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Theorem mapdpglem30 37826
Description: Lemma for mapdpg 37830. Baer p. 45 line 18: "Hence we deduce (from mapdpglem28 37825, using lvecindp2 17980) 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 37827? (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 2454 . . 3  |-  ( +g  `  C )  =  ( +g  `  C )
3 eqid 2454 . . 3  |-  (Scalar `  C )  =  (Scalar `  C )
4 eqid 2454 . . 3  |-  ( Base `  (Scalar `  C )
)  =  ( Base `  (Scalar `  C )
)
5 mapdpglem26.t . . 3  |-  .x.  =  ( .s `  C )
6 eqid 2454 . . 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 37715 . . 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 37819 . . . 4  |-  ( ph  ->  G  =/=  ( 0g
`  C ) )
25 eldifsn 4141 . . . 4  |-  ( G  e.  ( F  \  { ( 0g `  C ) } )  <-> 
( G  e.  F  /\  G  =/=  ( 0g `  C ) ) )
2612, 24, 25sylanbrc 662 . . 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 457 . . . 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 37820 . . . 4  |-  ( ph  ->  i  =/=  ( 0g
`  C ) )
31 eldifsn 4141 . . . 4  |-  ( i  e.  ( F  \  { ( 0g `  C ) } )  <-> 
( i  e.  F  /\  i  =/=  ( 0g `  C ) ) )
3228, 30, 31sylanbrc 662 . . 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 37724 . . . 4  |-  ( ph  ->  ( Base `  (Scalar `  C ) )  =  B )
3733, 36eleqtrrd 2545 . . 3  |-  ( ph  ->  v  e.  ( Base `  (Scalar `  C )
) )
388, 14, 10dvhlmod 37234 . . . . . 6  |-  ( ph  ->  U  e.  LMod )
3934lmodring 17715 . . . . . 6  |-  ( U  e.  LMod  ->  A  e. 
Ring )
4038, 39syl 16 . . . . 5  |-  ( ph  ->  A  e.  Ring )
41 ringgrp 17398 . . . . . . 7  |-  ( A  e.  Ring  ->  A  e. 
Grp )
4240, 41syl 16 . . . . . 6  |-  ( ph  ->  A  e.  Grp )
43 eqid 2454 . . . . . . . 8  |-  ( 1r
`  A )  =  ( 1r `  A
)
4435, 43ringidcl 17414 . . . . . . 7  |-  ( A  e.  Ring  ->  ( 1r
`  A )  e.  B )
4540, 44syl 16 . . . . . 6  |-  ( ph  ->  ( 1r `  A
)  e.  B )
46 eqid 2454 . . . . . . 7  |-  ( invg `  A )  =  ( invg `  A )
4735, 46grpinvcl 16294 . . . . . 6  |-  ( ( A  e.  Grp  /\  ( 1r `  A )  e.  B )  -> 
( ( invg `  A ) `  ( 1r `  A ) )  e.  B )
4842, 45, 47syl2anc 659 . . . . 5  |-  ( ph  ->  ( ( invg `  A ) `  ( 1r `  A ) )  e.  B )
49 eqid 2454 . . . . . 6  |-  ( .r
`  A )  =  ( .r `  A
)
5035, 49ringcl 17407 . . . . 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 1226 . . . 4  |-  ( ph  ->  ( v ( .r
`  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  e.  B )
5251, 36eleqtrrd 2545 . . 3  |-  ( ph  ->  ( v ( .r
`  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  e.  ( Base `  (Scalar `  C )
) )
5345, 36eleqtrrd 2545 . . 3  |-  ( ph  ->  ( 1r `  A
)  e.  ( Base `  (Scalar `  C )
) )
54 mapdpglem28.ue . . . . 5  |-  ( ph  ->  u  e.  B )
5535, 49ringcl 17407 . . . . 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 1226 . . . 4  |-  ( ph  ->  ( u ( .r
`  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  e.  B )
5756, 36eleqtrrd 2545 . . 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 37824 . . 3  |-  ( ph  ->  ( J `  { G } )  =/=  ( J `  { i } ) )
628, 14, 34, 35, 49, 9, 1, 5, 10, 48, 54, 28lcdvsass 37731 . . . . 5  |-  ( ph  ->  ( ( u ( .r `  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  .x.  i )  =  ( ( ( invg `  A
) `  ( 1r `  A ) )  .x.  ( u  .x.  i ) ) )
6362oveq2d 6286 . . . 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 37729 . . . . 5  |-  ( ph  ->  ( ( 1r `  A )  .x.  G
)  e.  F )
658, 14, 34, 35, 9, 1, 5, 10, 54, 28lcdvscl 37729 . . . . 5  |-  ( ph  ->  ( u  .x.  i
)  e.  F )
668, 14, 34, 46, 43, 9, 1, 2, 5, 19, 10, 64, 65lcdvsub 37741 . . . 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 37731 . . . . . 6  |-  ( ph  ->  ( ( v ( .r `  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  .x.  i )  =  ( ( ( invg `  A
) `  ( 1r `  A ) )  .x.  ( v  .x.  i
) ) )
6867oveq2d 6286 . . . . 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 37729 . . . . . 6  |-  ( ph  ->  ( v  .x.  G
)  e.  F )
708, 14, 34, 35, 9, 1, 5, 10, 33, 28lcdvscl 37729 . . . . . 6  |-  ( ph  ->  ( v  .x.  i
)  e.  F )
718, 14, 34, 46, 43, 9, 1, 2, 5, 19, 10, 69, 70lcdvsub 37741 . . . . 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 37825 . . . . . 6  |-  ( ph  ->  ( ( v  .x.  G ) R ( v  .x.  i ) )  =  ( G R ( u  .x.  i ) ) )
73 eqid 2454 . . . . . . . . . 10  |-  ( 1r
`  (Scalar `  C )
)  =  ( 1r
`  (Scalar `  C )
)
748, 14, 34, 43, 9, 3, 73, 10lcd1 37733 . . . . . . . . 9  |-  ( ph  ->  ( 1r `  (Scalar `  C ) )  =  ( 1r `  A
) )
7574oveq1d 6285 . . . . . . . 8  |-  ( ph  ->  ( ( 1r `  (Scalar `  C ) ) 
.x.  G )  =  ( ( 1r `  A )  .x.  G
) )
768, 9, 10lcdlmod 37716 . . . . . . . . 9  |-  ( ph  ->  C  e.  LMod )
771, 3, 5, 73lmodvs1 17735 . . . . . . . . 9  |-  ( ( C  e.  LMod  /\  G  e.  F )  ->  (
( 1r `  (Scalar `  C ) )  .x.  G )  =  G )
7876, 12, 77syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( ( 1r `  (Scalar `  C ) ) 
.x.  G )  =  G )
7975, 78eqtr3d 2497 . . . . . . 7  |-  ( ph  ->  ( ( 1r `  A )  .x.  G
)  =  G )
8079oveq1d 6285 . . . . . 6  |-  ( ph  ->  ( ( ( 1r
`  A )  .x.  G ) R ( u  .x.  i ) )  =  ( G R ( u  .x.  i ) ) )
8172, 80eqtr4d 2498 . . . . 5  |-  ( ph  ->  ( ( v  .x.  G ) R ( v  .x.  i ) )  =  ( ( ( 1r `  A
)  .x.  G ) R ( u  .x.  i ) ) )
8268, 71, 813eqtr2rd 2502 . . . 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 2502 . . 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 17980 . 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 17436 . . . . 5  |-  ( ph  ->  ( v ( .r
`  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  =  ( ( invg `  A
) `  v )
)
8635, 49, 43, 46, 40, 54rngnegr 17436 . . . . 5  |-  ( ph  ->  ( u ( .r
`  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  =  ( ( invg `  A
) `  u )
)
8785, 86eqeq12d 2476 . . . 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 16306 . . . 4  |-  ( ph  ->  ( ( ( invg `  A ) `
 v )  =  ( ( invg `  A ) `  u
)  <->  v  =  u ) )
8987, 88bitrd 253 . . 3  |-  ( ph  ->  ( ( v ( .r `  A ) ( ( invg `  A ) `  ( 1r `  A ) ) )  =  ( u ( .r `  A
) ( ( invg `  A ) `
 ( 1r `  A ) ) )  <-> 
v  =  u ) )
9089anbi2d 701 . 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 210 1  |-  ( ph  ->  ( v  =  ( 1r `  A )  /\  v  =  u ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649    \ cdif 3458   {csn 4016   ` cfv 5570  (class class class)co 6270   Basecbs 14716   +g cplusg 14784   .rcmulr 14785  Scalarcsca 14787   .scvsca 14788   0gc0g 14929   Grpcgrp 16252   invgcminusg 16253   -gcsg 16254   1rcur 17348   Ringcrg 17393   LModclmod 17707   LSpanclspn 17812   HLchlt 35472   LHypclh 36105   DVecHcdvh 37202  LCDualclcd 37710  mapdcmpd 37748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-riotaBAD 35081
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-tpos 6947  df-undef 6994  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-sca 14800  df-vsca 14801  df-0g 14931  df-mre 15075  df-mrc 15076  df-acs 15078  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397  df-cntz 16554  df-oppg 16580  df-lsm 16855  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-dvr 17527  df-drng 17593  df-lmod 17709  df-lss 17774  df-lsp 17813  df-lvec 17944  df-lsatoms 35098  df-lshyp 35099  df-lcv 35141  df-lfl 35180  df-lkr 35208  df-ldual 35246  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-llines 35619  df-lplanes 35620  df-lvols 35621  df-lines 35622  df-psubsp 35624  df-pmap 35625  df-padd 35917  df-lhyp 36109  df-laut 36110  df-ldil 36225  df-ltrn 36226  df-trl 36281  df-tgrp 36866  df-tendo 36878  df-edring 36880  df-dveca 37126  df-disoa 37153  df-dvech 37203  df-dib 37263  df-dic 37297  df-dih 37353  df-doch 37472  df-djh 37519  df-lcdual 37711  df-mapd 37749
This theorem is referenced by:  mapdpglem31  37827
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