Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lcfl7lem Structured version   Visualization version   Unicode version

Theorem lcfl7lem 35138
Description: Lemma for lcfl7N 35140. If two functionals  G and  J are equal, they are determined by the same vector. (Contributed by NM, 4-Jan-2015.)
Hypotheses
Ref Expression
lcfl7lem.h  |-  H  =  ( LHyp `  K
)
lcfl7lem.o  |-  ._|_  =  ( ( ocH `  K
) `  W )
lcfl7lem.u  |-  U  =  ( ( DVecH `  K
) `  W )
lcfl7lem.v  |-  V  =  ( Base `  U
)
lcfl7lem.a  |-  .+  =  ( +g  `  U )
lcfl7lem.t  |-  .x.  =  ( .s `  U )
lcfl7lem.s  |-  S  =  (Scalar `  U )
lcfl7lem.r  |-  R  =  ( Base `  S
)
lcfl7lem.z  |-  .0.  =  ( 0g `  U )
lcfl7lem.f  |-  F  =  (LFnl `  U )
lcfl7lem.l  |-  L  =  (LKer `  U )
lcfl7lem.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
lcfl7lem.g  |-  G  =  ( v  e.  V  |->  ( iota_ k  e.  R  E. w  e.  (  ._|_  `  { X }
) v  =  ( w  .+  ( k 
.x.  X ) ) ) )
lcfl7lem.j  |-  J  =  ( v  e.  V  |->  ( iota_ k  e.  R  E. w  e.  (  ._|_  `  { Y }
) v  =  ( w  .+  ( k 
.x.  Y ) ) ) )
lcfl7lem.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
lcfl7lem.x2  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
lcfl7lem.gj  |-  ( ph  ->  G  =  J )
Assertion
Ref Expression
lcfl7lem  |-  ( ph  ->  X  =  Y )
Distinct variable groups:    v, k, w,  .+    ._|_ , k, v, w   
w,  .0.    R, k, v    S, k, w    v, V    .x. , k, v, w    k, X, v, w    k, Y, v, w
Allowed substitution hints:    ph( w, v, k)    R( w)    S( v)    U( w, v, k)    F( w, v, k)    G( w, v, k)    H( w, v, k)    J( w, v, k)    K( w, v, k)    L( w, v, k)    V( w, k)    W( w, v, k)    .0. ( v, k)

Proof of Theorem lcfl7lem
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 lcfl7lem.h . . . . . 6  |-  H  =  ( LHyp `  K
)
2 lcfl7lem.o . . . . . 6  |-  ._|_  =  ( ( ocH `  K
) `  W )
3 lcfl7lem.u . . . . . 6  |-  U  =  ( ( DVecH `  K
) `  W )
4 lcfl7lem.v . . . . . 6  |-  V  =  ( Base `  U
)
5 lcfl7lem.z . . . . . 6  |-  .0.  =  ( 0g `  U )
6 lcfl7lem.a . . . . . 6  |-  .+  =  ( +g  `  U )
7 lcfl7lem.t . . . . . 6  |-  .x.  =  ( .s `  U )
8 lcfl7lem.l . . . . . 6  |-  L  =  (LKer `  U )
9 lcfl7lem.s . . . . . 6  |-  S  =  (Scalar `  U )
10 lcfl7lem.r . . . . . 6  |-  R  =  ( Base `  S
)
11 lcfl7lem.g . . . . . 6  |-  G  =  ( v  e.  V  |->  ( iota_ k  e.  R  E. w  e.  (  ._|_  `  { X }
) v  =  ( w  .+  ( k 
.x.  X ) ) ) )
12 lcfl7lem.k . . . . . 6  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
13 lcfl7lem.x . . . . . 6  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dochsnkr2cl 35113 . . . . 5  |-  ( ph  ->  X  e.  ( ( 
._|_  `  ( L `  G ) )  \  {  .0.  } ) )
1514eldifad 3402 . . . 4  |-  ( ph  ->  X  e.  (  ._|_  `  ( L `  G
) ) )
16 lcfl7lem.gj . . . . . . . 8  |-  ( ph  ->  G  =  J )
1716fveq2d 5883 . . . . . . 7  |-  ( ph  ->  ( L `  G
)  =  ( L `
 J ) )
18 lcfl7lem.j . . . . . . . 8  |-  J  =  ( v  e.  V  |->  ( iota_ k  e.  R  E. w  e.  (  ._|_  `  { Y }
) v  =  ( w  .+  ( k 
.x.  Y ) ) ) )
19 lcfl7lem.x2 . . . . . . . 8  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
201, 2, 3, 4, 5, 6, 7, 8, 9, 10, 18, 12, 19dochsnkr2 35112 . . . . . . 7  |-  ( ph  ->  ( L `  J
)  =  (  ._|_  `  { Y } ) )
2117, 20eqtrd 2505 . . . . . 6  |-  ( ph  ->  ( L `  G
)  =  (  ._|_  `  { Y } ) )
2221fveq2d 5883 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) )  =  (  ._|_  `  (  ._|_  `  { Y }
) ) )
23 eqid 2471 . . . . . . 7  |-  ( LSpan `  U )  =  (
LSpan `  U )
2419eldifad 3402 . . . . . . . 8  |-  ( ph  ->  Y  e.  V )
2524snssd 4108 . . . . . . 7  |-  ( ph  ->  { Y }  C_  V )
261, 3, 2, 4, 23, 12, 25dochocsp 35018 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( (
LSpan `  U ) `  { Y } ) )  =  (  ._|_  `  { Y } ) )
2726fveq2d 5883 . . . . 5  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( ( LSpan `  U
) `  { Y } ) ) )  =  (  ._|_  `  (  ._|_  `  { Y }
) ) )
28 eqid 2471 . . . . . . . 8  |-  ( (
DIsoH `  K ) `  W )  =  ( ( DIsoH `  K ) `  W )
291, 3, 4, 23, 28dihlsprn 34970 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  e.  V
)  ->  ( ( LSpan `  U ) `  { Y } )  e. 
ran  ( ( DIsoH `  K ) `  W
) )
3012, 24, 29syl2anc 673 . . . . . 6  |-  ( ph  ->  ( ( LSpan `  U
) `  { Y } )  e.  ran  ( ( DIsoH `  K
) `  W )
)
311, 28, 2dochoc 35006 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( LSpan `  U ) `  { Y } )  e.  ran  ( ( DIsoH `  K
) `  W )
)  ->  (  ._|_  `  (  ._|_  `  ( (
LSpan `  U ) `  { Y } ) ) )  =  ( (
LSpan `  U ) `  { Y } ) )
3212, 30, 31syl2anc 673 . . . . 5  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( ( LSpan `  U
) `  { Y } ) ) )  =  ( ( LSpan `  U ) `  { Y } ) )
3322, 27, 323eqtr2d 2511 . . . 4  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) )  =  ( ( LSpan `  U ) `  { Y } ) )
3415, 33eleqtrd 2551 . . 3  |-  ( ph  ->  X  e.  ( (
LSpan `  U ) `  { Y } ) )
351, 3, 12dvhlmod 34749 . . . 4  |-  ( ph  ->  U  e.  LMod )
369, 10, 4, 7, 23lspsnel 18304 . . . 4  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( X  e.  ( ( LSpan `  U ) `  { Y } )  <->  E. s  e.  R  X  =  ( s  .x.  Y
) ) )
3735, 24, 36syl2anc 673 . . 3  |-  ( ph  ->  ( X  e.  ( ( LSpan `  U ) `  { Y } )  <->  E. s  e.  R  X  =  ( s  .x.  Y ) ) )
3834, 37mpbid 215 . 2  |-  ( ph  ->  E. s  e.  R  X  =  ( s  .x.  Y ) )
39 simp3 1032 . . . 4  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  X  =  ( s  .x.  Y
) )
40 fveq2 5879 . . . . . . . . . 10  |-  ( X  =  ( s  .x.  Y )  ->  ( G `  X )  =  ( G `  ( s  .x.  Y
) ) )
41403ad2ant3 1053 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  X )  =  ( G `  ( s 
.x.  Y ) ) )
42 eqid 2471 . . . . . . . . . . . 12  |-  ( 1r
`  S )  =  ( 1r `  S
)
431, 2, 3, 4, 6, 7, 5, 9, 10, 42, 12, 19, 18dochfl1 35115 . . . . . . . . . . 11  |-  ( ph  ->  ( J `  Y
)  =  ( 1r
`  S ) )
4416fveq1d 5881 . . . . . . . . . . 11  |-  ( ph  ->  ( G `  Y
)  =  ( J `
 Y ) )
451, 2, 3, 4, 6, 7, 5, 9, 10, 42, 12, 13, 11dochfl1 35115 . . . . . . . . . . 11  |-  ( ph  ->  ( G `  X
)  =  ( 1r
`  S ) )
4643, 44, 453eqtr4rd 2516 . . . . . . . . . 10  |-  ( ph  ->  ( G `  X
)  =  ( G `
 Y ) )
47463ad2ant1 1051 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  X )  =  ( G `  Y ) )
48353ad2ant1 1051 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  U  e.  LMod )
49 lcfl7lem.f . . . . . . . . . . . 12  |-  F  =  (LFnl `  U )
501, 2, 3, 4, 5, 6, 7, 49, 9, 10, 11, 12, 13dochflcl 35114 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  F )
51503ad2ant1 1051 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  G  e.  F )
52 simp2 1031 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  s  e.  R )
53243ad2ant1 1051 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  Y  e.  V )
54 eqid 2471 . . . . . . . . . . 11  |-  ( .r
`  S )  =  ( .r `  S
)
559, 10, 54, 4, 7, 49lflmul 32705 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  G  e.  F  /\  (
s  e.  R  /\  Y  e.  V )
)  ->  ( G `  ( s  .x.  Y
) )  =  ( s ( .r `  S ) ( G `
 Y ) ) )
5648, 51, 52, 53, 55syl112anc 1296 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  ( s  .x.  Y
) )  =  ( s ( .r `  S ) ( G `
 Y ) ) )
5741, 47, 563eqtr3d 2513 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  Y )  =  ( s ( .r `  S ) ( G `
 Y ) ) )
5857oveq1d 6323 . . . . . . 7  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( ( G `  Y )
( .r `  S
) ( ( invr `  S ) `  ( G `  Y )
) )  =  ( ( s ( .r
`  S ) ( G `  Y ) ) ( .r `  S ) ( (
invr `  S ) `  ( G `  Y
) ) ) )
599lmodring 18177 . . . . . . . . . 10  |-  ( U  e.  LMod  ->  S  e. 
Ring )
6035, 59syl 17 . . . . . . . . 9  |-  ( ph  ->  S  e.  Ring )
61603ad2ant1 1051 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  S  e.  Ring )
629, 10, 4, 49lflcl 32701 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  G  e.  F  /\  Y  e.  V )  ->  ( G `  Y )  e.  R )
6335, 50, 24, 62syl3anc 1292 . . . . . . . . 9  |-  ( ph  ->  ( G `  Y
)  e.  R )
64633ad2ant1 1051 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  Y )  e.  R
)
651, 3, 12dvhlvec 34748 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  LVec )
669lvecdrng 18406 . . . . . . . . . . 11  |-  ( U  e.  LVec  ->  S  e.  DivRing )
6765, 66syl 17 . . . . . . . . . 10  |-  ( ph  ->  S  e.  DivRing )
6844, 43eqtrd 2505 . . . . . . . . . . 11  |-  ( ph  ->  ( G `  Y
)  =  ( 1r
`  S ) )
69 eqid 2471 . . . . . . . . . . . . 13  |-  ( 0g
`  S )  =  ( 0g `  S
)
7069, 42drngunz 18068 . . . . . . . . . . . 12  |-  ( S  e.  DivRing  ->  ( 1r `  S )  =/=  ( 0g `  S ) )
7167, 70syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( 1r `  S
)  =/=  ( 0g
`  S ) )
7268, 71eqnetrd 2710 . . . . . . . . . 10  |-  ( ph  ->  ( G `  Y
)  =/=  ( 0g
`  S ) )
73 eqid 2471 . . . . . . . . . . 11  |-  ( invr `  S )  =  (
invr `  S )
7410, 69, 73drnginvrcl 18070 . . . . . . . . . 10  |-  ( ( S  e.  DivRing  /\  ( G `  Y )  e.  R  /\  ( G `  Y )  =/=  ( 0g `  S
) )  ->  (
( invr `  S ) `  ( G `  Y
) )  e.  R
)
7567, 63, 72, 74syl3anc 1292 . . . . . . . . 9  |-  ( ph  ->  ( ( invr `  S
) `  ( G `  Y ) )  e.  R )
76753ad2ant1 1051 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( ( invr `  S ) `  ( G `  Y ) )  e.  R )
7710, 54ringass 17875 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  (
s  e.  R  /\  ( G `  Y )  e.  R  /\  (
( invr `  S ) `  ( G `  Y
) )  e.  R
) )  ->  (
( s ( .r
`  S ) ( G `  Y ) ) ( .r `  S ) ( (
invr `  S ) `  ( G `  Y
) ) )  =  ( s ( .r
`  S ) ( ( G `  Y
) ( .r `  S ) ( (
invr `  S ) `  ( G `  Y
) ) ) ) )
7861, 52, 64, 76, 77syl13anc 1294 . . . . . . 7  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( (
s ( .r `  S ) ( G `
 Y ) ) ( .r `  S
) ( ( invr `  S ) `  ( G `  Y )
) )  =  ( s ( .r `  S ) ( ( G `  Y ) ( .r `  S
) ( ( invr `  S ) `  ( G `  Y )
) ) ) )
7910, 69, 54, 42, 73drnginvrr 18073 . . . . . . . . . 10  |-  ( ( S  e.  DivRing  /\  ( G `  Y )  e.  R  /\  ( G `  Y )  =/=  ( 0g `  S
) )  ->  (
( G `  Y
) ( .r `  S ) ( (
invr `  S ) `  ( G `  Y
) ) )  =  ( 1r `  S
) )
8067, 63, 72, 79syl3anc 1292 . . . . . . . . 9  |-  ( ph  ->  ( ( G `  Y ) ( .r
`  S ) ( ( invr `  S
) `  ( G `  Y ) ) )  =  ( 1r `  S ) )
81803ad2ant1 1051 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( ( G `  Y )
( .r `  S
) ( ( invr `  S ) `  ( G `  Y )
) )  =  ( 1r `  S ) )
8281oveq2d 6324 . . . . . . 7  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( s
( .r `  S
) ( ( G `
 Y ) ( .r `  S ) ( ( invr `  S
) `  ( G `  Y ) ) ) )  =  ( s ( .r `  S
) ( 1r `  S ) ) )
8358, 78, 823eqtrrd 2510 . . . . . 6  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( s
( .r `  S
) ( 1r `  S ) )  =  ( ( G `  Y ) ( .r
`  S ) ( ( invr `  S
) `  ( G `  Y ) ) ) )
8410, 54, 42ringridm 17883 . . . . . . 7  |-  ( ( S  e.  Ring  /\  s  e.  R )  ->  (
s ( .r `  S ) ( 1r
`  S ) )  =  s )
8561, 52, 84syl2anc 673 . . . . . 6  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( s
( .r `  S
) ( 1r `  S ) )  =  s )
8683, 85, 813eqtr3d 2513 . . . . 5  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  s  =  ( 1r `  S ) )
87 oveq1 6315 . . . . . 6  |-  ( s  =  ( 1r `  S )  ->  (
s  .x.  Y )  =  ( ( 1r
`  S )  .x.  Y ) )
884, 9, 7, 42lmodvs1 18197 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  (
( 1r `  S
)  .x.  Y )  =  Y )
8935, 24, 88syl2anc 673 . . . . . . 7  |-  ( ph  ->  ( ( 1r `  S )  .x.  Y
)  =  Y )
90893ad2ant1 1051 . . . . . 6  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( ( 1r `  S )  .x.  Y )  =  Y )
9187, 90sylan9eqr 2527 . . . . 5  |-  ( ( ( ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y
) )  /\  s  =  ( 1r `  S ) )  -> 
( s  .x.  Y
)  =  Y )
9286, 91mpdan 681 . . . 4  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( s  .x.  Y )  =  Y )
9339, 92eqtrd 2505 . . 3  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  X  =  Y )
9493rexlimdv3a 2873 . 2  |-  ( ph  ->  ( E. s  e.  R  X  =  ( s  .x.  Y )  ->  X  =  Y ) )
9538, 94mpd 15 1  |-  ( ph  ->  X  =  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757    \ cdif 3387   {csn 3959    |-> cmpt 4454   ran crn 4840   ` cfv 5589   iota_crio 6269  (class class class)co 6308   Basecbs 15199   +g cplusg 15268   .rcmulr 15269  Scalarcsca 15271   .scvsca 15272   0gc0g 15416   1rcur 17813   Ringcrg 17858   invrcinvr 17977   DivRingcdr 18053   LModclmod 18169   LSpanclspn 18272   LVecclvec 18403  LFnlclfn 32694  LKerclk 32722   HLchlt 32987   LHypclh 33620   DVecHcdvh 34717   DIsoHcdih 34867   ocHcoch 34986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-riotaBAD 32589
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-tpos 6991  df-undef 7038  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-sca 15284  df-vsca 15285  df-0g 15418  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-p1 16364  df-lat 16370  df-clat 16432  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-subg 16892  df-cntz 17049  df-lsm 17366  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-dvr 17989  df-drng 18055  df-lmod 18171  df-lss 18234  df-lsp 18273  df-lvec 18404  df-lsatoms 32613  df-lshyp 32614  df-lfl 32695  df-lkr 32723  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-llines 33134  df-lplanes 33135  df-lvols 33136  df-lines 33137  df-psubsp 33139  df-pmap 33140  df-padd 33432  df-lhyp 33624  df-laut 33625  df-ldil 33740  df-ltrn 33741  df-trl 33796  df-tgrp 34381  df-tendo 34393  df-edring 34395  df-dveca 34641  df-disoa 34668  df-dvech 34718  df-dib 34778  df-dic 34812  df-dih 34868  df-doch 34987  df-djh 35034
This theorem is referenced by:  lcfl7N  35140  lcfrlem9  35189
  Copyright terms: Public domain W3C validator