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Theorem hdmap1l6lem1 35772
Description: Lemma for hdmap1l6 35786. Part (6) in [Baer] p. 47, lines 16-18. (Contributed by NM, 13-Apr-2015.)
Hypotheses
Ref Expression
hdmap1l6.h  |-  H  =  ( LHyp `  K
)
hdmap1l6.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmap1l6.v  |-  V  =  ( Base `  U
)
hdmap1l6.p  |-  .+  =  ( +g  `  U )
hdmap1l6.s  |-  .-  =  ( -g `  U )
hdmap1l6c.o  |-  .0.  =  ( 0g `  U )
hdmap1l6.n  |-  N  =  ( LSpan `  U )
hdmap1l6.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmap1l6.d  |-  D  =  ( Base `  C
)
hdmap1l6.a  |-  .+b  =  ( +g  `  C )
hdmap1l6.r  |-  R  =  ( -g `  C
)
hdmap1l6.q  |-  Q  =  ( 0g `  C
)
hdmap1l6.l  |-  L  =  ( LSpan `  C )
hdmap1l6.m  |-  M  =  ( (mapd `  K
) `  W )
hdmap1l6.i  |-  I  =  ( (HDMap1 `  K
) `  W )
hdmap1l6.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmap1l6.f  |-  ( ph  ->  F  e.  D )
hdmap1l6cl.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
hdmap1l6.mn  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
hdmap1l6e.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
hdmap1l6e.z  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
hdmap1l6e.xn  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
hdmap1l6.yz  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
hdmap1l6.fg  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
hdmap1l6.fe  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )
Assertion
Ref Expression
hdmap1l6lem1  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( L `  { ( F R ( G 
.+b  E ) ) } ) )

Proof of Theorem hdmap1l6lem1
StepHypRef Expression
1 hdmap1l6.h . . . 4  |-  H  =  ( LHyp `  K
)
2 hdmap1l6.m . . . 4  |-  M  =  ( (mapd `  K
) `  W )
3 hdmap1l6.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
4 eqid 2452 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 hdmap1l6.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 35074 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 hdmap1l6cl.x . . . . . . . 8  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
87eldifad 3443 . . . . . . 7  |-  ( ph  ->  X  e.  V )
9 hdmap1l6e.y . . . . . . . 8  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
109eldifad 3443 . . . . . . 7  |-  ( ph  ->  Y  e.  V )
11 hdmap1l6.v . . . . . . . 8  |-  V  =  ( Base `  U
)
12 hdmap1l6.s . . . . . . . 8  |-  .-  =  ( -g `  U )
1311, 12lmodvsubcl 17108 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .-  Y )  e.  V )
146, 8, 10, 13syl3anc 1219 . . . . . 6  |-  ( ph  ->  ( X  .-  Y
)  e.  V )
15 hdmap1l6.n . . . . . . 7  |-  N  =  ( LSpan `  U )
1611, 4, 15lspsncl 17176 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Y )  e.  V )  ->  ( N `  { ( X  .-  Y ) } )  e.  ( LSubSp `  U ) )
176, 14, 16syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Y
) } )  e.  ( LSubSp `  U )
)
18 hdmap1l6e.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
1918eldifad 3443 . . . . . 6  |-  ( ph  ->  Z  e.  V )
2011, 4, 15lspsncl 17176 . . . . . 6  |-  ( ( U  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
216, 19, 20syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
22 eqid 2452 . . . . . 6  |-  ( LSSum `  U )  =  (
LSSum `  U )
234, 22lsmcl 17282 . . . . 5  |-  ( ( U  e.  LMod  /\  ( N `  { ( X  .-  Y ) } )  e.  ( LSubSp `  U )  /\  ( N `  { Z } )  e.  (
LSubSp `  U ) )  ->  ( ( N `
 { ( X 
.-  Y ) } ) ( LSSum `  U
) ( N `  { Z } ) )  e.  ( LSubSp `  U
) )
246, 17, 21, 23syl3anc 1219 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  e.  ( LSubSp `  U )
)
2511, 12lmodvsubcl 17108 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Z  e.  V )  ->  ( X  .-  Z )  e.  V )
266, 8, 19, 25syl3anc 1219 . . . . . 6  |-  ( ph  ->  ( X  .-  Z
)  e.  V )
2711, 4, 15lspsncl 17176 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Z )  e.  V )  ->  ( N `  { ( X  .-  Z ) } )  e.  ( LSubSp `  U ) )
286, 26, 27syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Z
) } )  e.  ( LSubSp `  U )
)
2911, 4, 15lspsncl 17176 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
306, 10, 29syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
314, 22lsmcl 17282 . . . . 5  |-  ( ( U  e.  LMod  /\  ( N `  { ( X  .-  Z ) } )  e.  ( LSubSp `  U )  /\  ( N `  { Y } )  e.  (
LSubSp `  U ) )  ->  ( ( N `
 { ( X 
.-  Z ) } ) ( LSSum `  U
) ( N `  { Y } ) )  e.  ( LSubSp `  U
) )
326, 28, 30, 31syl3anc 1219 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) )  e.  ( LSubSp `  U )
)
331, 2, 3, 4, 5, 24, 32mapdin 35626 . . 3  |-  ( ph  ->  ( M `  (
( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )  =  ( ( M `  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  i^i  ( M `  (
( N `  {
( X  .-  Z
) } ) (
LSSum `  U ) ( N `  { Y } ) ) ) ) )
34 hdmap1l6.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
35 eqid 2452 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
361, 2, 3, 4, 22, 34, 35, 5, 17, 21mapdlsm 35628 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { Z } ) ) )  =  ( ( M `
 ( N `  { ( X  .-  Y ) } ) ) ( LSSum `  C
) ( M `  ( N `  { Z } ) ) ) )
371, 2, 3, 4, 22, 34, 35, 5, 28, 30mapdlsm 35628 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  Z
) } ) (
LSSum `  U ) ( N `  { Y } ) ) )  =  ( ( M `
 ( N `  { ( X  .-  Z ) } ) ) ( LSSum `  C
) ( M `  ( N `  { Y } ) ) ) )
3836, 37ineq12d 3656 . . . 4  |-  ( ph  ->  ( ( M `  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) ) )  i^i  ( M `  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )  =  ( ( ( M `  ( N `  { ( X  .-  Y ) } ) ) ( LSSum `  C ) ( M `
 ( N `  { Z } ) ) )  i^i  ( ( M `  ( N `
 { ( X 
.-  Z ) } ) ) ( LSSum `  C ) ( M `
 ( N `  { Y } ) ) ) ) )
39 hdmap1l6.fg . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
40 hdmap1l6c.o . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
41 hdmap1l6.d . . . . . . . . 9  |-  D  =  ( Base `  C
)
42 hdmap1l6.r . . . . . . . . 9  |-  R  =  ( -g `  C
)
43 hdmap1l6.l . . . . . . . . 9  |-  L  =  ( LSpan `  C )
44 hdmap1l6.i . . . . . . . . 9  |-  I  =  ( (HDMap1 `  K
) `  W )
45 hdmap1l6.f . . . . . . . . 9  |-  ( ph  ->  F  e.  D )
46 hdmap1l6.mn . . . . . . . . . . 11  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
471, 3, 5dvhlvec 35073 . . . . . . . . . . . . 13  |-  ( ph  ->  U  e.  LVec )
48 hdmap1l6.yz . . . . . . . . . . . . 13  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
49 hdmap1l6e.xn . . . . . . . . . . . . 13  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
5011, 40, 15, 47, 10, 18, 8, 48, 49lspindp2 17334 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  -.  Z  e.  ( N `  { X ,  Y } ) ) )
5150simpld 459 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
521, 3, 11, 40, 15, 34, 41, 43, 2, 44, 5, 45, 46, 51, 7, 10hdmap1cl 35769 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5339, 52eqeltrrd 2541 . . . . . . . . 9  |-  ( ph  ->  G  e.  D )
541, 3, 11, 12, 40, 15, 34, 41, 42, 43, 2, 44, 5, 7, 45, 9, 53, 51, 46hdmap1eq 35766 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) ) )
5539, 54mpbid 210 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) )
5655simprd 463 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `  { ( F R G ) } ) )
57 hdmap1l6.fe . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )
5811, 40, 15, 47, 9, 19, 8, 48, 49lspindp1 17332 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Z } )  /\  -.  Y  e.  ( N `  { X ,  Z } ) ) )
5958simpld 459 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z } ) )
601, 3, 11, 40, 15, 34, 41, 43, 2, 44, 5, 45, 46, 59, 7, 19hdmap1cl 35769 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6157, 60eqeltrrd 2541 . . . . . . . . 9  |-  ( ph  ->  E  e.  D )
621, 3, 11, 12, 40, 15, 34, 41, 42, 43, 2, 44, 5, 7, 45, 18, 61, 59, 46hdmap1eq 35766 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Z >. )  =  E  <-> 
( ( M `  ( N `  { Z } ) )  =  ( L `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( L `
 { ( F R E ) } ) ) ) )
6357, 62mpbid 210 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Z } ) )  =  ( L `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( L `
 { ( F R E ) } ) ) )
6463simpld 459 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Z } ) )  =  ( L `  { E } ) )
6556, 64oveq12d 6213 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  Y ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( L `  { ( F R G ) } ) ( LSSum `  C )
( L `  { E } ) ) )
6663simprd 463 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( L `  { ( F R E ) } ) )
6755simpld 459 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( L `  { G } ) )
6866, 67oveq12d 6213 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  Z ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  =  ( ( L `  { ( F R E ) } ) ( LSSum `  C )
( L `  { G } ) ) )
6965, 68ineq12d 3656 . . . 4  |-  ( ph  ->  ( ( ( M `
 ( N `  { ( X  .-  Y ) } ) ) ( LSSum `  C
) ( M `  ( N `  { Z } ) ) )  i^i  ( ( M `
 ( N `  { ( X  .-  Z ) } ) ) ( LSSum `  C
) ( M `  ( N `  { Y } ) ) ) )  =  ( ( ( L `  {
( F R G ) } ) (
LSSum `  C ) ( L `  { E } ) )  i^i  ( ( L `  { ( F R E ) } ) ( LSSum `  C )
( L `  { G } ) ) ) )
7038, 69eqtrd 2493 . . 3  |-  ( ph  ->  ( ( M `  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) ) )  i^i  ( M `  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )  =  ( ( ( L `  {
( F R G ) } ) (
LSSum `  C ) ( L `  { E } ) )  i^i  ( ( L `  { ( F R E ) } ) ( LSSum `  C )
( L `  { G } ) ) ) )
7133, 70eqtrd 2493 . 2  |-  ( ph  ->  ( M `  (
( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )  =  ( ( ( L `  {
( F R G ) } ) (
LSSum `  C ) ( L `  { E } ) )  i^i  ( ( L `  { ( F R E ) } ) ( LSSum `  C )
( L `  { G } ) ) ) )
72 hdmap1l6.p . . . 4  |-  .+  =  ( +g  `  U )
7311, 12, 40, 22, 15, 47, 8, 49, 48, 9, 18, 72baerlem5a 35678 . . 3  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  Z ) ) } )  =  ( ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )
7473fveq2d 5798 . 2  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( M `  ( ( ( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) ) )
75 hdmap1l6.q . . 3  |-  Q  =  ( 0g `  C
)
761, 34, 5lcdlvec 35555 . . 3  |-  ( ph  ->  C  e.  LVec )
771, 2, 3, 11, 15, 34, 41, 43, 5, 45, 46, 8, 10, 53, 67, 19, 61, 64, 49mapdindp 35635 . . 3  |-  ( ph  ->  -.  F  e.  ( L `  { G ,  E } ) )
781, 2, 3, 11, 15, 34, 41, 43, 5, 53, 67, 10, 19, 61, 64, 48mapdncol 35634 . . 3  |-  ( ph  ->  ( L `  { G } )  =/=  ( L `  { E } ) )
791, 2, 3, 11, 15, 34, 41, 43, 5, 53, 67, 40, 75, 9mapdn0 35633 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { Q }
) )
801, 2, 3, 11, 15, 34, 41, 43, 5, 61, 64, 40, 75, 18mapdn0 35633 . . 3  |-  ( ph  ->  E  e.  ( D 
\  { Q }
) )
81 hdmap1l6.a . . 3  |-  .+b  =  ( +g  `  C )
8241, 42, 75, 35, 43, 76, 45, 77, 78, 79, 80, 81baerlem5a 35678 . 2  |-  ( ph  ->  ( L `  {
( F R ( G  .+b  E )
) } )  =  ( ( ( L `
 { ( F R G ) } ) ( LSSum `  C
) ( L `  { E } ) )  i^i  ( ( L `
 { ( F R E ) } ) ( LSSum `  C
) ( L `  { G } ) ) ) )
8371, 74, 823eqtr4d 2503 1  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( L `  { ( F R ( G 
.+b  E ) ) } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2645    \ cdif 3428    i^i cin 3430   {csn 3980   {cpr 3982   <.cotp 3988   ` cfv 5521  (class class class)co 6195   Basecbs 14287   +g cplusg 14352   0gc0g 14492   -gcsg 15527   LSSumclsm 16249   LModclmod 17066   LSubSpclss 17131   LSpanclspn 17170   HLchlt 33314   LHypclh 33947   DVecHcdvh 35042  LCDualclcd 35550  mapdcmpd 35588  HDMap1chdma1 35756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-riotaBAD 32923
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-ot 3989  df-uni 4195  df-int 4232  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-of 6425  df-om 6582  df-1st 6682  df-2nd 6683  df-tpos 6850  df-undef 6897  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-n0 10686  df-z 10753  df-uz 10968  df-fz 11550  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-sca 14368  df-vsca 14369  df-0g 14494  df-mre 14638  df-mrc 14639  df-acs 14641  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-p1 15324  df-lat 15330  df-clat 15392  df-mnd 15529  df-submnd 15579  df-grp 15659  df-minusg 15660  df-sbg 15661  df-subg 15792  df-cntz 15949  df-oppg 15975  df-lsm 16251  df-cmn 16395  df-abl 16396  df-mgp 16709  df-ur 16721  df-rng 16765  df-oppr 16833  df-dvdsr 16851  df-unit 16852  df-invr 16882  df-dvr 16893  df-drng 16952  df-lmod 17068  df-lss 17132  df-lsp 17171  df-lvec 17302  df-lsatoms 32940  df-lshyp 32941  df-lcv 32983  df-lfl 33022  df-lkr 33050  df-ldual 33088  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-llines 33461  df-lplanes 33462  df-lvols 33463  df-lines 33464  df-psubsp 33466  df-pmap 33467  df-padd 33759  df-lhyp 33951  df-laut 33952  df-ldil 34067  df-ltrn 34068  df-trl 34122  df-tgrp 34706  df-tendo 34718  df-edring 34720  df-dveca 34966  df-disoa 34993  df-dvech 35043  df-dib 35103  df-dic 35137  df-dih 35193  df-doch 35312  df-djh 35359  df-lcdual 35551  df-mapd 35589  df-hdmap1 35758
This theorem is referenced by:  hdmap1l6lem2  35773  hdmap1l6a  35774
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