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Theorem dochkr1 35429
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 33021. (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 2451 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
2 eqid 2451 . . . 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 35061 . . . 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 35407 . . . . 5  |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =/=  V  <->  (  ._|_  `  ( L `  G
) )  e.  (LSAtoms `  U ) ) )
147, 13mpbid 210 . . . 4  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) )  e.  (LSAtoms `  U
) )
151, 2, 6, 14lsateln0 32946 . . 3  |-  ( ph  ->  E. z  e.  ( 
._|_  `  ( L `  G ) ) z  =/=  ( 0g `  U ) )
16 dochkr1.r . . . . . 6  |-  R  =  (Scalar `  U )
17 eqid 2451 . . . . . 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 4098 . . . . . . . 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 35428 . . . . 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 2924 . . 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 33047 . . . . . . . . 9  |-  ( ph  ->  ( L `  G
)  C_  V )
28 eqid 2451 . . . . . . . . . 10  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
293, 4, 9, 28, 8dochlss 35305 . . . . . . . . 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 1009 . . . . . 6  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( U  e.  LMod  /\  (  ._|_  `  ( L `
 G ) )  e.  ( LSubSp `  U
) ) )
333, 4, 5dvhlvec 35060 . . . . . . . . . 10  |-  ( ph  ->  U  e.  LVec )
34333ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  U  e.  LVec )
3516lvecdrng 17292 . . . . . . . . 9  |-  ( U  e.  LVec  ->  R  e.  DivRing )
3634, 35syl 16 . . . . . . . 8  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  R  e.  DivRing )
3763ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  U  e.  LMod )
38123ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  G  e.  F )
393, 4, 9, 8dochssv 35306 . . . . . . . . . . . 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 3454 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  z  e.  V
)
42413adant3 1008 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
z  e.  V )
43 eqid 2451 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
4416, 43, 9, 10lflcl 33015 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  G  e.  F  /\  z  e.  V )  ->  ( G `  z )  e.  ( Base `  R
) )
4537, 38, 42, 44syl3anc 1219 . . . . . . . 8  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( G `  z
)  e.  ( Base `  R ) )
46 simp3 990 . . . . . . . 8  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( G `  z
)  =/=  ( 0g
`  R ) )
47 eqid 2451 . . . . . . . . 9  |-  ( invr `  R )  =  (
invr `  R )
4843, 17, 47drnginvrcl 16955 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  ( G `  z )  e.  ( Base `  R
)  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  ( ( invr `  R ) `  ( G `  z )
)  e.  ( Base `  R ) )
4936, 45, 46, 48syl3anc 1219 . . . . . . 7  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( invr `  R
) `  ( G `  z ) )  e.  ( Base `  R
) )
50 simp2 989 . . . . . . 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 2451 . . . . . . 7  |-  ( .s
`  U )  =  ( .s `  U
)
5316, 52, 43, 28lssvscl 17142 . . . . . 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 16956 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  ( G `  z )  e.  ( Base `  R
)  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  ( ( invr `  R ) `  ( G `  z )
)  =/=  ( 0g
`  R ) )
5636, 45, 46, 55syl3anc 1219 . . . . . 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 33016 . . . . . . . . . 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 5789 . . . . . . . . . 10  |-  ( z  =  .0.  ->  ( G `  z )  =  ( G `  .0.  ) )
6362eqeq1d 2453 . . . . . . . . 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 2672 . . . . . . 7  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  ( ( G `
 z )  =/=  ( 0g `  R
)  ->  z  =/=  .0.  ) )
66653impia 1185 . . . . . 6  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
z  =/=  .0.  )
679, 52, 16, 43, 17, 59, 34, 49, 42lvecvsn0 17296 . . . . . 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 913 . . . . 5  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( ( invr `  R ) `  ( G `  z )
) ( .s `  U ) z )  =/=  .0.  )
69 eldifsn 4098 . . . . 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 2451 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
7216, 43, 71, 9, 52, 10lflmul 33019 . . . . . 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 1223 . . . . 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 16957 . . . . . 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 1219 . . . . 5  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( ( invr `  R ) `  ( G `  z )
) ( .r `  R ) ( G `
 z ) )  =  .1.  )
7773, 76eqtrd 2492 . . . 4  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( G `  (
( ( invr `  R
) `  ( G `  z ) ) ( .s `  U ) z ) )  =  .1.  )
78 fveq2 5789 . . . . . 6  |-  ( x  =  ( ( (
invr `  R ) `  ( G `  z
) ) ( .s
`  U ) z )  ->  ( G `  x )  =  ( G `  ( ( ( invr `  R
) `  ( G `  z ) ) ( .s `  U ) z ) ) )
7978eqeq1d 2453 . . . . 5  |-  ( x  =  ( ( (
invr `  R ) `  ( G `  z
) ) ( .s
`  U ) z )  ->  ( ( G `  x )  =  .1.  <->  ( G `  ( ( ( invr `  R ) `  ( G `  z )
) ( .s `  U ) z ) )  =  .1.  )
)
8079rspcev 3169 . . . 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 2939 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2644   E.wrex 2796    \ cdif 3423    C_ wss 3426   {csn 3975   ` cfv 5516  (class class class)co 6190   Basecbs 14276   .rcmulr 14341  Scalarcsca 14343   .scvsca 14344   0gc0g 14480   1rcur 16708   invrcinvr 16869   DivRingcdr 16938   LModclmod 17054   LSubSpclss 17119   LVecclvec 17289  LSAtomsclsa 32925  LFnlclfn 33008  LKerclk 33036   HLchlt 33301   LHypclh 33934   DVecHcdvh 35029   ocHcoch 35298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-riotaBAD 32910
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-tpos 6845  df-undef 6892  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-n0 10681  df-z 10748  df-uz 10963  df-fz 11539  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-sca 14356  df-vsca 14357  df-0g 14482  df-poset 15218  df-plt 15230  df-lub 15246  df-glb 15247  df-join 15248  df-meet 15249  df-p0 15311  df-p1 15312  df-lat 15318  df-clat 15380  df-mnd 15517  df-submnd 15567  df-grp 15647  df-minusg 15648  df-sbg 15649  df-subg 15780  df-cntz 15937  df-lsm 16239  df-cmn 16383  df-abl 16384  df-mgp 16697  df-ur 16709  df-rng 16753  df-oppr 16821  df-dvdsr 16839  df-unit 16840  df-invr 16870  df-dvr 16881  df-drng 16940  df-lmod 17056  df-lss 17120  df-lsp 17159  df-lvec 17290  df-lsatoms 32927  df-lshyp 32928  df-lfl 33009  df-lkr 33037  df-oposet 33127  df-ol 33129  df-oml 33130  df-covers 33217  df-ats 33218  df-atl 33249  df-cvlat 33273  df-hlat 33302  df-llines 33448  df-lplanes 33449  df-lvols 33450  df-lines 33451  df-psubsp 33453  df-pmap 33454  df-padd 33746  df-lhyp 33938  df-laut 33939  df-ldil 34054  df-ltrn 34055  df-trl 34109  df-tgrp 34693  df-tendo 34705  df-edring 34707  df-dveca 34953  df-disoa 34980  df-dvech 35030  df-dib 35090  df-dic 35124  df-dih 35180  df-doch 35299  df-djh 35346
This theorem is referenced by:  lcfl6  35451
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