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Theorem hgmapmul 30847
Description: Part 15 of [Baer] p. 50 line 16. The multiplication is reversed after converting to the dual space scalar to the vector space scalar. (Contributed by NM, 7-Jun-2015.)
Hypotheses
Ref Expression
hgmapmul.h  |-  H  =  ( LHyp `  K
)
hgmapmul.u  |-  U  =  ( ( DVecH `  K
) `  W )
hgmapmul.r  |-  R  =  (Scalar `  U )
hgmapmul.b  |-  B  =  ( Base `  R
)
hgmapmul.t  |-  .x.  =  ( .r `  R )
hgmapmul.g  |-  G  =  ( (HGMap `  K
) `  W )
hgmapmul.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hgmapmul.x  |-  ( ph  ->  X  e.  B )
hgmapmul.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
hgmapmul  |-  ( ph  ->  ( G `  ( X  .x.  Y ) )  =  ( ( G `
 Y )  .x.  ( G `  X ) ) )

Proof of Theorem hgmapmul
StepHypRef Expression
1 hgmapmul.h . . . 4  |-  H  =  ( LHyp `  K
)
2 hgmapmul.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
3 eqid 2253 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
4 eqid 2253 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
5 hgmapmul.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 2, 3, 4, 5dvh1dim 30391 . . 3  |-  ( ph  ->  E. t  e.  (
Base `  U )
t  =/=  ( 0g
`  U ) )
7 eqid 2253 . . . . . . . . 9  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
81, 7, 5lcdlmod 30541 . . . . . . . 8  |-  ( ph  ->  ( (LCDual `  K
) `  W )  e.  LMod )
983ad2ant1 981 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( (LCDual `  K ) `  W
)  e.  LMod )
10 hgmapmul.r . . . . . . . . 9  |-  R  =  (Scalar `  U )
11 hgmapmul.b . . . . . . . . 9  |-  B  =  ( Base `  R
)
12 eqid 2253 . . . . . . . . 9  |-  (Scalar `  ( (LCDual `  K ) `  W ) )  =  (Scalar `  ( (LCDual `  K ) `  W
) )
13 eqid 2253 . . . . . . . . 9  |-  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) )  =  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) )
14 hgmapmul.g . . . . . . . . 9  |-  G  =  ( (HGMap `  K
) `  W )
15 hgmapmul.x . . . . . . . . 9  |-  ( ph  ->  X  e.  B )
161, 2, 10, 11, 7, 12, 13, 14, 5, 15hgmapdcl 30842 . . . . . . . 8  |-  ( ph  ->  ( G `  X
)  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
17163ad2ant1 981 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( G `  X )  e.  (
Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )
18 hgmapmul.y . . . . . . . . 9  |-  ( ph  ->  Y  e.  B )
191, 2, 10, 11, 7, 12, 13, 14, 5, 18hgmapdcl 30842 . . . . . . . 8  |-  ( ph  ->  ( G `  Y
)  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
20193ad2ant1 981 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( G `  Y )  e.  (
Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )
21 eqid 2253 . . . . . . . 8  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  ( Base `  ( (LCDual `  K
) `  W )
)
22 eqid 2253 . . . . . . . 8  |-  ( (HDMap `  K ) `  W
)  =  ( (HDMap `  K ) `  W
)
2353ad2ant1 981 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
24 simp2 961 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  t  e.  (
Base `  U )
)
251, 2, 3, 7, 21, 22, 23, 24hdmapcl 30782 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  t )  e.  ( Base `  (
(LCDual `  K ) `  W ) ) )
26 eqid 2253 . . . . . . . 8  |-  ( .s
`  ( (LCDual `  K ) `  W
) )  =  ( .s `  ( (LCDual `  K ) `  W
) )
27 eqid 2253 . . . . . . . 8  |-  ( .r
`  (Scalar `  ( (LCDual `  K ) `  W
) ) )  =  ( .r `  (Scalar `  ( (LCDual `  K
) `  W )
) )
2821, 12, 26, 13, 27lmodvsass 15489 . . . . . . 7  |-  ( ( ( (LCDual `  K
) `  W )  e.  LMod  /\  ( ( G `  X )  e.  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) )  /\  ( G `  Y )  e.  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) )  /\  (
( (HDMap `  K
) `  W ) `  t )  e.  (
Base `  ( (LCDual `  K ) `  W
) ) ) )  ->  ( ( ( G `  X ) ( .r `  (Scalar `  ( (LCDual `  K
) `  W )
) ) ( G `
 Y ) ) ( .s `  (
(LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  t ) )  =  ( ( G `  X ) ( .s
`  ( (LCDual `  K ) `  W
) ) ( ( G `  Y ) ( .s `  (
(LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  t ) ) ) )
299, 17, 20, 25, 28syl13anc 1189 . . . . . 6  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( ( G `  X ) ( .r `  (Scalar `  ( (LCDual `  K
) `  W )
) ) ( G `
 Y ) ) ( .s `  (
(LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  t ) )  =  ( ( G `  X ) ( .s
`  ( (LCDual `  K ) `  W
) ) ( ( G `  Y ) ( .s `  (
(LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  t ) ) ) )
301, 2, 5dvhlmod 30059 . . . . . . . . . 10  |-  ( ph  ->  U  e.  LMod )
31303ad2ant1 981 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  U  e.  LMod )
32153ad2ant1 981 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  X  e.  B
)
33183ad2ant1 981 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  Y  e.  B
)
34 eqid 2253 . . . . . . . . . 10  |-  ( .s
`  U )  =  ( .s `  U
)
35 hgmapmul.t . . . . . . . . . 10  |-  .x.  =  ( .r `  R )
363, 10, 34, 11, 35lmodvsass 15489 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  ( X  e.  B  /\  Y  e.  B  /\  t  e.  ( Base `  U ) ) )  ->  ( ( X 
.x.  Y ) ( .s `  U ) t )  =  ( X ( .s `  U ) ( Y ( .s `  U
) t ) ) )
3731, 32, 33, 24, 36syl13anc 1189 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( X 
.x.  Y ) ( .s `  U ) t )  =  ( X ( .s `  U ) ( Y ( .s `  U
) t ) ) )
3837fveq2d 5381 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  ( ( X  .x.  Y ) ( .s `  U ) t ) )  =  ( ( (HDMap `  K ) `  W
) `  ( X
( .s `  U
) ( Y ( .s `  U ) t ) ) ) )
393, 10, 34, 11lmodvscl 15479 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  Y  e.  B  /\  t  e.  ( Base `  U
) )  ->  ( Y ( .s `  U ) t )  e.  ( Base `  U
) )
4031, 33, 24, 39syl3anc 1187 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( Y ( .s `  U ) t )  e.  (
Base `  U )
)
411, 2, 3, 34, 10, 11, 7, 26, 22, 14, 23, 40, 32hgmapvs 30843 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  ( X
( .s `  U
) ( Y ( .s `  U ) t ) ) )  =  ( ( G `
 X ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  ( Y ( .s `  U ) t ) ) ) )
421, 2, 3, 34, 10, 11, 7, 26, 22, 14, 23, 24, 33hgmapvs 30843 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  ( Y
( .s `  U
) t ) )  =  ( ( G `
 Y ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  t
) ) )
4342oveq2d 5726 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( G `
 X ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  ( Y ( .s `  U ) t ) ) )  =  ( ( G `  X
) ( .s `  ( (LCDual `  K ) `  W ) ) ( ( G `  Y
) ( .s `  ( (LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  t ) ) ) )
4438, 41, 433eqtrd 2289 . . . . . 6  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  ( ( X  .x.  Y ) ( .s `  U ) t ) )  =  ( ( G `  X ) ( .s
`  ( (LCDual `  K ) `  W
) ) ( ( G `  Y ) ( .s `  (
(LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  t ) ) ) )
4510, 11, 35lmodmcl 15474 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  Y )  e.  B )
4630, 15, 18, 45syl3anc 1187 . . . . . . . 8  |-  ( ph  ->  ( X  .x.  Y
)  e.  B )
47463ad2ant1 981 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( X  .x.  Y )  e.  B
)
481, 2, 3, 34, 10, 11, 7, 26, 22, 14, 23, 24, 47hgmapvs 30843 . . . . . 6  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  ( ( X  .x.  Y ) ( .s `  U ) t ) )  =  ( ( G `  ( X  .x.  Y ) ) ( .s `  ( (LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  t ) ) )
4929, 44, 483eqtr2rd 2292 . . . . 5  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( G `
 ( X  .x.  Y ) ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  t
) )  =  ( ( ( G `  X ) ( .r
`  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ( G `  Y ) ) ( .s `  ( (LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  t ) ) )
50 eqid 2253 . . . . . 6  |-  ( 0g
`  ( (LCDual `  K ) `  W
) )  =  ( 0g `  ( (LCDual `  K ) `  W
) )
511, 7, 5lcdlvec 30540 . . . . . . 7  |-  ( ph  ->  ( (LCDual `  K
) `  W )  e.  LVec )
52513ad2ant1 981 . . . . . 6  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( (LCDual `  K ) `  W
)  e.  LVec )
531, 2, 10, 11, 7, 12, 13, 14, 5, 46hgmapdcl 30842 . . . . . . 7  |-  ( ph  ->  ( G `  ( X  .x.  Y ) )  e.  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) ) )
54533ad2ant1 981 . . . . . 6  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( G `  ( X  .x.  Y ) )  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
5512, 13, 27lmodmcl 15474 . . . . . . . 8  |-  ( ( ( (LCDual `  K
) `  W )  e.  LMod  /\  ( G `  X )  e.  (
Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) )  /\  ( G `  Y )  e.  (
Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )  ->  ( ( G `  X )
( .r `  (Scalar `  ( (LCDual `  K
) `  W )
) ) ( G `
 Y ) )  e.  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) ) )
568, 16, 19, 55syl3anc 1187 . . . . . . 7  |-  ( ph  ->  ( ( G `  X ) ( .r
`  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ( G `  Y ) )  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
57563ad2ant1 981 . . . . . 6  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( G `
 X ) ( .r `  (Scalar `  ( (LCDual `  K ) `  W ) ) ) ( G `  Y
) )  e.  (
Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )
58 simp3 962 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  t  =/=  ( 0g `  U ) )
591, 2, 3, 4, 7, 50, 22, 23, 24hdmapeq0 30796 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( ( (HDMap `  K ) `  W ) `  t
)  =  ( 0g
`  ( (LCDual `  K ) `  W
) )  <->  t  =  ( 0g `  U ) ) )
6059necon3bid 2447 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( ( (HDMap `  K ) `  W ) `  t
)  =/=  ( 0g
`  ( (LCDual `  K ) `  W
) )  <->  t  =/=  ( 0g `  U ) ) )
6158, 60mpbird 225 . . . . . 6  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  t )  =/=  ( 0g `  (
(LCDual `  K ) `  W ) ) )
6221, 26, 12, 13, 50, 52, 54, 57, 25, 61lvecvscan2 15700 . . . . 5  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( ( G `  ( X 
.x.  Y ) ) ( .s `  (
(LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  t ) )  =  ( ( ( G `
 X ) ( .r `  (Scalar `  ( (LCDual `  K ) `  W ) ) ) ( G `  Y
) ) ( .s
`  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  t
) )  <->  ( G `  ( X  .x.  Y
) )  =  ( ( G `  X
) ( .r `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ( G `  Y ) ) ) )
6349, 62mpbid 203 . . . 4  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( G `  ( X  .x.  Y ) )  =  ( ( G `  X ) ( .r `  (Scalar `  ( (LCDual `  K
) `  W )
) ) ( G `
 Y ) ) )
6463rexlimdv3a 2631 . . 3  |-  ( ph  ->  ( E. t  e.  ( Base `  U
) t  =/=  ( 0g `  U )  -> 
( G `  ( X  .x.  Y ) )  =  ( ( G `
 X ) ( .r `  (Scalar `  ( (LCDual `  K ) `  W ) ) ) ( G `  Y
) ) ) )
656, 64mpd 16 . 2  |-  ( ph  ->  ( G `  ( X  .x.  Y ) )  =  ( ( G `
 X ) ( .r `  (Scalar `  ( (LCDual `  K ) `  W ) ) ) ( G `  Y
) ) )
661, 2, 10, 11, 14, 5, 15hgmapcl 30841 . . 3  |-  ( ph  ->  ( G `  X
)  e.  B )
671, 2, 10, 11, 14, 5, 18hgmapcl 30841 . . 3  |-  ( ph  ->  ( G `  Y
)  e.  B )
681, 2, 10, 11, 35, 7, 12, 27, 5, 66, 67lcdsmul 30551 . 2  |-  ( ph  ->  ( ( G `  X ) ( .r
`  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ( G `  Y ) )  =  ( ( G `  Y ) 
.x.  ( G `  X ) ) )
6965, 68eqtrd 2285 1  |-  ( ph  ->  ( G `  ( X  .x.  Y ) )  =  ( ( G `
 Y )  .x.  ( G `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   E.wrex 2510   ` cfv 4592  (class class class)co 5710   Basecbs 13022   .rcmulr 13083  Scalarcsca 13085   .scvsca 13086   0gc0g 13274   LModclmod 15462   LVecclvec 15690   HLchlt 28299   LHypclh 28932   DVecHcdvh 30027  LCDualclcd 30535  HDMapchdma 30742  HGMapchg 30835
This theorem is referenced by:  hgmapvvlem1  30875  hdmapglem7  30881  hlhilsrnglem  30905
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-ot 3554  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-of 5930  df-1st 5974  df-2nd 5975  df-tpos 6086  df-iota 6143  df-undef 6182  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-n0 9845  df-z 9904  df-uz 10110  df-fz 10661  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-sca 13098  df-vsca 13099  df-0g 13278  df-mre 13361  df-mrc 13362  df-acs 13363  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-mnd 14202  df-submnd 14251  df-grp 14324  df-minusg 14325  df-sbg 14326  df-subg 14453  df-cntz 14628  df-oppg 14654  df-lsm 14782  df-cmn 14926  df-abl 14927  df-mgp 15161  df-ring 15175  df-ur 15177  df-oppr 15240  df-dvdsr 15258  df-unit 15259  df-invr 15289  df-dvr 15300  df-drng 15349  df-lmod 15464  df-lss 15525  df-lsp 15564  df-lvec 15691  df-lsatoms 27925  df-lshyp 27926  df-lcv 27968  df-lfl 28007  df-lkr 28035  df-ldual 28073  df-oposet 28125  df-ol 28127  df-oml 28128  df-covers 28215  df-ats 28216  df-atl 28247  df-cvlat 28271  df-hlat 28300  df-llines 28446  df-lplanes 28447  df-lvols 28448  df-lines 28449  df-psubsp 28451  df-pmap 28452  df-padd 28744  df-lhyp 28936  df-laut 28937  df-ldil 29052  df-ltrn 29053  df-trl 29107  df-tgrp 29691  df-tendo 29703  df-edring 29705  df-dveca 29951  df-disoa 29978  df-dvech 30028  df-dib 30088  df-dic 30122  df-dih 30178  df-doch 30297  df-djh 30344  df-lcdual 30536  df-mapd 30574  df-hvmap 30706  df-hdmap1 30743  df-hdmap 30744  df-hgmap 30836
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