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Theorem mapdpglem30 35341
Description: Lemma for mapdpg 35345. Baer p. 45 line 18: "Hence we deduce (from mapdpglem28 35340, using lvecindp2 18440) 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 35342? (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 2471 . . 3  |-  ( +g  `  C )  =  ( +g  `  C )
3 eqid 2471 . . 3  |-  (Scalar `  C )  =  (Scalar `  C )
4 eqid 2471 . . 3  |-  ( Base `  (Scalar `  C )
)  =  ( Base `  (Scalar `  C )
)
5 mapdpglem26.t . . 3  |-  .x.  =  ( .s `  C )
6 eqid 2471 . . 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 35230 . . 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 35334 . . . 4  |-  ( ph  ->  G  =/=  ( 0g
`  C ) )
25 eldifsn 4088 . . . 4  |-  ( G  e.  ( F  \  { ( 0g `  C ) } )  <-> 
( G  e.  F  /\  G  =/=  ( 0g `  C ) ) )
2612, 24, 25sylanbrc 677 . . 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 466 . . . 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 35335 . . . 4  |-  ( ph  ->  i  =/=  ( 0g
`  C ) )
31 eldifsn 4088 . . . 4  |-  ( i  e.  ( F  \  { ( 0g `  C ) } )  <-> 
( i  e.  F  /\  i  =/=  ( 0g `  C ) ) )
3228, 30, 31sylanbrc 677 . . 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 35239 . . . 4  |-  ( ph  ->  ( Base `  (Scalar `  C ) )  =  B )
3733, 36eleqtrrd 2552 . . 3  |-  ( ph  ->  v  e.  ( Base `  (Scalar `  C )
) )
388, 14, 10dvhlmod 34749 . . . . . 6  |-  ( ph  ->  U  e.  LMod )
3934lmodring 18177 . . . . . 6  |-  ( U  e.  LMod  ->  A  e. 
Ring )
4038, 39syl 17 . . . . 5  |-  ( ph  ->  A  e.  Ring )
41 ringgrp 17863 . . . . . . 7  |-  ( A  e.  Ring  ->  A  e. 
Grp )
4240, 41syl 17 . . . . . 6  |-  ( ph  ->  A  e.  Grp )
43 eqid 2471 . . . . . . . 8  |-  ( 1r
`  A )  =  ( 1r `  A
)
4435, 43ringidcl 17879 . . . . . . 7  |-  ( A  e.  Ring  ->  ( 1r
`  A )  e.  B )
4540, 44syl 17 . . . . . 6  |-  ( ph  ->  ( 1r `  A
)  e.  B )
46 eqid 2471 . . . . . . 7  |-  ( invg `  A )  =  ( invg `  A )
4735, 46grpinvcl 16789 . . . . . 6  |-  ( ( A  e.  Grp  /\  ( 1r `  A )  e.  B )  -> 
( ( invg `  A ) `  ( 1r `  A ) )  e.  B )
4842, 45, 47syl2anc 673 . . . . 5  |-  ( ph  ->  ( ( invg `  A ) `  ( 1r `  A ) )  e.  B )
49 eqid 2471 . . . . . 6  |-  ( .r
`  A )  =  ( .r `  A
)
5035, 49ringcl 17872 . . . . 5  |-  ( ( A  e.  Ring  /\  v  e.  B  /\  (
( invg `  A ) `  ( 1r `  A ) )  e.  B )  -> 
( v ( .r
`  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  e.  B )
5140, 33, 48, 50syl3anc 1292 . . . 4  |-  ( ph  ->  ( v ( .r
`  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  e.  B )
5251, 36eleqtrrd 2552 . . 3  |-  ( ph  ->  ( v ( .r
`  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  e.  ( Base `  (Scalar `  C )
) )
5345, 36eleqtrrd 2552 . . 3  |-  ( ph  ->  ( 1r `  A
)  e.  ( Base `  (Scalar `  C )
) )
54 mapdpglem28.ue . . . . 5  |-  ( ph  ->  u  e.  B )
5535, 49ringcl 17872 . . . . 5  |-  ( ( A  e.  Ring  /\  u  e.  B  /\  (
( invg `  A ) `  ( 1r `  A ) )  e.  B )  -> 
( u ( .r
`  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  e.  B )
5640, 54, 48, 55syl3anc 1292 . . . 4  |-  ( ph  ->  ( u ( .r
`  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  e.  B )
5756, 36eleqtrrd 2552 . . 3  |-  ( ph  ->  ( u ( .r
`  A ) ( ( invg `  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 35339 . . 3  |-  ( ph  ->  ( J `  { G } )  =/=  ( J `  { i } ) )
628, 14, 34, 35, 49, 9, 1, 5, 10, 48, 54, 28lcdvsass 35246 . . . . 5  |-  ( ph  ->  ( ( u ( .r `  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  .x.  i )  =  ( ( ( invg `  A
) `  ( 1r `  A ) )  .x.  ( u  .x.  i ) ) )
6362oveq2d 6324 . . . 4  |-  ( ph  ->  ( ( ( 1r
`  A )  .x.  G ) ( +g  `  C ) ( ( u ( .r `  A ) ( ( invg `  A
) `  ( 1r `  A ) ) ) 
.x.  i ) )  =  ( ( ( 1r `  A ) 
.x.  G ) ( +g  `  C ) ( ( ( invg `  A ) `
 ( 1r `  A ) )  .x.  ( u  .x.  i ) ) ) )
648, 14, 34, 35, 9, 1, 5, 10, 45, 12lcdvscl 35244 . . . . 5  |-  ( ph  ->  ( ( 1r `  A )  .x.  G
)  e.  F )
658, 14, 34, 35, 9, 1, 5, 10, 54, 28lcdvscl 35244 . . . . 5  |-  ( ph  ->  ( u  .x.  i
)  e.  F )
668, 14, 34, 46, 43, 9, 1, 2, 5, 19, 10, 64, 65lcdvsub 35256 . . . 4  |-  ( ph  ->  ( ( ( 1r
`  A )  .x.  G ) R ( u  .x.  i ) )  =  ( ( ( 1r `  A
)  .x.  G )
( +g  `  C ) ( ( ( invg `  A ) `
 ( 1r `  A ) )  .x.  ( u  .x.  i ) ) ) )
678, 14, 34, 35, 49, 9, 1, 5, 10, 48, 33, 28lcdvsass 35246 . . . . . 6  |-  ( ph  ->  ( ( v ( .r `  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  .x.  i )  =  ( ( ( invg `  A
) `  ( 1r `  A ) )  .x.  ( v  .x.  i
) ) )
6867oveq2d 6324 . . . . 5  |-  ( ph  ->  ( ( v  .x.  G ) ( +g  `  C ) ( ( v ( .r `  A ) ( ( invg `  A
) `  ( 1r `  A ) ) ) 
.x.  i ) )  =  ( ( v 
.x.  G ) ( +g  `  C ) ( ( ( invg `  A ) `
 ( 1r `  A ) )  .x.  ( v  .x.  i
) ) ) )
698, 14, 34, 35, 9, 1, 5, 10, 33, 12lcdvscl 35244 . . . . . 6  |-  ( ph  ->  ( v  .x.  G
)  e.  F )
708, 14, 34, 35, 9, 1, 5, 10, 33, 28lcdvscl 35244 . . . . . 6  |-  ( ph  ->  ( v  .x.  i
)  e.  F )
718, 14, 34, 46, 43, 9, 1, 2, 5, 19, 10, 69, 70lcdvsub 35256 . . . . 5  |-  ( ph  ->  ( ( v  .x.  G ) R ( v  .x.  i ) )  =  ( ( v  .x.  G ) ( +g  `  C
) ( ( ( invg `  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 35340 . . . . . 6  |-  ( ph  ->  ( ( v  .x.  G ) R ( v  .x.  i ) )  =  ( G R ( u  .x.  i ) ) )
73 eqid 2471 . . . . . . . . . 10  |-  ( 1r
`  (Scalar `  C )
)  =  ( 1r
`  (Scalar `  C )
)
748, 14, 34, 43, 9, 3, 73, 10lcd1 35248 . . . . . . . . 9  |-  ( ph  ->  ( 1r `  (Scalar `  C ) )  =  ( 1r `  A
) )
7574oveq1d 6323 . . . . . . . 8  |-  ( ph  ->  ( ( 1r `  (Scalar `  C ) ) 
.x.  G )  =  ( ( 1r `  A )  .x.  G
) )
768, 9, 10lcdlmod 35231 . . . . . . . . 9  |-  ( ph  ->  C  e.  LMod )
771, 3, 5, 73lmodvs1 18197 . . . . . . . . 9  |-  ( ( C  e.  LMod  /\  G  e.  F )  ->  (
( 1r `  (Scalar `  C ) )  .x.  G )  =  G )
7876, 12, 77syl2anc 673 . . . . . . . 8  |-  ( ph  ->  ( ( 1r `  (Scalar `  C ) ) 
.x.  G )  =  G )
7975, 78eqtr3d 2507 . . . . . . 7  |-  ( ph  ->  ( ( 1r `  A )  .x.  G
)  =  G )
8079oveq1d 6323 . . . . . 6  |-  ( ph  ->  ( ( ( 1r
`  A )  .x.  G ) R ( u  .x.  i ) )  =  ( G R ( u  .x.  i ) ) )
8172, 80eqtr4d 2508 . . . . 5  |-  ( ph  ->  ( ( v  .x.  G ) R ( v  .x.  i ) )  =  ( ( ( 1r `  A
)  .x.  G ) R ( u  .x.  i ) ) )
8268, 71, 813eqtr2rd 2512 . . . 4  |-  ( ph  ->  ( ( ( 1r
`  A )  .x.  G ) R ( u  .x.  i ) )  =  ( ( v  .x.  G ) ( +g  `  C
) ( ( v ( .r `  A
) ( ( invg `  A ) `
 ( 1r `  A ) ) ) 
.x.  i ) ) )
8363, 66, 823eqtr2rd 2512 . . 3  |-  ( ph  ->  ( ( v  .x.  G ) ( +g  `  C ) ( ( v ( .r `  A ) ( ( invg `  A
) `  ( 1r `  A ) ) ) 
.x.  i ) )  =  ( ( ( 1r `  A ) 
.x.  G ) ( +g  `  C ) ( ( u ( .r `  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  .x.  i ) ) )
841, 2, 3, 4, 5, 6, 7, 11, 26, 32, 37, 52, 53, 57, 61, 83lvecindp2 18440 . 2  |-  ( ph  ->  ( v  =  ( 1r `  A )  /\  ( v ( .r `  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  =  ( u ( .r `  A
) ( ( invg `  A ) `
 ( 1r `  A ) ) ) ) )
8535, 49, 43, 46, 40, 33rngnegr 17901 . . . . 5  |-  ( ph  ->  ( v ( .r
`  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  =  ( ( invg `  A
) `  v )
)
8635, 49, 43, 46, 40, 54rngnegr 17901 . . . . 5  |-  ( ph  ->  ( u ( .r
`  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  =  ( ( invg `  A
) `  u )
)
8785, 86eqeq12d 2486 . . . 4  |-  ( ph  ->  ( ( v ( .r `  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  =  ( u ( .r `  A
) ( ( invg `  A ) `
 ( 1r `  A ) ) )  <-> 
( ( invg `  A ) `  v
)  =  ( ( invg `  A
) `  u )
) )
8835, 46, 42, 33, 54grpinv11 16801 . . . 4  |-  ( ph  ->  ( ( ( invg `  A ) `
 v )  =  ( ( invg `  A ) `  u
)  <->  v  =  u ) )
8987, 88bitrd 261 . . 3  |-  ( ph  ->  ( ( v ( .r `  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  =  ( u ( .r `  A
) ( ( invg `  A ) `
 ( 1r `  A ) ) )  <-> 
v  =  u ) )
9089anbi2d 718 . 2  |-  ( ph  ->  ( ( v  =  ( 1r `  A
)  /\  ( v
( .r `  A
) ( ( invg `  A ) `
 ( 1r `  A ) ) )  =  ( u ( .r `  A ) ( ( invg `  A ) `  ( 1r `  A ) ) ) )  <->  ( v  =  ( 1r `  A )  /\  v  =  u ) ) )
9184, 90mpbid 215 1  |-  ( ph  ->  ( v  =  ( 1r `  A )  /\  v  =  u ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641    \ cdif 3387   {csn 3959   ` cfv 5589  (class class class)co 6308   Basecbs 15199   +g cplusg 15268   .rcmulr 15269  Scalarcsca 15271   .scvsca 15272   0gc0g 15416   Grpcgrp 16747   invgcminusg 16748   -gcsg 16749   1rcur 17813   Ringcrg 17858   LModclmod 18169   LSpanclspn 18272   HLchlt 32987   LHypclh 33620   DVecHcdvh 34717  LCDualclcd 35225  mapdcmpd 35263
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-of 6550  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-mre 15570  df-mrc 15571  df-acs 15573  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-oppg 17075  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-lcv 32656  df-lfl 32695  df-lkr 32723  df-ldual 32761  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  df-lcdual 35226  df-mapd 35264
This theorem is referenced by:  mapdpglem31  35342
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