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Theorem lcfl7lem 32311
Description: Lemma for lcfl7N 32313. 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 32286 . . . . 5  |-  ( ph  ->  X  e.  ( ( 
._|_  `  ( L `  G ) )  \  {  .0.  } ) )
15 eldifi 3311 . . . . 5  |-  ( X  e.  ( (  ._|_  `  ( L `  G
) )  \  {  .0.  } )  ->  X  e.  (  ._|_  `  ( L `  G )
) )
1614, 15syl 15 . . . 4  |-  ( ph  ->  X  e.  (  ._|_  `  ( L `  G
) ) )
17 lcfl7lem.gj . . . . . . . 8  |-  ( ph  ->  G  =  J )
1817fveq2d 5545 . . . . . . 7  |-  ( ph  ->  ( L `  G
)  =  ( L `
 J ) )
19 lcfl7lem.j . . . . . . . 8  |-  J  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { Y }
) v  =  ( w  .+  ( k 
.x.  Y ) ) ) )
20 lcfl7lem.x2 . . . . . . . 8  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
211, 2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 12, 20dochsnkr2 32285 . . . . . . 7  |-  ( ph  ->  ( L `  J
)  =  (  ._|_  `  { Y } ) )
2218, 21eqtrd 2328 . . . . . 6  |-  ( ph  ->  ( L `  G
)  =  (  ._|_  `  { Y } ) )
2322fveq2d 5545 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) )  =  (  ._|_  `  (  ._|_  `  { Y }
) ) )
24 eqid 2296 . . . . . . 7  |-  ( LSpan `  U )  =  (
LSpan `  U )
25 eldifi 3311 . . . . . . . . 9  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
2620, 25syl 15 . . . . . . . 8  |-  ( ph  ->  Y  e.  V )
2726snssd 3776 . . . . . . 7  |-  ( ph  ->  { Y }  C_  V )
281, 3, 2, 4, 24, 12, 27dochocsp 32191 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( (
LSpan `  U ) `  { Y } ) )  =  (  ._|_  `  { Y } ) )
2928fveq2d 5545 . . . . 5  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( ( LSpan `  U
) `  { Y } ) ) )  =  (  ._|_  `  (  ._|_  `  { Y }
) ) )
30 eqid 2296 . . . . . . . 8  |-  ( (
DIsoH `  K ) `  W )  =  ( ( DIsoH `  K ) `  W )
311, 3, 4, 24, 30dihlsprn 32143 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  e.  V
)  ->  ( ( LSpan `  U ) `  { Y } )  e. 
ran  ( ( DIsoH `  K ) `  W
) )
3212, 26, 31syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( LSpan `  U
) `  { Y } )  e.  ran  ( ( DIsoH `  K
) `  W )
)
331, 30, 2dochoc 32179 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( LSpan `  U ) `  { Y } )  e.  ran  ( ( DIsoH `  K
) `  W )
)  ->  (  ._|_  `  (  ._|_  `  ( (
LSpan `  U ) `  { Y } ) ) )  =  ( (
LSpan `  U ) `  { Y } ) )
3412, 32, 33syl2anc 642 . . . . 5  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( ( LSpan `  U
) `  { Y } ) ) )  =  ( ( LSpan `  U ) `  { Y } ) )
3523, 29, 343eqtr2d 2334 . . . 4  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) )  =  ( ( LSpan `  U ) `  { Y } ) )
3616, 35eleqtrd 2372 . . 3  |-  ( ph  ->  X  e.  ( (
LSpan `  U ) `  { Y } ) )
371, 3, 12dvhlmod 31922 . . . 4  |-  ( ph  ->  U  e.  LMod )
389, 10, 4, 7, 24lspsnel 15776 . . . 4  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( X  e.  ( ( LSpan `  U ) `  { Y } )  <->  E. s  e.  R  X  =  ( s  .x.  Y
) ) )
3937, 26, 38syl2anc 642 . . 3  |-  ( ph  ->  ( X  e.  ( ( LSpan `  U ) `  { Y } )  <->  E. s  e.  R  X  =  ( s  .x.  Y ) ) )
4036, 39mpbid 201 . 2  |-  ( ph  ->  E. s  e.  R  X  =  ( s  .x.  Y ) )
41 simp3 957 . . . 4  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  X  =  ( s  .x.  Y
) )
42 fveq2 5541 . . . . . . . . . 10  |-  ( X  =  ( s  .x.  Y )  ->  ( G `  X )  =  ( G `  ( s  .x.  Y
) ) )
43423ad2ant3 978 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  X )  =  ( G `  ( s 
.x.  Y ) ) )
44 eqid 2296 . . . . . . . . . . . 12  |-  ( 1r
`  S )  =  ( 1r `  S
)
451, 2, 3, 4, 6, 7, 5, 9, 10, 44, 12, 20, 19dochfl1 32288 . . . . . . . . . . 11  |-  ( ph  ->  ( J `  Y
)  =  ( 1r
`  S ) )
4617fveq1d 5543 . . . . . . . . . . 11  |-  ( ph  ->  ( G `  Y
)  =  ( J `
 Y ) )
471, 2, 3, 4, 6, 7, 5, 9, 10, 44, 12, 13, 11dochfl1 32288 . . . . . . . . . . 11  |-  ( ph  ->  ( G `  X
)  =  ( 1r
`  S ) )
4845, 46, 473eqtr4rd 2339 . . . . . . . . . 10  |-  ( ph  ->  ( G `  X
)  =  ( G `
 Y ) )
49483ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  X )  =  ( G `  Y ) )
50373ad2ant1 976 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  U  e.  LMod )
51 lcfl7lem.f . . . . . . . . . . . 12  |-  F  =  (LFnl `  U )
521, 2, 3, 4, 5, 6, 7, 51, 9, 10, 11, 12, 13dochflcl 32287 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  F )
53523ad2ant1 976 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  G  e.  F )
54 simp2 956 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  s  e.  R )
55263ad2ant1 976 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  Y  e.  V )
56 eqid 2296 . . . . . . . . . . 11  |-  ( .r
`  S )  =  ( .r `  S
)
579, 10, 56, 4, 7, 51lflmul 29880 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  G  e.  F  /\  (
s  e.  R  /\  Y  e.  V )
)  ->  ( G `  ( s  .x.  Y
) )  =  ( s ( .r `  S ) ( G `
 Y ) ) )
5850, 53, 54, 55, 57syl112anc 1186 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  ( s  .x.  Y
) )  =  ( s ( .r `  S ) ( G `
 Y ) ) )
5943, 49, 583eqtr3d 2336 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  Y )  =  ( s ( .r `  S ) ( G `
 Y ) ) )
6059oveq1d 5889 . . . . . . 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
) ) ) )
619lmodrng 15651 . . . . . . . . . 10  |-  ( U  e.  LMod  ->  S  e. 
Ring )
6237, 61syl 15 . . . . . . . . 9  |-  ( ph  ->  S  e.  Ring )
63623ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  S  e.  Ring )
649, 10, 4, 51lflcl 29876 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  G  e.  F  /\  Y  e.  V )  ->  ( G `  Y )  e.  R )
6537, 52, 26, 64syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( G `  Y
)  e.  R )
66653ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  Y )  e.  R
)
671, 3, 12dvhlvec 31921 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  LVec )
689lvecdrng 15874 . . . . . . . . . . 11  |-  ( U  e.  LVec  ->  S  e.  DivRing )
6967, 68syl 15 . . . . . . . . . 10  |-  ( ph  ->  S  e.  DivRing )
7046, 45eqtrd 2328 . . . . . . . . . . 11  |-  ( ph  ->  ( G `  Y
)  =  ( 1r
`  S ) )
71 eqid 2296 . . . . . . . . . . . . 13  |-  ( 0g
`  S )  =  ( 0g `  S
)
7271, 44drngunz 15543 . . . . . . . . . . . 12  |-  ( S  e.  DivRing  ->  ( 1r `  S )  =/=  ( 0g `  S ) )
7369, 72syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( 1r `  S
)  =/=  ( 0g
`  S ) )
7470, 73eqnetrd 2477 . . . . . . . . . 10  |-  ( ph  ->  ( G `  Y
)  =/=  ( 0g
`  S ) )
75 eqid 2296 . . . . . . . . . . 11  |-  ( invr `  S )  =  (
invr `  S )
7610, 71, 75drnginvrcl 15545 . . . . . . . . . 10  |-  ( ( S  e.  DivRing  /\  ( G `  Y )  e.  R  /\  ( G `  Y )  =/=  ( 0g `  S
) )  ->  (
( invr `  S ) `  ( G `  Y
) )  e.  R
)
7769, 65, 74, 76syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( ( invr `  S
) `  ( G `  Y ) )  e.  R )
78773ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( ( invr `  S ) `  ( G `  Y ) )  e.  R )
7910, 56rngass 15373 . . . . . . . 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
) ) ) ) )
8063, 54, 66, 78, 79syl13anc 1184 . . . . . . 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 )
) ) ) )
8110, 71, 56, 44, 75drnginvrr 15548 . . . . . . . . . 10  |-  ( ( S  e.  DivRing  /\  ( G `  Y )  e.  R  /\  ( G `  Y )  =/=  ( 0g `  S
) )  ->  (
( G `  Y
) ( .r `  S ) ( (
invr `  S ) `  ( G `  Y
) ) )  =  ( 1r `  S
) )
8269, 65, 74, 81syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( ( G `  Y ) ( .r
`  S ) ( ( invr `  S
) `  ( G `  Y ) ) )  =  ( 1r `  S ) )
83823ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( ( G `  Y )
( .r `  S
) ( ( invr `  S ) `  ( G `  Y )
) )  =  ( 1r `  S ) )
8483oveq2d 5890 . . . . . . 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 ) ) )
8560, 80, 843eqtrrd 2333 . . . . . 6  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( s
( .r `  S
) ( 1r `  S ) )  =  ( ( G `  Y ) ( .r
`  S ) ( ( invr `  S
) `  ( G `  Y ) ) ) )
8610, 56, 44rngridm 15381 . . . . . . 7  |-  ( ( S  e.  Ring  /\  s  e.  R )  ->  (
s ( .r `  S ) ( 1r
`  S ) )  =  s )
8763, 54, 86syl2anc 642 . . . . . 6  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( s
( .r `  S
) ( 1r `  S ) )  =  s )
8885, 87, 833eqtr3d 2336 . . . . 5  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  s  =  ( 1r `  S ) )
89 oveq1 5881 . . . . . 6  |-  ( s  =  ( 1r `  S )  ->  (
s  .x.  Y )  =  ( ( 1r
`  S )  .x.  Y ) )
904, 9, 7, 44lmodvs1 15674 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  (
( 1r `  S
)  .x.  Y )  =  Y )
9137, 26, 90syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( 1r `  S )  .x.  Y
)  =  Y )
92913ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( ( 1r `  S )  .x.  Y )  =  Y )
9389, 92sylan9eqr 2350 . . . . 5  |-  ( ( ( ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y
) )  /\  s  =  ( 1r `  S ) )  -> 
( s  .x.  Y
)  =  Y )
9488, 93mpdan 649 . . . 4  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( s  .x.  Y )  =  Y )
9541, 94eqtrd 2328 . . 3  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  X  =  Y )
9695rexlimdv3a 2682 . 2  |-  ( ph  ->  ( E. s  e.  R  X  =  ( s  .x.  Y )  ->  X  =  Y ) )
9740, 96mpd 14 1  |-  ( ph  ->  X  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    \ cdif 3162   {csn 3653    e. cmpt 4093   ran crn 4706   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164   +g cplusg 13224   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228   0gc0g 13416   Ringcrg 15353   1rcur 15355   invrcinvr 15469   DivRingcdr 15528   LModclmod 15643   LSpanclspn 15744   LVecclvec 15871  LFnlclfn 29869  LKerclk 29897   HLchlt 30162   LHypclh 30795   DVecHcdvh 31890   DIsoHcdih 32040   ocHcoch 32159
This theorem is referenced by:  lcfl7N  32313  lcfrlem9  32362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-undef 6314  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872  df-lsatoms 29788  df-lshyp 29789  df-lfl 29870  df-lkr 29898  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tgrp 31554  df-tendo 31566  df-edring 31568  df-dveca 31814  df-disoa 31841  df-dvech 31891  df-dib 31951  df-dic 31985  df-dih 32041  df-doch 32160  df-djh 32207
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