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Theorem mapdpglem30 32427
Description: Lemma for mapdpg 32431. Baer p. 45 line 18: "Hence we deduce (from mapdpglem28 32426, using lvecindp2 16203) 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 32428? (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 2435 . . 3  |-  ( +g  `  C )  =  ( +g  `  C )
3 eqid 2435 . . 3  |-  (Scalar `  C )  =  (Scalar `  C )
4 eqid 2435 . . 3  |-  ( Base `  (Scalar `  C )
)  =  ( Base `  (Scalar `  C )
)
5 mapdpglem26.t . . 3  |-  .x.  =  ( .s `  C )
6 eqid 2435 . . 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 32316 . . 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 32420 . . . 4  |-  ( ph  ->  G  =/=  ( 0g
`  C ) )
25 eldifsn 3919 . . . 4  |-  ( G  e.  ( F  \  { ( 0g `  C ) } )  <-> 
( G  e.  F  /\  G  =/=  ( 0g `  C ) ) )
2612, 24, 25sylanbrc 646 . . 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 446 . . . 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 32421 . . . 4  |-  ( ph  ->  i  =/=  ( 0g
`  C ) )
31 eldifsn 3919 . . . 4  |-  ( i  e.  ( F  \  { ( 0g `  C ) } )  <-> 
( i  e.  F  /\  i  =/=  ( 0g `  C ) ) )
3228, 30, 31sylanbrc 646 . . 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 32325 . . . 4  |-  ( ph  ->  ( Base `  (Scalar `  C ) )  =  B )
3733, 36eleqtrrd 2512 . . 3  |-  ( ph  ->  v  e.  ( Base `  (Scalar `  C )
) )
388, 14, 10dvhlmod 31835 . . . . . 6  |-  ( ph  ->  U  e.  LMod )
3934lmodrng 15950 . . . . . 6  |-  ( U  e.  LMod  ->  A  e. 
Ring )
4038, 39syl 16 . . . . 5  |-  ( ph  ->  A  e.  Ring )
41 rnggrp 15661 . . . . . . 7  |-  ( A  e.  Ring  ->  A  e. 
Grp )
4240, 41syl 16 . . . . . 6  |-  ( ph  ->  A  e.  Grp )
43 eqid 2435 . . . . . . . 8  |-  ( 1r
`  A )  =  ( 1r `  A
)
4435, 43rngidcl 15676 . . . . . . 7  |-  ( A  e.  Ring  ->  ( 1r
`  A )  e.  B )
4540, 44syl 16 . . . . . 6  |-  ( ph  ->  ( 1r `  A
)  e.  B )
46 eqid 2435 . . . . . . 7  |-  ( inv g `  A )  =  ( inv g `  A )
4735, 46grpinvcl 14842 . . . . . 6  |-  ( ( A  e.  Grp  /\  ( 1r `  A )  e.  B )  -> 
( ( inv g `  A ) `  ( 1r `  A ) )  e.  B )
4842, 45, 47syl2anc 643 . . . . 5  |-  ( ph  ->  ( ( inv g `  A ) `  ( 1r `  A ) )  e.  B )
49 eqid 2435 . . . . . 6  |-  ( .r
`  A )  =  ( .r `  A
)
5035, 49rngcl 15669 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( v ( .r
`  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  e.  B )
5251, 36eleqtrrd 2512 . . 3  |-  ( ph  ->  ( v ( .r
`  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  e.  ( Base `  (Scalar `  C )
) )
5345, 36eleqtrrd 2512 . . 3  |-  ( ph  ->  ( 1r `  A
)  e.  ( Base `  (Scalar `  C )
) )
54 mapdpglem28.ue . . . . 5  |-  ( ph  ->  u  e.  B )
5535, 49rngcl 15669 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( u ( .r
`  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  e.  B )
5756, 36eleqtrrd 2512 . . 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 32425 . . 3  |-  ( ph  ->  ( J `  { G } )  =/=  ( J `  { i } ) )
628, 14, 34, 35, 49, 9, 1, 5, 10, 48, 54, 28lcdvsass 32332 . . . . 5  |-  ( ph  ->  ( ( u ( .r `  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  .x.  i )  =  ( ( ( inv g `  A
) `  ( 1r `  A ) )  .x.  ( u  .x.  i ) ) )
6362oveq2d 6089 . . . 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 32330 . . . . 5  |-  ( ph  ->  ( ( 1r `  A )  .x.  G
)  e.  F )
658, 14, 34, 35, 9, 1, 5, 10, 54, 28lcdvscl 32330 . . . . 5  |-  ( ph  ->  ( u  .x.  i
)  e.  F )
668, 14, 34, 46, 43, 9, 1, 2, 5, 19, 10, 64, 65lcdvsub 32342 . . . 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 32332 . . . . . 6  |-  ( ph  ->  ( ( v ( .r `  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  .x.  i )  =  ( ( ( inv g `  A
) `  ( 1r `  A ) )  .x.  ( v  .x.  i
) ) )
6867oveq2d 6089 . . . . 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 32330 . . . . . 6  |-  ( ph  ->  ( v  .x.  G
)  e.  F )
708, 14, 34, 35, 9, 1, 5, 10, 33, 28lcdvscl 32330 . . . . . 6  |-  ( ph  ->  ( v  .x.  i
)  e.  F )
718, 14, 34, 46, 43, 9, 1, 2, 5, 19, 10, 69, 70lcdvsub 32342 . . . . 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 32426 . . . . . 6  |-  ( ph  ->  ( ( v  .x.  G ) R ( v  .x.  i ) )  =  ( G R ( u  .x.  i ) ) )
73 eqid 2435 . . . . . . . . . 10  |-  ( 1r
`  (Scalar `  C )
)  =  ( 1r
`  (Scalar `  C )
)
748, 14, 34, 43, 9, 3, 73, 10lcd1 32334 . . . . . . . . 9  |-  ( ph  ->  ( 1r `  (Scalar `  C ) )  =  ( 1r `  A
) )
7574oveq1d 6088 . . . . . . . 8  |-  ( ph  ->  ( ( 1r `  (Scalar `  C ) ) 
.x.  G )  =  ( ( 1r `  A )  .x.  G
) )
768, 9, 10lcdlmod 32317 . . . . . . . . 9  |-  ( ph  ->  C  e.  LMod )
771, 3, 5, 73lmodvs1 15970 . . . . . . . . 9  |-  ( ( C  e.  LMod  /\  G  e.  F )  ->  (
( 1r `  (Scalar `  C ) )  .x.  G )  =  G )
7876, 12, 77syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( ( 1r `  (Scalar `  C ) ) 
.x.  G )  =  G )
7975, 78eqtr3d 2469 . . . . . . 7  |-  ( ph  ->  ( ( 1r `  A )  .x.  G
)  =  G )
8079oveq1d 6088 . . . . . 6  |-  ( ph  ->  ( ( ( 1r
`  A )  .x.  G ) R ( u  .x.  i ) )  =  ( G R ( u  .x.  i ) ) )
8172, 80eqtr4d 2470 . . . . 5  |-  ( ph  ->  ( ( v  .x.  G ) R ( v  .x.  i ) )  =  ( ( ( 1r `  A
)  .x.  G ) R ( u  .x.  i ) ) )
8268, 71, 813eqtr2rd 2474 . . . 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 2474 . . 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 16203 . 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 15696 . . . . 5  |-  ( ph  ->  ( v ( .r
`  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  =  ( ( inv g `  A
) `  v )
)
8635, 49, 43, 46, 40, 54rngnegr 15696 . . . . 5  |-  ( ph  ->  ( u ( .r
`  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  =  ( ( inv g `  A
) `  u )
)
8785, 86eqeq12d 2449 . . . 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 14852 . . . 4  |-  ( ph  ->  ( ( ( inv g `  A ) `
 v )  =  ( ( inv g `  A ) `  u
)  <->  v  =  u ) )
8987, 88bitrd 245 . . 3  |-  ( ph  ->  ( ( v ( .r `  A ) ( ( inv g `  A ) `  ( 1r `  A ) ) )  =  ( u ( .r `  A
) ( ( inv g `  A ) `
 ( 1r `  A ) ) )  <-> 
v  =  u ) )
9089anbi2d 685 . 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 202 1  |-  ( ph  ->  ( v  =  ( 1r `  A )  /\  v  =  u ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309   {csn 3806   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   .rcmulr 13522  Scalarcsca 13524   .scvsca 13525   0gc0g 13715   Grpcgrp 14677   inv gcminusg 14678   -gcsg 14680   Ringcrg 15652   1rcur 15654   LModclmod 15942   LSpanclspn 16039   HLchlt 30075   LHypclh 30708   DVecHcdvh 31803  LCDualclcd 32311  mapdcmpd 32349
This theorem is referenced by:  mapdpglem31  32428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-undef 6535  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-0g 13719  df-mre 13803  df-mrc 13804  df-acs 13806  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-cntz 15108  df-oppg 15134  df-lsm 15262  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-dvr 15780  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167  df-lsatoms 29701  df-lshyp 29702  df-lcv 29744  df-lfl 29783  df-lkr 29811  df-ldual 29849  df-oposet 29901  df-ol 29903  df-oml 29904  df-covers 29991  df-ats 29992  df-atl 30023  df-cvlat 30047  df-hlat 30076  df-llines 30222  df-lplanes 30223  df-lvols 30224  df-lines 30225  df-psubsp 30227  df-pmap 30228  df-padd 30520  df-lhyp 30712  df-laut 30713  df-ldil 30828  df-ltrn 30829  df-trl 30883  df-tgrp 31467  df-tendo 31479  df-edring 31481  df-dveca 31727  df-disoa 31754  df-dvech 31804  df-dib 31864  df-dic 31898  df-dih 31954  df-doch 32073  df-djh 32120  df-lcdual 32312  df-mapd 32350
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