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Theorem mapdpglem30 31892
Description: Lemma for mapdpg 31896. Baer p. 45 line 18: "Hence we deduce (from mapdpglem28 31891, using lvecindp2 15892) that v = 1 and v = u...". TODO: would it be shorter to have only the  v  =  ( 1r `  A ) part and use mapdpglem28.u2 in mapdpglem31 31893? (Contributed by NM, 22-Mar-2015.)
Hypotheses
Ref Expression
mapdpg.h  |-  H  =  ( LHyp `  K
)
mapdpg.m  |-  M  =  ( (mapd `  K
) `  W )
mapdpg.u  |-  U  =  ( ( DVecH `  K
) `  W )
mapdpg.v  |-  V  =  ( Base `  U
)
mapdpg.s  |-  .-  =  ( -g `  U )
mapdpg.z  |-  .0.  =  ( 0g `  U )
mapdpg.n  |-  N  =  ( LSpan `  U )
mapdpg.c  |-  C  =  ( (LCDual `  K
) `  W )
mapdpg.f  |-  F  =  ( Base `  C
)
mapdpg.r  |-  R  =  ( -g `  C
)
mapdpg.j  |-  J  =  ( LSpan `  C )
mapdpg.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
mapdpg.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
mapdpg.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
mapdpg.g  |-  ( ph  ->  G  e.  F )
mapdpg.ne  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
mapdpg.e  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )
mapdpgem25.h1  |-  ( ph  ->  ( h  e.  F  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) ) ) )
mapdpgem25.i1  |-  ( ph  ->  ( i  e.  F  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )
mapdpglem26.a  |-  A  =  (Scalar `  U )
mapdpglem26.b  |-  B  =  ( Base `  A
)
mapdpglem26.t  |-  .x.  =  ( .s `  C )
mapdpglem26.o  |-  O  =  ( 0g `  A
)
mapdpglem28.ve  |-  ( ph  ->  v  e.  B )
mapdpglem28.u1  |-  ( ph  ->  h  =  ( u 
.x.  i ) )
mapdpglem28.u2  |-  ( ph  ->  ( G R h )  =  ( v 
.x.  ( G R i ) ) )
mapdpglem28.ue  |-  ( ph  ->  u  e.  B )
Assertion
Ref Expression
mapdpglem30  |-  ( ph  ->  ( v  =  ( 1r `  A )  /\  v  =  u ) )
Distinct variable groups:    h, i, u, v    u, B, v   
u, C, v    u, O, v    u,  .x. , v    v, G    v, R
Allowed substitution hints:    ph( v, u, h, i)    A( v, u, h, i)    B( h, i)    C( h, i)    R( u, h, i)    .x. ( h, i)    U( v, u, h, i)    F( v, u, h, i)    G( u, h, i)    H( v, u, h, i)    J( v, u, h, i)    K( v, u, h, i)    M( v, u, h, i)    .- ( v, u, h, i)    N( v, u, h, i)    O( h, i)    V( v, u, h, i)    W( v, u, h, i)    X( v, u, h, i)    Y( v, u, h, i)    .0. ( v, u, h, i)

Proof of Theorem mapdpglem30
StepHypRef Expression
1 mapdpg.f . . 3  |-  F  =  ( Base `  C
)
2 eqid 2283 . . 3  |-  ( +g  `  C )  =  ( +g  `  C )
3 eqid 2283 . . 3  |-  (Scalar `  C )  =  (Scalar `  C )
4 eqid 2283 . . 3  |-  ( Base `  (Scalar `  C )
)  =  ( Base `  (Scalar `  C )
)
5 mapdpglem26.t . . 3  |-  .x.  =  ( .s `  C )
6 eqid 2283 . . 3  |-  ( 0g
`  C )  =  ( 0g `  C
)
7 mapdpg.j . . 3  |-  J  =  ( LSpan `  C )
8 mapdpg.h . . . 4  |-  H  =  ( LHyp `  K
)
9 mapdpg.c . . . 4  |-  C  =  ( (LCDual `  K
) `  W )
10 mapdpg.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
118, 9, 10lcdlvec 31781 . . 3  |-  ( ph  ->  C  e.  LVec )
12 mapdpg.g . . . 4  |-  ( ph  ->  G  e.  F )
13 mapdpg.m . . . . 5  |-  M  =  ( (mapd `  K
) `  W )
14 mapdpg.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
15 mapdpg.v . . . . 5  |-  V  =  ( Base `  U
)
16 mapdpg.s . . . . 5  |-  .-  =  ( -g `  U )
17 mapdpg.z . . . . 5  |-  .0.  =  ( 0g `  U )
18 mapdpg.n . . . . 5  |-  N  =  ( LSpan `  U )
19 mapdpg.r . . . . 5  |-  R  =  ( -g `  C
)
20 mapdpg.x . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
21 mapdpg.y . . . . 5  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
22 mapdpg.ne . . . . 5  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
23 mapdpg.e . . . . 5  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )
248, 13, 14, 15, 16, 17, 18, 9, 1, 19, 7, 10, 20, 21, 12, 22, 23mapdpglem30a 31885 . . . 4  |-  ( ph  ->  G  =/=  ( 0g
`  C ) )
25 eldifsn 3749 . . . 4  |-  ( G  e.  ( F  \  { ( 0g `  C ) } )  <-> 
( G  e.  F  /\  G  =/=  ( 0g `  C ) ) )
2612, 24, 25sylanbrc 645 . . 3  |-  ( ph  ->  G  e.  ( F 
\  { ( 0g
`  C ) } ) )
27 mapdpgem25.i1 . . . . 5  |-  ( ph  ->  ( i  e.  F  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )
2827simpld 445 . . . 4  |-  ( ph  ->  i  e.  F )
29 mapdpgem25.h1 . . . . 5  |-  ( ph  ->  ( h  e.  F  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) ) ) )
308, 13, 14, 15, 16, 17, 18, 9, 1, 19, 7, 10, 20, 21, 12, 22, 23, 29, 27mapdpglem30b 31886 . . . 4  |-  ( ph  ->  i  =/=  ( 0g
`  C ) )
31 eldifsn 3749 . . . 4  |-  ( i  e.  ( F  \  { ( 0g `  C ) } )  <-> 
( i  e.  F  /\  i  =/=  ( 0g `  C ) ) )
3228, 30, 31sylanbrc 645 . . 3  |-  ( ph  ->  i  e.  ( F 
\  { ( 0g
`  C ) } ) )
33 mapdpglem28.ve . . . 4  |-  ( ph  ->  v  e.  B )
34 mapdpglem26.a . . . . 5  |-  A  =  (Scalar `  U )
35 mapdpglem26.b . . . . 5  |-  B  =  ( Base `  A
)
368, 14, 34, 35, 9, 3, 4, 10lcdsbase 31790 . . . 4  |-  ( ph  ->  ( Base `  (Scalar `  C ) )  =  B )
3733, 36eleqtrrd 2360 . . 3  |-  ( ph  ->  v  e.  ( Base `  (Scalar `  C )
) )
388, 14, 10dvhlmod 31300 . . . . . 6  |-  ( ph  ->  U  e.  LMod )
3934lmodrng 15635 . . . . . 6  |-  ( U  e.  LMod  ->  A  e. 
Ring )
4038, 39syl 15 . . . . 5  |-  ( ph  ->  A  e.  Ring )
41 rnggrp 15346 . . . . . . 7  |-  ( A  e.  Ring  ->  A  e. 
Grp )
4240, 41syl 15 . . . . . 6  |-  ( ph  ->  A  e.  Grp )
43 eqid 2283 . . . . . . . 8  |-  ( 1r
`  A )  =  ( 1r `  A
)
4435, 43rngidcl 15361 . . . . . . 7  |-  ( A  e.  Ring  ->  ( 1r
`  A )  e.  B )
4540, 44syl 15 . . . . . 6  |-  ( ph  ->  ( 1r `  A
)  e.  B )
46 eqid 2283 . . . . . . 7  |-  ( inv g `  A )  =  ( inv g `  A )
4735, 46grpinvcl 14527 . . . . . 6  |-  ( ( A  e.  Grp  /\  ( 1r `  A )  e.  B )  -> 
( ( inv g `  A ) `  ( 1r `  A ) )  e.  B )
4842, 45, 47syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( inv g `  A ) `  ( 1r `  A ) )  e.  B )
49 eqid 2283 . . . . . 6  |-  ( .r
`  A )  =  ( .r `  A
)
5035, 49rngcl 15354 . . . . 5  |-  ( ( A  e.  Ring  /\  v  e.  B  /\  (
( inv g `  A ) `  ( 1r `  A ) )  e.  B )  -> 
( v ( .r
`  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  e.  B )
5140, 33, 48, 50syl3anc 1182 . . . 4  |-  ( ph  ->  ( v ( .r
`  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  e.  B )
5251, 36eleqtrrd 2360 . . 3  |-  ( ph  ->  ( v ( .r
`  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  e.  ( Base `  (Scalar `  C )
) )
5345, 36eleqtrrd 2360 . . 3  |-  ( ph  ->  ( 1r `  A
)  e.  ( Base `  (Scalar `  C )
) )
54 mapdpglem28.ue . . . . 5  |-  ( ph  ->  u  e.  B )
5535, 49rngcl 15354 . . . . 5  |-  ( ( A  e.  Ring  /\  u  e.  B  /\  (
( inv g `  A ) `  ( 1r `  A ) )  e.  B )  -> 
( u ( .r
`  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  e.  B )
5640, 54, 48, 55syl3anc 1182 . . . 4  |-  ( ph  ->  ( u ( .r
`  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  e.  B )
5756, 36eleqtrrd 2360 . . 3  |-  ( ph  ->  ( u ( .r
`  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  e.  ( Base `  (Scalar `  C )
) )
58 mapdpglem26.o . . . 4  |-  O  =  ( 0g `  A
)
59 mapdpglem28.u1 . . . 4  |-  ( ph  ->  h  =  ( u 
.x.  i ) )
60 mapdpglem28.u2 . . . 4  |-  ( ph  ->  ( G R h )  =  ( v 
.x.  ( G R i ) ) )
618, 13, 14, 15, 16, 17, 18, 9, 1, 19, 7, 10, 20, 21, 12, 22, 23, 29, 27, 34, 35, 5, 58, 33, 59, 60mapdpglem29 31890 . . 3  |-  ( ph  ->  ( J `  { G } )  =/=  ( J `  { i } ) )
628, 14, 34, 35, 49, 9, 1, 5, 10, 48, 54, 28lcdvsass 31797 . . . . 5  |-  ( ph  ->  ( ( u ( .r `  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  .x.  i )  =  ( ( ( inv g `  A
) `  ( 1r `  A ) )  .x.  ( u  .x.  i ) ) )
6362oveq2d 5874 . . . 4  |-  ( ph  ->  ( ( ( 1r
`  A )  .x.  G ) ( +g  `  C ) ( ( u ( .r `  A ) ( ( inv g `  A
) `  ( 1r `  A ) ) ) 
.x.  i ) )  =  ( ( ( 1r `  A ) 
.x.  G ) ( +g  `  C ) ( ( ( inv g `  A ) `
 ( 1r `  A ) )  .x.  ( u  .x.  i ) ) ) )
648, 14, 34, 35, 9, 1, 5, 10, 45, 12lcdvscl 31795 . . . . 5  |-  ( ph  ->  ( ( 1r `  A )  .x.  G
)  e.  F )
658, 14, 34, 35, 9, 1, 5, 10, 54, 28lcdvscl 31795 . . . . 5  |-  ( ph  ->  ( u  .x.  i
)  e.  F )
668, 14, 34, 46, 43, 9, 1, 2, 5, 19, 10, 64, 65lcdvsub 31807 . . . 4  |-  ( ph  ->  ( ( ( 1r
`  A )  .x.  G ) R ( u  .x.  i ) )  =  ( ( ( 1r `  A
)  .x.  G )
( +g  `  C ) ( ( ( inv g `  A ) `
 ( 1r `  A ) )  .x.  ( u  .x.  i ) ) ) )
678, 14, 34, 35, 49, 9, 1, 5, 10, 48, 33, 28lcdvsass 31797 . . . . . 6  |-  ( ph  ->  ( ( v ( .r `  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  .x.  i )  =  ( ( ( inv g `  A
) `  ( 1r `  A ) )  .x.  ( v  .x.  i
) ) )
6867oveq2d 5874 . . . . 5  |-  ( ph  ->  ( ( v  .x.  G ) ( +g  `  C ) ( ( v ( .r `  A ) ( ( inv g `  A
) `  ( 1r `  A ) ) ) 
.x.  i ) )  =  ( ( v 
.x.  G ) ( +g  `  C ) ( ( ( inv g `  A ) `
 ( 1r `  A ) )  .x.  ( v  .x.  i
) ) ) )
698, 14, 34, 35, 9, 1, 5, 10, 33, 12lcdvscl 31795 . . . . . 6  |-  ( ph  ->  ( v  .x.  G
)  e.  F )
708, 14, 34, 35, 9, 1, 5, 10, 33, 28lcdvscl 31795 . . . . . 6  |-  ( ph  ->  ( v  .x.  i
)  e.  F )
718, 14, 34, 46, 43, 9, 1, 2, 5, 19, 10, 69, 70lcdvsub 31807 . . . . 5  |-  ( ph  ->  ( ( v  .x.  G ) R ( v  .x.  i ) )  =  ( ( v  .x.  G ) ( +g  `  C
) ( ( ( inv g `  A
) `  ( 1r `  A ) )  .x.  ( v  .x.  i
) ) ) )
728, 13, 14, 15, 16, 17, 18, 9, 1, 19, 7, 10, 20, 21, 12, 22, 23, 29, 27, 34, 35, 5, 58, 33, 59, 60mapdpglem28 31891 . . . . . 6  |-  ( ph  ->  ( ( v  .x.  G ) R ( v  .x.  i ) )  =  ( G R ( u  .x.  i ) ) )
73 eqid 2283 . . . . . . . . . 10  |-  ( 1r
`  (Scalar `  C )
)  =  ( 1r
`  (Scalar `  C )
)
748, 14, 34, 43, 9, 3, 73, 10lcd1 31799 . . . . . . . . 9  |-  ( ph  ->  ( 1r `  (Scalar `  C ) )  =  ( 1r `  A
) )
7574oveq1d 5873 . . . . . . . 8  |-  ( ph  ->  ( ( 1r `  (Scalar `  C ) ) 
.x.  G )  =  ( ( 1r `  A )  .x.  G
) )
768, 9, 10lcdlmod 31782 . . . . . . . . 9  |-  ( ph  ->  C  e.  LMod )
771, 3, 5, 73lmodvs1 15658 . . . . . . . . 9  |-  ( ( C  e.  LMod  /\  G  e.  F )  ->  (
( 1r `  (Scalar `  C ) )  .x.  G )  =  G )
7876, 12, 77syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( ( 1r `  (Scalar `  C ) ) 
.x.  G )  =  G )
7975, 78eqtr3d 2317 . . . . . . 7  |-  ( ph  ->  ( ( 1r `  A )  .x.  G
)  =  G )
8079oveq1d 5873 . . . . . 6  |-  ( ph  ->  ( ( ( 1r
`  A )  .x.  G ) R ( u  .x.  i ) )  =  ( G R ( u  .x.  i ) ) )
8172, 80eqtr4d 2318 . . . . 5  |-  ( ph  ->  ( ( v  .x.  G ) R ( v  .x.  i ) )  =  ( ( ( 1r `  A
)  .x.  G ) R ( u  .x.  i ) ) )
8268, 71, 813eqtr2rd 2322 . . . 4  |-  ( ph  ->  ( ( ( 1r
`  A )  .x.  G ) R ( u  .x.  i ) )  =  ( ( v  .x.  G ) ( +g  `  C
) ( ( v ( .r `  A
) ( ( inv g `  A ) `
 ( 1r `  A ) ) ) 
.x.  i ) ) )
8363, 66, 823eqtr2rd 2322 . . 3  |-  ( ph  ->  ( ( v  .x.  G ) ( +g  `  C ) ( ( v ( .r `  A ) ( ( inv g `  A
) `  ( 1r `  A ) ) ) 
.x.  i ) )  =  ( ( ( 1r `  A ) 
.x.  G ) ( +g  `  C ) ( ( u ( .r `  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  .x.  i ) ) )
841, 2, 3, 4, 5, 6, 7, 11, 26, 32, 37, 52, 53, 57, 61, 83lvecindp2 15892 . 2  |-  ( ph  ->  ( v  =  ( 1r `  A )  /\  ( v ( .r `  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  =  ( u ( .r `  A
) ( ( inv g `  A ) `
 ( 1r `  A ) ) ) ) )
8535, 49, 43, 46, 40, 33rngnegr 15381 . . . . 5  |-  ( ph  ->  ( v ( .r
`  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  =  ( ( inv g `  A
) `  v )
)
8635, 49, 43, 46, 40, 54rngnegr 15381 . . . . 5  |-  ( ph  ->  ( u ( .r
`  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  =  ( ( inv g `  A
) `  u )
)
8785, 86eqeq12d 2297 . . . 4  |-  ( ph  ->  ( ( v ( .r `  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  =  ( u ( .r `  A
) ( ( inv g `  A ) `
 ( 1r `  A ) ) )  <-> 
( ( inv g `  A ) `  v
)  =  ( ( inv g `  A
) `  u )
) )
8835, 46, 42, 33, 54grpinv11 14537 . . . 4  |-  ( ph  ->  ( ( ( inv g `  A ) `
 v )  =  ( ( inv g `  A ) `  u
)  <->  v  =  u ) )
8987, 88bitrd 244 . . 3  |-  ( ph  ->  ( ( v ( .r `  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  =  ( u ( .r `  A
) ( ( inv g `  A ) `
 ( 1r `  A ) ) )  <-> 
v  =  u ) )
9089anbi2d 684 . 2  |-  ( ph  ->  ( ( v  =  ( 1r `  A
)  /\  ( v
( .r `  A
) ( ( inv g `  A ) `
 ( 1r `  A ) ) )  =  ( u ( .r `  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) ) )  <->  ( v  =  ( 1r `  A )  /\  v  =  u ) ) )
9184, 90mpbid 201 1  |-  ( ph  ->  ( v  =  ( 1r `  A )  /\  v  =  u ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363   -gcsg 14365   Ringcrg 15337   1rcur 15339   LModclmod 15627   LSpanclspn 15728   HLchlt 29540   LHypclh 30173   DVecHcdvh 31268  LCDualclcd 31776  mapdcmpd 31814
This theorem is referenced by:  mapdpglem31  31893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-oppg 14819  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 29166  df-lshyp 29167  df-lcv 29209  df-lfl 29248  df-lkr 29276  df-ldual 29314  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tgrp 30932  df-tendo 30944  df-edring 30946  df-dveca 31192  df-disoa 31219  df-dvech 31269  df-dib 31329  df-dic 31363  df-dih 31419  df-doch 31538  df-djh 31585  df-lcdual 31777  df-mapd 31815
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