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Theorem dochkr1 34965
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 32555. (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 2422 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
2 eqid 2422 . . . 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 34597 . . . 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 34943 . . . . 5  |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =/=  V  <->  (  ._|_  `  ( L `  G
) )  e.  (LSAtoms `  U ) ) )
147, 13mpbid 213 . . . 4  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) )  e.  (LSAtoms `  U
) )
151, 2, 6, 14lsateln0 32480 . . 3  |-  ( ph  ->  E. z  e.  ( 
._|_  `  ( L `  G ) ) z  =/=  ( 0g `  U ) )
16 dochkr1.r . . . . . 6  |-  R  =  (Scalar `  U )
17 eqid 2422 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
185ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  z  e.  (  ._|_  `  ( L `  G )
) )  /\  z  =/=  ( 0g `  U
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
1912ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  z  e.  (  ._|_  `  ( L `  G )
) )  /\  z  =/=  ( 0g `  U
) )  ->  G  e.  F )
20 eldifsn 4122 . . . . . . . 8  |-  ( z  e.  ( (  ._|_  `  ( L `  G
) )  \  {
( 0g `  U
) } )  <->  ( z  e.  (  ._|_  `  ( L `  G )
)  /\  z  =/=  ( 0g `  U ) ) )
2120biimpri 209 . . . . . . 7  |-  ( ( z  e.  (  ._|_  `  ( L `  G
) )  /\  z  =/=  ( 0g `  U
) )  ->  z  e.  ( (  ._|_  `  ( L `  G )
)  \  { ( 0g `  U ) } ) )
2221adantll 718 . . . . . 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 34964 . . . . 5  |-  ( ( ( ph  /\  z  e.  (  ._|_  `  ( L `  G )
) )  /\  z  =/=  ( 0g `  U
) )  ->  ( G `  z )  =/=  ( 0g `  R
) )
2423ex 435 . . . 4  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  ( z  =/=  ( 0g `  U
)  ->  ( G `  z )  =/=  ( 0g `  R ) ) )
2524reximdva 2900 . . 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 32581 . . . . . . . . 9  |-  ( ph  ->  ( L `  G
)  C_  V )
28 eqid 2422 . . . . . . . . . 10  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
293, 4, 9, 28, 8dochlss 34841 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( L `  G )  C_  V
)  ->  (  ._|_  `  ( L `  G
) )  e.  (
LSubSp `  U ) )
305, 27, 29syl2anc 665 . . . . . . . 8  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) )  e.  ( LSubSp `  U
) )
316, 30jca 534 . . . . . . 7  |-  ( ph  ->  ( U  e.  LMod  /\  (  ._|_  `  ( L `
 G ) )  e.  ( LSubSp `  U
) ) )
32313ad2ant1 1026 . . . . . 6  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( U  e.  LMod  /\  (  ._|_  `  ( L `
 G ) )  e.  ( LSubSp `  U
) ) )
333, 4, 5dvhlvec 34596 . . . . . . . . . 10  |-  ( ph  ->  U  e.  LVec )
34333ad2ant1 1026 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  U  e.  LVec )
3516lvecdrng 18316 . . . . . . . . 9  |-  ( U  e.  LVec  ->  R  e.  DivRing )
3634, 35syl 17 . . . . . . . 8  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  R  e.  DivRing )
3763ad2ant1 1026 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  U  e.  LMod )
38123ad2ant1 1026 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  G  e.  F )
393, 4, 9, 8dochssv 34842 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( L `  G )  C_  V
)  ->  (  ._|_  `  ( L `  G
) )  C_  V
)
405, 27, 39syl2anc 665 . . . . . . . . . . 11  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) ) 
C_  V )
4140sselda 3464 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  z  e.  V
)
42413adant3 1025 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
z  e.  V )
43 eqid 2422 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
4416, 43, 9, 10lflcl 32549 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  G  e.  F  /\  z  e.  V )  ->  ( G `  z )  e.  ( Base `  R
) )
4537, 38, 42, 44syl3anc 1264 . . . . . . . 8  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( G `  z
)  e.  ( Base `  R ) )
46 simp3 1007 . . . . . . . 8  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( G `  z
)  =/=  ( 0g
`  R ) )
47 eqid 2422 . . . . . . . . 9  |-  ( invr `  R )  =  (
invr `  R )
4843, 17, 47drnginvrcl 17980 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  ( G `  z )  e.  ( Base `  R
)  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  ( ( invr `  R ) `  ( G `  z )
)  e.  ( Base `  R ) )
4936, 45, 46, 48syl3anc 1264 . . . . . . 7  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( invr `  R
) `  ( G `  z ) )  e.  ( Base `  R
) )
50 simp2 1006 . . . . . . 7  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
z  e.  (  ._|_  `  ( L `  G
) ) )
5149, 50jca 534 . . . . . 6  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( ( invr `  R ) `  ( G `  z )
)  e.  ( Base `  R )  /\  z  e.  (  ._|_  `  ( L `  G )
) ) )
52 eqid 2422 . . . . . . 7  |-  ( .s
`  U )  =  ( .s `  U
)
5316, 52, 43, 28lssvscl 18166 . . . . . 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 665 . . . . 5  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( ( invr `  R ) `  ( G `  z )
) ( .s `  U ) z )  e.  (  ._|_  `  ( L `  G )
) )
5543, 17, 47drnginvrn0 17981 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  ( G `  z )  e.  ( Base `  R
)  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  ( ( invr `  R ) `  ( G `  z )
)  =/=  ( 0g
`  R ) )
5636, 45, 46, 55syl3anc 1264 . . . . . 6  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( invr `  R
) `  ( G `  z ) )  =/=  ( 0g `  R
) )
576adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  U  e.  LMod )
5812adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  G  e.  F
)
59 dochkr1.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  U )
6016, 17, 59, 10lfl0 32550 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  G  e.  F )  ->  ( G `  .0.  )  =  ( 0g `  R
) )
6157, 58, 60syl2anc 665 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  ( G `  .0.  )  =  ( 0g `  R ) )
62 fveq2 5878 . . . . . . . . . 10  |-  ( z  =  .0.  ->  ( G `  z )  =  ( G `  .0.  ) )
6362eqeq1d 2424 . . . . . . . . 9  |-  ( z  =  .0.  ->  (
( G `  z
)  =  ( 0g
`  R )  <->  ( G `  .0.  )  =  ( 0g `  R ) ) )
6461, 63syl5ibrcom 225 . . . . . . . 8  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  ( z  =  .0.  ->  ( G `  z )  =  ( 0g `  R ) ) )
6564necon3d 2648 . . . . . . 7  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) ) )  ->  ( ( G `
 z )  =/=  ( 0g `  R
)  ->  z  =/=  .0.  ) )
66653impia 1202 . . . . . 6  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
z  =/=  .0.  )
679, 52, 16, 43, 17, 59, 34, 49, 42lvecvsn0 18320 . . . . . 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 930 . . . . 5  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( ( invr `  R ) `  ( G `  z )
) ( .s `  U ) z )  =/=  .0.  )
69 eldifsn 4122 . . . . 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 668 . . . 4  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( ( invr `  R ) `  ( G `  z )
) ( .s `  U ) z )  e.  ( (  ._|_  `  ( L `  G
) )  \  {  .0.  } ) )
71 eqid 2422 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
7216, 43, 71, 9, 52, 10lflmul 32553 . . . . . 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 1268 . . . . 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 17982 . . . . . 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 1264 . . . . 5  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( ( ( invr `  R ) `  ( G `  z )
) ( .r `  R ) ( G `
 z ) )  =  .1.  )
7773, 76eqtrd 2463 . . . 4  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  -> 
( G `  (
( ( invr `  R
) `  ( G `  z ) ) ( .s `  U ) z ) )  =  .1.  )
78 fveq2 5878 . . . . . 6  |-  ( x  =  ( ( (
invr `  R ) `  ( G `  z
) ) ( .s
`  U ) z )  ->  ( G `  x )  =  ( G `  ( ( ( invr `  R
) `  ( G `  z ) ) ( .s `  U ) z ) ) )
7978eqeq1d 2424 . . . . 5  |-  ( x  =  ( ( (
invr `  R ) `  ( G `  z
) ) ( .s
`  U ) z )  ->  ( ( G `  x )  =  .1.  <->  ( G `  ( ( ( invr `  R ) `  ( G `  z )
) ( .s `  U ) z ) )  =  .1.  )
)
8079rspcev 3182 . . . 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 665 . . 3  |-  ( (
ph  /\  z  e.  (  ._|_  `  ( L `  G ) )  /\  ( G `  z )  =/=  ( 0g `  R ) )  ->  E. x  e.  (
(  ._|_  `  ( L `  G ) )  \  {  .0.  } ) ( G `  x )  =  .1.  )
8281rexlimdv3a 2919 . 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 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   E.wrex 2776    \ cdif 3433    C_ wss 3436   {csn 3996   ` cfv 5598  (class class class)co 6302   Basecbs 15109   .rcmulr 15179  Scalarcsca 15181   .scvsca 15182   0gc0g 15326   1rcur 17723   invrcinvr 17887   DivRingcdr 17963   LModclmod 18079   LSubSpclss 18143   LVecclvec 18313  LSAtomsclsa 32459  LFnlclfn 32542  LKerclk 32570   HLchlt 32835   LHypclh 33468   DVecHcdvh 34565   ocHcoch 34834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-riotaBAD 32444
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-1st 6804  df-2nd 6805  df-tpos 6978  df-undef 7025  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-n0 10871  df-z 10939  df-uz 11161  df-fz 11786  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-sca 15194  df-vsca 15195  df-0g 15328  df-preset 16161  df-poset 16179  df-plt 16192  df-lub 16208  df-glb 16209  df-join 16210  df-meet 16211  df-p0 16273  df-p1 16274  df-lat 16280  df-clat 16342  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-submnd 16571  df-grp 16661  df-minusg 16662  df-sbg 16663  df-subg 16802  df-cntz 16959  df-lsm 17276  df-cmn 17420  df-abl 17421  df-mgp 17712  df-ur 17724  df-ring 17770  df-oppr 17839  df-dvdsr 17857  df-unit 17858  df-invr 17888  df-dvr 17899  df-drng 17965  df-lmod 18081  df-lss 18144  df-lsp 18183  df-lvec 18314  df-lsatoms 32461  df-lshyp 32462  df-lfl 32543  df-lkr 32571  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-llines 32982  df-lplanes 32983  df-lvols 32984  df-lines 32985  df-psubsp 32987  df-pmap 32988  df-padd 33280  df-lhyp 33472  df-laut 33473  df-ldil 33588  df-ltrn 33589  df-trl 33644  df-tgrp 34229  df-tendo 34241  df-edring 34243  df-dveca 34489  df-disoa 34516  df-dvech 34566  df-dib 34626  df-dic 34660  df-dih 34716  df-doch 34835  df-djh 34882
This theorem is referenced by:  lcfl6  34987
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