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Theorem hdmap1l6lem2 35762
Description: Lemma for hdmap1l6 35775. Part (6) in [Baer] p. 47, lines 20-22. (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
hdmap1l6lem2  |-  ( ph  ->  ( M `  ( N `  { ( Y  .+  Z ) } ) )  =  ( L `  { ( G  .+b  E ) } ) )

Proof of Theorem hdmap1l6lem2
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 2451 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 hdmap1l6.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 35063 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 hdmap1l6e.y . . . . . . 7  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
87eldifad 3440 . . . . . 6  |-  ( ph  ->  Y  e.  V )
9 hdmap1l6.v . . . . . . 7  |-  V  =  ( Base `  U
)
10 hdmap1l6.n . . . . . . 7  |-  N  =  ( LSpan `  U )
119, 4, 10lspsncl 17166 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
126, 8, 11syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
13 hdmap1l6e.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
1413eldifad 3440 . . . . . 6  |-  ( ph  ->  Z  e.  V )
159, 4, 10lspsncl 17166 . . . . . 6  |-  ( ( U  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
166, 14, 15syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
17 eqid 2451 . . . . . 6  |-  ( LSSum `  U )  =  (
LSSum `  U )
184, 17lsmcl 17272 . . . . 5  |-  ( ( U  e.  LMod  /\  ( N `  { Y } )  e.  (
LSubSp `  U )  /\  ( N `  { Z } )  e.  (
LSubSp `  U ) )  ->  ( ( N `
 { Y }
) ( LSSum `  U
) ( N `  { Z } ) )  e.  ( LSubSp `  U
) )
196, 12, 16, 18syl3anc 1219 . . . 4  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  e.  ( LSubSp `  U )
)
20 hdmap1l6cl.x . . . . . . . 8  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
2120eldifad 3440 . . . . . . 7  |-  ( ph  ->  X  e.  V )
22 hdmap1l6.p . . . . . . . . 9  |-  .+  =  ( +g  `  U )
239, 22lmodvacl 17070 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  Y  e.  V  /\  Z  e.  V )  ->  ( Y  .+  Z )  e.  V )
246, 8, 14, 23syl3anc 1219 . . . . . . 7  |-  ( ph  ->  ( Y  .+  Z
)  e.  V )
25 hdmap1l6.s . . . . . . . 8  |-  .-  =  ( -g `  U )
269, 25lmodvsubcl 17098 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  ( Y  .+  Z )  e.  V )  ->  ( X  .-  ( Y  .+  Z ) )  e.  V )
276, 21, 24, 26syl3anc 1219 . . . . . 6  |-  ( ph  ->  ( X  .-  ( Y  .+  Z ) )  e.  V )
289, 4, 10lspsncl 17166 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  ( Y  .+  Z ) )  e.  V )  ->  ( N `  { ( X  .-  ( Y  .+  Z ) ) } )  e.  ( LSubSp `  U ) )
296, 27, 28syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  Z ) ) } )  e.  (
LSubSp `  U ) )
309, 4, 10lspsncl 17166 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
316, 21, 30syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
324, 17lsmcl 17272 . . . . 5  |-  ( ( U  e.  LMod  /\  ( N `  { ( X  .-  ( Y  .+  Z ) ) } )  e.  ( LSubSp `  U )  /\  ( N `  { X } )  e.  (
LSubSp `  U ) )  ->  ( ( N `
 { ( X 
.-  ( Y  .+  Z ) ) } ) ( LSSum `  U
) ( N `  { X } ) )  e.  ( LSubSp `  U
) )
336, 29, 31, 32syl3anc 1219 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) (
LSSum `  U ) ( N `  { X } ) )  e.  ( LSubSp `  U )
)
341, 2, 3, 4, 5, 19, 33mapdin 35615 . . 3  |-  ( ph  ->  ( M `  (
( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) (
LSSum `  U ) ( N `  { X } ) ) ) )  =  ( ( M `  ( ( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  i^i  ( M `  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) ) ) )
35 hdmap1l6.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
36 eqid 2451 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
371, 2, 3, 4, 17, 35, 36, 5, 12, 16mapdlsm 35617 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  =  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) ) )
38 hdmap1l6.fg . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
39 hdmap1l6c.o . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
40 hdmap1l6.d . . . . . . . . 9  |-  D  =  ( Base `  C
)
41 hdmap1l6.r . . . . . . . . 9  |-  R  =  ( -g `  C
)
42 hdmap1l6.l . . . . . . . . 9  |-  L  =  ( LSpan `  C )
43 hdmap1l6.i . . . . . . . . 9  |-  I  =  ( (HDMap1 `  K
) `  W )
44 hdmap1l6.f . . . . . . . . 9  |-  ( ph  ->  F  e.  D )
45 hdmap1l6.mn . . . . . . . . . . 11  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
461, 3, 5dvhlvec 35062 . . . . . . . . . . . . 13  |-  ( ph  ->  U  e.  LVec )
47 hdmap1l6.yz . . . . . . . . . . . . 13  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
48 hdmap1l6e.xn . . . . . . . . . . . . 13  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
499, 39, 10, 46, 8, 13, 21, 47, 48lspindp2 17324 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  -.  Z  e.  ( N `  { X ,  Y } ) ) )
5049simpld 459 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
511, 3, 9, 39, 10, 35, 40, 42, 2, 43, 5, 44, 45, 50, 20, 8hdmap1cl 35758 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5238, 51eqeltrrd 2540 . . . . . . . . 9  |-  ( ph  ->  G  e.  D )
531, 3, 9, 25, 39, 10, 35, 40, 41, 42, 2, 43, 5, 20, 44, 7, 52, 50, 45hdmap1eq 35755 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) ) )
5438, 53mpbid 210 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) )
5554simpld 459 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( L `  { G } ) )
56 hdmap1l6.fe . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )
579, 39, 10, 46, 7, 14, 21, 47, 48lspindp1 17322 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Z } )  /\  -.  Y  e.  ( N `  { X ,  Z } ) ) )
5857simpld 459 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z } ) )
591, 3, 9, 39, 10, 35, 40, 42, 2, 43, 5, 44, 45, 58, 20, 14hdmap1cl 35758 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6056, 59eqeltrrd 2540 . . . . . . . . 9  |-  ( ph  ->  E  e.  D )
611, 3, 9, 25, 39, 10, 35, 40, 41, 42, 2, 43, 5, 20, 44, 13, 60, 58, 45hdmap1eq 35755 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Z >. )  =  E  <-> 
( ( M `  ( N `  { Z } ) )  =  ( L `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( L `
 { ( F R E ) } ) ) ) )
6256, 61mpbid 210 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Z } ) )  =  ( L `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( L `
 { ( F R E ) } ) ) )
6362simpld 459 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Z } ) )  =  ( L `  { E } ) )
6455, 63oveq12d 6210 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( L `  { G } ) (
LSSum `  C ) ( L `  { E } ) ) )
6537, 64eqtrd 2492 . . . 4  |-  ( ph  ->  ( M `  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  =  ( ( L `  { G } ) (
LSSum `  C ) ( L `  { E } ) ) )
661, 2, 3, 4, 17, 35, 36, 5, 29, 31mapdlsm 35617 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) )  =  ( ( M `  ( N `  { ( X  .-  ( Y 
.+  Z ) ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { X }
) ) ) )
67 hdmap1l6.a . . . . . . 7  |-  .+b  =  ( +g  `  C )
68 hdmap1l6.q . . . . . . 7  |-  Q  =  ( 0g `  C
)
691, 3, 9, 22, 25, 39, 10, 35, 40, 67, 41, 68, 42, 2, 43, 5, 44, 20, 45, 7, 13, 48, 47, 38, 56hdmap1l6lem1 35761 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( L `  { ( F R ( G 
.+b  E ) ) } ) )
7069, 45oveq12d 6210 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  ( Y 
.+  Z ) ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { X }
) ) )  =  ( ( L `  { ( F R ( G  .+b  E
) ) } ) ( LSSum `  C )
( L `  { F } ) ) )
7166, 70eqtrd 2492 . . . 4  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) )  =  ( ( L `  { ( F R ( G  .+b  E
) ) } ) ( LSSum `  C )
( L `  { F } ) ) )
7265, 71ineq12d 3653 . . 3  |-  ( ph  ->  ( ( M `  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) ) )  i^i  ( M `  ( ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) (
LSSum `  U ) ( N `  { X } ) ) ) )  =  ( ( ( L `  { G } ) ( LSSum `  C ) ( L `
 { E }
) )  i^i  (
( L `  {
( F R ( G  .+b  E )
) } ) (
LSSum `  C ) ( L `  { F } ) ) ) )
7334, 72eqtrd 2492 . 2  |-  ( ph  ->  ( M `  (
( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) (
LSSum `  U ) ( N `  { X } ) ) ) )  =  ( ( ( L `  { G } ) ( LSSum `  C ) ( L `
 { E }
) )  i^i  (
( L `  {
( F R ( G  .+b  E )
) } ) (
LSSum `  C ) ( L `  { F } ) ) ) )
749, 25, 39, 17, 10, 46, 21, 48, 47, 7, 13, 22baerlem5b 35668 . . 3  |-  ( ph  ->  ( N `  {
( Y  .+  Z
) } )  =  ( ( ( N `
 { Y }
) ( LSSum `  U
) ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  ( Y  .+  Z ) ) } ) ( LSSum `  U
) ( N `  { X } ) ) ) )
7574fveq2d 5795 . 2  |-  ( ph  ->  ( M `  ( N `  { ( Y  .+  Z ) } ) )  =  ( M `  ( ( ( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) )  i^i  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) ) ) )
761, 35, 5lcdlvec 35544 . . 3  |-  ( ph  ->  C  e.  LVec )
771, 2, 3, 9, 10, 35, 40, 42, 5, 44, 45, 21, 8, 52, 55, 14, 60, 63, 48mapdindp 35624 . . 3  |-  ( ph  ->  -.  F  e.  ( L `  { G ,  E } ) )
781, 2, 3, 9, 10, 35, 40, 42, 5, 52, 55, 8, 14, 60, 63, 47mapdncol 35623 . . 3  |-  ( ph  ->  ( L `  { G } )  =/=  ( L `  { E } ) )
791, 2, 3, 9, 10, 35, 40, 42, 5, 52, 55, 39, 68, 7mapdn0 35622 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { Q }
) )
801, 2, 3, 9, 10, 35, 40, 42, 5, 60, 63, 39, 68, 13mapdn0 35622 . . 3  |-  ( ph  ->  E  e.  ( D 
\  { Q }
) )
8140, 41, 68, 36, 42, 76, 44, 77, 78, 79, 80, 67baerlem5b 35668 . 2  |-  ( ph  ->  ( L `  {
( G  .+b  E
) } )  =  ( ( ( L `
 { G }
) ( LSSum `  C
) ( L `  { E } ) )  i^i  ( ( L `
 { ( F R ( G  .+b  E ) ) } ) ( LSSum `  C )
( L `  { F } ) ) ) )
8273, 75, 813eqtr4d 2502 1  |-  ( ph  ->  ( M `  ( N `  { ( Y  .+  Z ) } ) )  =  ( L `  { ( G  .+b  E ) } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644    \ cdif 3425    i^i cin 3427   {csn 3977   {cpr 3979   <.cotp 3985   ` cfv 5518  (class class class)co 6192   Basecbs 14278   +g cplusg 14342   0gc0g 14482   -gcsg 15517   LSSumclsm 16239   LModclmod 17056   LSubSpclss 17121   LSpanclspn 17160   HLchlt 33303   LHypclh 33936   DVecHcdvh 35031  LCDualclcd 35539  mapdcmpd 35577  HDMap1chdma1 35745
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 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-riotaBAD 32912
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-ot 3986  df-uni 4192  df-int 4229  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-of 6422  df-om 6579  df-1st 6679  df-2nd 6680  df-tpos 6847  df-undef 6894  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-plusg 14355  df-mulr 14356  df-sca 14358  df-vsca 14359  df-0g 14484  df-mre 14628  df-mrc 14629  df-acs 14631  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-p1 15314  df-lat 15320  df-clat 15382  df-mnd 15519  df-submnd 15569  df-grp 15649  df-minusg 15650  df-sbg 15651  df-subg 15782  df-cntz 15939  df-oppg 15965  df-lsm 16241  df-cmn 16385  df-abl 16386  df-mgp 16699  df-ur 16711  df-rng 16755  df-oppr 16823  df-dvdsr 16841  df-unit 16842  df-invr 16872  df-dvr 16883  df-drng 16942  df-lmod 17058  df-lss 17122  df-lsp 17161  df-lvec 17292  df-lsatoms 32929  df-lshyp 32930  df-lcv 32972  df-lfl 33011  df-lkr 33039  df-ldual 33077  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304  df-llines 33450  df-lplanes 33451  df-lvols 33452  df-lines 33453  df-psubsp 33455  df-pmap 33456  df-padd 33748  df-lhyp 33940  df-laut 33941  df-ldil 34056  df-ltrn 34057  df-trl 34111  df-tgrp 34695  df-tendo 34707  df-edring 34709  df-dveca 34955  df-disoa 34982  df-dvech 35032  df-dib 35092  df-dic 35126  df-dih 35182  df-doch 35301  df-djh 35348  df-lcdual 35540  df-mapd 35578  df-hdmap1 35747
This theorem is referenced by:  hdmap1l6a  35763
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