  V  =  ( Base `  U
)
dochkr1.r  |-  R  =  (Scalar `  U )
dochkr1.z  |-  .0.  =  ( 0g `  U )
dochkr1.i  |-  .1.  =  ( 1r `  R )
dochkr1.f  |-  F  =  (LFnl `  U )
dochkr1.l  |-  L  =  (LKer `  U )
dochkr1.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
dochkr1.g  |-  ( ph  ->  G  e.  F )
dochkr1.n  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  G
) ) )  =/= 
V )
Assertion
Ref Expression
dochkr1  |-  ( ph  ->  E. x  e.  ( (  ._|_  `  ( L `
 G ) ) 
\  {  .0.  }
) ( G `  x )  =  .1.  )
Distinct variable groups:    x,  .0.    x, G    x, L    x, R    x, U    x,  ._|_    x,  .1.
Allowed substitution hints:    ph( x)    F( x)    H( x)    K( x)    V( x)    W( x)

Proof of Theorem dochkr1
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqid 2316 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
2 eqid 2316 . . . 4  |-  (LSAtoms `  U
)  =  (LSAtoms `  U
)
3 dochkr1.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 dochkr1.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
5 dochkr1.k . . . . 5  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
63, 4, 5dvhlmod 31118 . . . 4  |-  ( ph  ->  U  e.  LMod )
7 dochkr1.n . . . . 5  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  G
) ) )  =/= 
V )
8 dochkr1.o . . . . . 6  |-  ._|_  =  ( ( ocH `  K
) `  W )
9 dochkr1.v . . . . . 6  |-  V  =  ( Base `  U
)
10 dochkr1.f . . . . . 6  |-  F  =  (LFnl `  U )
11 dochkr1.l . . . . . 6  |-  L  =  (LKer `  U )
12 dochkr1.g . . . . . 6  |-  ( ph  ->  G  e.  F )
133, 8, 4, 9, 2, 10, 11, 5, 12dochkrsat2 31464 . . . . 5  |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =/=  V  <->  (  ._|_  `  ( L `  G
) )  e.  (LSAtoms `  U ) ) )
147, 13mpbid 201 . . . 4  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) )  e.  (LSAtoms `  U
) )
151, 2, 6, 14lsateln0 29003 . . 3  |-  ( ph  ->  E. z  e.  ( 
._|_  `  ( L `  G ) ) z  =/=  ( 0g `  U ) )
16 dochkr1.r . . . . . 6  |-  R  =  (Scalar `  U )
17 eqid 2316 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
185ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  z  e.  (  ._|_  `  ( L `  G )
) )  /\  z  =/=  ( 0g `  U
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
1912ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  z  e.  (  ._|_  `  ( L `  G )
) )  /\  z  =/=  ( 0g `  U
) )  ->  G  e.  F )
20 eldifsn 3783 . . . . . . . 8  |-  ( z  e.  ( (  ._|_  `  ( L `  G
) )  \  {
( 0g `  U
) } )  <->  ( z  e.  (  ._|_  `  ( L `  G )
)  /\  z  =/=  ( 0g `  U ) ) )
2120biimpri 197 . . . . . . 7  |-  ( ( z  e.  (  ._|_  `  ( L `  G
) )  /\  z  =/=  ( 0g `  U
) )  ->  z  e.  ( (  ._|_  `  ( L `  G )
)  \  { ( 0g `  U ) } ) )
2221adantll 694 . . . . . 6  |-  ( ( ( ph  /\  z  e.  (  ._|_  `  ( L `  G )
) )  /\  z  =/=  ( 0g `  U
) )  ->  z  e.  ( (  ._|_  `  ( L `  G )
)  \  { ( 0g `  U ) } ) )
233, 8, 4, 9, 16, 17, 1, 10, 11, 18, 19, 22dochfln0 31485 . . . . 5  |-  ( ( ( ph  /\  z  e.  (  ._|_  `  ( L `  G )
) )  /\  z  =/=  ( 0g `  U
) )  ->  ( G `  z )  =/=  ( 0g `  R
) )
2423ex 423 . . . 4  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  ( z  =/=  ( 0g `  U
)  ->  ( G `  z )  =/=  ( 0g `  R ) ) )
2524reximdva 2689 . . 3  |-  ( ph  ->  ( E. z  e.  (  ._|_  `  ( L `
 G ) ) z  =/=  ( 0g
`  U )  ->  E. z  e.  (  ._|_  `  ( L `  G ) ) ( G `  z )  =/=  ( 0g `  R ) ) )
2615, 25mpd 14 . 2  |-  ( ph  ->  E. z  e.  ( 
._|_  `  ( L `  G ) ) ( G `  z )  =/=  ( 0g `  R ) )
279, 10, 11, 6, 12lkrssv 29104 . . . . . . . . 9  |-  ( ph  ->  ( L `  G
)  C_  V )
28 eqid 2316 . . . . . . . . . 10  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
293, 4, 9, 28, 8dochlss 31362 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( L `  G )  C_  V
)  ->  (  ._|_  `  ( L `  G
) )  e.  (
LSubSp `  U ) )
305, 27, 29syl2anc 642 . . . . . . . 8  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) )  e.  ( LSubSp `  U
) )
316, 30jca 518 . . . . . . 7  |-  ( ph  ->  ( U  e.  LMod  /\  (  ._|_  `  ( L `
 G ) )  e.  ( LSubSp `  U
) ) )
32313ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( U  e.  LMod  /\  (  ._|_  `  ( L `
 G ) )  e.  ( LSubSp `  U
) ) )
333, 4, 5dvhlvec 31117 . . . . . . . . . 10  |-  ( ph  ->  U  e.  LVec )
34333ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  U  e.  LVec )
3516lvecdrng 15907 . . . . . . . . 9  |-  ( U  e.  LVec  ->  R  e.  DivRing )
3634, 35syl 15 . . . . . . . 8  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  R  e.  DivRing )
3763ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  U  e.  LMod )
38123ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  G  e.  F )
393, 4, 9, 8dochssv 31363 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( L `  G )  C_  V
)  ->  (  ._|_  `  ( L `  G
) )  C_  V
)
405, 27, 39syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) ) 
C_  V )
4140sselda 3214 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  z  e.  V
)
42413adant3 975 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
z  e.  V )
43 eqid 2316 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
4416, 43, 9, 10lflcl 29072 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  G  e.  F  /\  z  e.  V )  ->  ( G `  z )  e.  ( Base `  R
) )
4537, 38, 42, 44syl3anc 1182 . . . . . . . 8  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( G `  z
)  e.  ( Base `  R ) )
46 simp3 957 . . . . . . . 8  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( G `  z
)  =/=  ( 0g
`  R ) )
47 eqid 2316 . . . . . . . . 9  |-  ( invr `  R )  =  (
invr `  R )
4843, 17, 47drnginvrcl 15578 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  ( G `  z )  e.  ( Base `  R
)  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  ( ( invr `  R ) `  ( G `  z )
)  e.  ( Base `  R ) )
4936, 45, 46, 48syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( invr `  R
) `  ( G `  z ) )  e.  ( Base `  R
) )
50 simp2 956 . . . . . . 7  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
z  e.  (  ._|_  `  ( L `  G
) ) )
5149, 50jca 518 . . . . . 6  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( ( invr `  R ) `  ( G `  z )
)  e.  ( Base `  R )  /\  z  e.  (  ._|_  `  ( L `  G )
) ) )
52 eqid 2316 . . . . . . 7  |-  ( .s
`  U )  =  ( .s `  U
)
5316, 52, 43, 28lssvscl 15761 . . . . . 6  |-  ( ( ( U  e.  LMod  /\  (  ._|_  `  ( L `
 G ) )  e.  ( LSubSp `  U
) )  /\  (
( ( invr `  R
) `  ( G `  z ) )  e.  ( Base `  R
)  /\  z  e.  (  ._|_  `  ( L `  G ) ) ) )  ->  ( (
( invr `  R ) `  ( G `  z
) ) ( .s
`  U ) z )  e.  (  ._|_  `  ( L `  G
) ) )
5432, 51, 53syl2anc 642 . . . . 5  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( ( invr `  R ) `  ( G `  z )
) ( .s `  U ) z )  e.  (  ._|_  `  ( L `  G )
) )
5543, 17, 47drnginvrn0 15579 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  ( G `  z )  e.  ( Base `  R
)  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  ( ( invr `  R ) `  ( G `  z )
)  =/=  ( 0g
`  R ) )
5636, 45, 46, 55syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( invr `  R
) `  ( G `  z ) )  =/=  ( 0g `  R
) )
576adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  U  e.  LMod )
5812adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  G  e.  F
)
59 dochkr1.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  U )
6016, 17, 59, 10lfl0 29073 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  G  e.  F )  ->  ( G `  .0.  )  =  ( 0g `  R
) )
6157, 58, 60syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  ( G `  .0.  )  =  ( 0g `  R ) )
62 fveq2 5563 . . . . . . . . . 10  |-  ( z  =  .0.  ->  ( G `  z )  =  ( G `  .0.  ) )
6362eqeq1d 2324 . . . . . . . . 9  |-  ( z  =  .0.  ->  (
( G `  z
)  =  ( 0g
`  R )  <->  ( G `  .0.  )  =  ( 0g `  R ) ) )
6461, 63syl5ibrcom 213 . . . . . . . 8  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  ( z  =  .0.  ->  ( G `  z )  =  ( 0g `  R ) ) )
6564necon3d 2517 . . . . . . 7  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  ( ( G `
 z )  =/=  ( 0g `  R
)  ->  z  =/=  .0.  ) )
66653impia 1148 . . . . . 6  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
z  =/=  .0.  )
679, 52, 16, 43, 17, 59, 34, 49, 42lvecvsn0 15911 . . . . . 6  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( ( (
invr `  R ) `  ( G `  z
) ) ( .s
`  U ) z )  =/=  .0.  <->  ( (
( invr `  R ) `  ( G `  z
) )  =/=  ( 0g `  R )  /\  z  =/=  .0.  ) ) )
6856, 66, 67mpbir2and 888 . . . . 5  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( ( invr `  R ) `  ( G `  z )
) ( .s `  U ) z )  =/=  .0.  )
69 eldifsn 3783 . . . . 5  |-  ( ( ( ( invr `  R
) `  ( G `  z ) ) ( .s `  U ) z )  e.  ( (  ._|_  `  ( L `
 G ) ) 
\  {  .0.  }
)  <->  ( ( ( ( invr `  R
) `  ( G `  z ) ) ( .s `  U ) z )  e.  ( 
._|_  `  ( L `  G ) )  /\  ( ( ( invr `  R ) `  ( G `  z )
) ( .s `  U ) z )  =/=  .0.  ) )
7054, 68, 69sylanbrc 645 . . . 4  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( ( invr `  R ) `  ( G `  z )
) ( .s `  U ) z )  e.  ( (  ._|_  `  ( L `  G
) )  \  {  .0.  } ) )
71 eqid 2316 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
7216, 43, 71, 9, 52, 10lflmul 29076 . . . . . 6  |-  ( ( U  e.  LMod  /\  G  e.  F  /\  (
( ( invr `  R
) `  ( G `  z ) )  e.  ( Base `  R
)  /\  z  e.  V ) )  -> 
( G `  (
( ( invr `  R
) `  ( G `  z ) ) ( .s `  U ) z ) )  =  ( ( ( invr `  R ) `  ( G `  z )
) ( .r `  R ) ( G `
 z ) ) )
7337, 38, 49, 42, 72syl112anc 1186 . . . . 5  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( G `  (
( ( invr `  R
) `  ( G `  z ) ) ( .s `  U ) z ) )  =  ( ( ( invr `  R ) `  ( G `  z )
) ( .r `  R ) ( G `
 z ) ) )
74 dochkr1.i . . . . . . 7  |-  .1.  =  ( 1r `  R )
7543, 17, 71, 74, 47drnginvrl 15580 . . . . . 6  |-  ( ( R  e.  DivRing  /\  ( G `  z )  e.  ( Base `  R
)  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  ( ( (
invr `  R ) `  ( G `  z
) ) ( .r
`  R ) ( G `  z ) )  =  .1.  )
7636, 45, 46, 75syl3anc 1182 . . . . 5  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( ( invr `  R ) `  ( G `  z )
) ( .r `  R ) ( G `
 z ) )  =  .1.  )
7773, 76eqtrd 2348 . . . 4  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( G `  (
( ( invr `  R
) `  ( G `  z ) ) ( .s `  U ) z ) )  =  .1.  )
78 fveq2 5563 . . . . . 6  |-  ( x  =  ( ( (
invr `  R ) `  ( G `  z
) ) ( .s
`  U ) z )  ->  ( G `  x )  =  ( G `  ( ( ( invr `  R
) `  ( G `  z ) ) ( .s `  U ) z ) ) )
7978eqeq1d 2324 . . . . 5  |-  ( x  =  ( ( (
invr `  R ) `  ( G `  z
) ) ( .s
`  U ) z )  ->  ( ( G `  x )  =  .1.  <->  ( G `  ( ( ( invr `  R ) `  ( G `  z )
) ( .s `  U ) z ) )  =  .1.  )
)
8079rspcev 2918 . . . 4  |-  ( ( ( ( ( invr `  R ) `  ( G `  z )
) ( .s `  U ) z )  e.  ( (  ._|_  `  ( L `  G
) )  \  {  .0.  } )  /\  ( G `  ( (
( invr `  R ) `  ( G `  z
) ) ( .s
`  U ) z ) )  =  .1.  )  ->  E. x  e.  ( (  ._|_  `  ( L `  G )
)  \  {  .0.  } ) ( G `  x )  =  .1.  )
8170, 77, 80syl2anc 642 . . 3  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  E. x  e.  (
(  ._|_  `  ( L `  G ) )  \  {  .0.  } ) ( G `  x )  =  .1.  )
8281rexlimdv3a 2703 . 2  |-  ( ph  ->  ( E. z  e.  (  ._|_  `  ( L `
 G ) ) ( G `  z
)  =/=  ( 0g
`  R )  ->  E. x  e.  (
(  ._|_  `  ( L `  G ) )  \  {  .0.  } ) ( G `  x )  =  .1.  ) )
8326, 82mpd 14 1  |-  ( ph  ->  E. x  e.  ( (  ._|_  `  ( L `
 G ) ) 
\  {  .0.  }
) ( G `  x )  =  .1.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   E.wrex 2578    \ cdif 3183    C_ wss 3186   {csn 3674   ` cfv 5292  (class class class)co 5900   Basecbs 13195   .rcmulr 13256  Scalarcsca 13258   .scvsca 13259   0gc0g 13449   1rcur 15388   invrcinvr 15502   DivRingcdr 15561   LModclmod 15676   LSubSpclss 15738   LVecclvec 15904  LSAtomsclsa 28982  LFnlclfn 29065  LKerclk 29093   HLchlt 29358   LHypclh 29991   DVecHcdvh 31086   ocHcoch 31355
This theorem is referenced by:  lcfl6  31508
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-tpos 6276  df-undef 6340  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-sca 13271  df-vsca 13272  df-0g 13453  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-p1 14195  df-lat 14201  df-clat 14263  df-mnd 14416  df-submnd 14465  df-grp 14538  df-minusg 14539  df-sbg 14540  df-subg 14667  df-cntz 14842  df-lsm 14996  df-cmn 15140  df-abl 15141  df-mgp 15375  df-rng 15389  df-ur 15391  df-oppr 15454  df-dvdsr 15472  df-unit 15473  df-invr 15503  df-dvr 15514  df-drng 15563  df-lmod 15678  df-lss 15739  df-lsp 15778  df-lvec 15905  df-lsatoms 28984  df-lshyp 28985  df-lfl 29066  df-lkr 29094  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-llines 29505  df-lplanes 29506  df-lvols 29507  df-lines 29508  df-psubsp 29510  df-pmap 29511  df-padd 29803  df-lhyp 29995  df-laut 29996  df-ldil 30111  df-ltrn 30112  df-trl 30166  df-tgrp 30750  df-tendo 30762  df-edring 30764  df-dveca 31010  df-disoa 31037  df-dvech 31087  df-dib 31147  df-dic 31181  df-dih 31237  df-doch 31356  df-djh 31403
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