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Theorem dochkr1 36275
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 33867. (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 2467 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
2 eqid 2467 . . . 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 35907 . . . 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 36253 . . . . 5  |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =/=  V  <->  (  ._|_  `  ( L `  G
) )  e.  (LSAtoms `  U ) ) )
147, 13mpbid 210 . . . 4  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) )  e.  (LSAtoms `  U
) )
151, 2, 6, 14lsateln0 33792 . . 3  |-  ( ph  ->  E. z  e.  ( 
._|_  `  ( L `  G ) ) z  =/=  ( 0g `  U ) )
16 dochkr1.r . . . . . 6  |-  R  =  (Scalar `  U )
17 eqid 2467 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
185ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  z  e.  (  ._|_  `  ( L `  G )
) )  /\  z  =/=  ( 0g `  U
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
1912ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  z  e.  (  ._|_  `  ( L `  G )
) )  /\  z  =/=  ( 0g `  U
) )  ->  G  e.  F )
20 eldifsn 4152 . . . . . . . 8  |-  ( z  e.  ( (  ._|_  `  ( L `  G
) )  \  {
( 0g `  U
) } )  <->  ( z  e.  (  ._|_  `  ( L `  G )
)  /\  z  =/=  ( 0g `  U ) ) )
2120biimpri 206 . . . . . . 7  |-  ( ( z  e.  (  ._|_  `  ( L `  G
) )  /\  z  =/=  ( 0g `  U
) )  ->  z  e.  ( (  ._|_  `  ( L `  G )
)  \  { ( 0g `  U ) } ) )
2221adantll 713 . . . . . 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 36274 . . . . 5  |-  ( ( ( ph  /\  z  e.  (  ._|_  `  ( L `  G )
) )  /\  z  =/=  ( 0g `  U
) )  ->  ( G `  z )  =/=  ( 0g `  R
) )
2423ex 434 . . . 4  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  ( z  =/=  ( 0g `  U
)  ->  ( G `  z )  =/=  ( 0g `  R ) ) )
2524reximdva 2938 . . 3  |-  ( ph  ->  ( E. z  e.  (  ._|_  `  ( L `
 G ) ) z  =/=  ( 0g
`  U )  ->  E. z  e.  (  ._|_  `  ( L `  G ) ) ( G `  z )  =/=  ( 0g `  R ) ) )
2615, 25mpd 15 . 2  |-  ( ph  ->  E. z  e.  ( 
._|_  `  ( L `  G ) ) ( G `  z )  =/=  ( 0g `  R ) )
279, 10, 11, 6, 12lkrssv 33893 . . . . . . . . 9  |-  ( ph  ->  ( L `  G
)  C_  V )
28 eqid 2467 . . . . . . . . . 10  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
293, 4, 9, 28, 8dochlss 36151 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( L `  G )  C_  V
)  ->  (  ._|_  `  ( L `  G
) )  e.  (
LSubSp `  U ) )
305, 27, 29syl2anc 661 . . . . . . . 8  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) )  e.  ( LSubSp `  U
) )
316, 30jca 532 . . . . . . 7  |-  ( ph  ->  ( U  e.  LMod  /\  (  ._|_  `  ( L `
 G ) )  e.  ( LSubSp `  U
) ) )
32313ad2ant1 1017 . . . . . 6  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( U  e.  LMod  /\  (  ._|_  `  ( L `
 G ) )  e.  ( LSubSp `  U
) ) )
333, 4, 5dvhlvec 35906 . . . . . . . . . 10  |-  ( ph  ->  U  e.  LVec )
34333ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  U  e.  LVec )
3516lvecdrng 17534 . . . . . . . . 9  |-  ( U  e.  LVec  ->  R  e.  DivRing )
3634, 35syl 16 . . . . . . . 8  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  R  e.  DivRing )
3763ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  U  e.  LMod )
38123ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  G  e.  F )
393, 4, 9, 8dochssv 36152 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( L `  G )  C_  V
)  ->  (  ._|_  `  ( L `  G
) )  C_  V
)
405, 27, 39syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) ) 
C_  V )
4140sselda 3504 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  z  e.  V
)
42413adant3 1016 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
z  e.  V )
43 eqid 2467 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
4416, 43, 9, 10lflcl 33861 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  G  e.  F  /\  z  e.  V )  ->  ( G `  z )  e.  ( Base `  R
) )
4537, 38, 42, 44syl3anc 1228 . . . . . . . 8  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( G `  z
)  e.  ( Base `  R ) )
46 simp3 998 . . . . . . . 8  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( G `  z
)  =/=  ( 0g
`  R ) )
47 eqid 2467 . . . . . . . . 9  |-  ( invr `  R )  =  (
invr `  R )
4843, 17, 47drnginvrcl 17196 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  ( G `  z )  e.  ( Base `  R
)  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  ( ( invr `  R ) `  ( G `  z )
)  e.  ( Base `  R ) )
4936, 45, 46, 48syl3anc 1228 . . . . . . 7  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( invr `  R
) `  ( G `  z ) )  e.  ( Base `  R
) )
50 simp2 997 . . . . . . 7  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
z  e.  (  ._|_  `  ( L `  G
) ) )
5149, 50jca 532 . . . . . 6  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( ( invr `  R ) `  ( G `  z )
)  e.  ( Base `  R )  /\  z  e.  (  ._|_  `  ( L `  G )
) ) )
52 eqid 2467 . . . . . . 7  |-  ( .s
`  U )  =  ( .s `  U
)
5316, 52, 43, 28lssvscl 17384 . . . . . 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 661 . . . . 5  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( ( invr `  R ) `  ( G `  z )
) ( .s `  U ) z )  e.  (  ._|_  `  ( L `  G )
) )
5543, 17, 47drnginvrn0 17197 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  ( G `  z )  e.  ( Base `  R
)  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  ( ( invr `  R ) `  ( G `  z )
)  =/=  ( 0g
`  R ) )
5636, 45, 46, 55syl3anc 1228 . . . . . 6  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( invr `  R
) `  ( G `  z ) )  =/=  ( 0g `  R
) )
576adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  U  e.  LMod )
5812adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  G  e.  F
)
59 dochkr1.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  U )
6016, 17, 59, 10lfl0 33862 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  G  e.  F )  ->  ( G `  .0.  )  =  ( 0g `  R
) )
6157, 58, 60syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  ( G `  .0.  )  =  ( 0g `  R ) )
62 fveq2 5864 . . . . . . . . . 10  |-  ( z  =  .0.  ->  ( G `  z )  =  ( G `  .0.  ) )
6362eqeq1d 2469 . . . . . . . . 9  |-  ( z  =  .0.  ->  (
( G `  z
)  =  ( 0g
`  R )  <->  ( G `  .0.  )  =  ( 0g `  R ) ) )
6461, 63syl5ibrcom 222 . . . . . . . 8  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  ( z  =  .0.  ->  ( G `  z )  =  ( 0g `  R ) ) )
6564necon3d 2691 . . . . . . 7  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  ( ( G `
 z )  =/=  ( 0g `  R
)  ->  z  =/=  .0.  ) )
66653impia 1193 . . . . . 6  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
z  =/=  .0.  )
679, 52, 16, 43, 17, 59, 34, 49, 42lvecvsn0 17538 . . . . . 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 920 . . . . 5  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( ( invr `  R ) `  ( G `  z )
) ( .s `  U ) z )  =/=  .0.  )
69 eldifsn 4152 . . . . 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 664 . . . 4  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( ( invr `  R ) `  ( G `  z )
) ( .s `  U ) z )  e.  ( (  ._|_  `  ( L `  G
) )  \  {  .0.  } ) )
71 eqid 2467 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
7216, 43, 71, 9, 52, 10lflmul 33865 . . . . . 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 1232 . . . . 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 17198 . . . . . 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 1228 . . . . 5  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( ( invr `  R ) `  ( G `  z )
) ( .r `  R ) ( G `
 z ) )  =  .1.  )
7773, 76eqtrd 2508 . . . 4  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( G `  (
( ( invr `  R
) `  ( G `  z ) ) ( .s `  U ) z ) )  =  .1.  )
78 fveq2 5864 . . . . . 6  |-  ( x  =  ( ( (
invr `  R ) `  ( G `  z
) ) ( .s
`  U ) z )  ->  ( G `  x )  =  ( G `  ( ( ( invr `  R
) `  ( G `  z ) ) ( .s `  U ) z ) ) )
7978eqeq1d 2469 . . . . 5  |-  ( x  =  ( ( (
invr `  R ) `  ( G `  z
) ) ( .s
`  U ) z )  ->  ( ( G `  x )  =  .1.  <->  ( G `  ( ( ( invr `  R ) `  ( G `  z )
) ( .s `  U ) z ) )  =  .1.  )
)
8079rspcev 3214 . . . 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 661 . . 3  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  E. x  e.  (
(  ._|_  `  ( L `  G ) )  \  {  .0.  } ) ( G `  x )  =  .1.  )
8281rexlimdv3a 2957 . 2  |-  ( ph  ->  ( E. z  e.  (  ._|_  `  ( L `
 G ) ) ( G `  z
)  =/=  ( 0g
`  R )  ->  E. x  e.  (
(  ._|_  `  ( L `  G ) )  \  {  .0.  } ) ( G `  x )  =  .1.  ) )
8326, 82mpd 15 1  |-  ( ph  ->  E. x  e.  ( (  ._|_  `  ( L `
 G ) ) 
\  {  .0.  }
) ( G `  x )  =  .1.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    \ cdif 3473    C_ wss 3476   {csn 4027   ` cfv 5586  (class class class)co 6282   Basecbs 14486   .rcmulr 14552  Scalarcsca 14554   .scvsca 14555   0gc0g 14691   1rcur 16943   invrcinvr 17104   DivRingcdr 17179   LModclmod 17295   LSubSpclss 17361   LVecclvec 17531  LSAtomsclsa 33771  LFnlclfn 33854  LKerclk 33882   HLchlt 34147   LHypclh 34780   DVecHcdvh 35875   ocHcoch 36144
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-riotaBAD 33756
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-undef 6999  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-0g 14693  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-mnd 15728  df-submnd 15778  df-grp 15858  df-minusg 15859  df-sbg 15860  df-subg 15993  df-cntz 16150  df-lsm 16452  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-oppr 17056  df-dvdsr 17074  df-unit 17075  df-invr 17105  df-dvr 17116  df-drng 17181  df-lmod 17297  df-lss 17362  df-lsp 17401  df-lvec 17532  df-lsatoms 33773  df-lshyp 33774  df-lfl 33855  df-lkr 33883  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lvols 34296  df-lines 34297  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955  df-tgrp 35539  df-tendo 35551  df-edring 35553  df-dveca 35799  df-disoa 35826  df-dvech 35876  df-dib 35936  df-dic 35970  df-dih 36026  df-doch 36145  df-djh 36192
This theorem is referenced by:  lcfl6  36297
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