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Theorem dicvscacl 34728
Description: Membership in value of the partial isomorphism C is closed under scalar product. (Contributed by NM, 16-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
dicvscacl.l  |-  .<_  =  ( le `  K )
dicvscacl.a  |-  A  =  ( Atoms `  K )
dicvscacl.h  |-  H  =  ( LHyp `  K
)
dicvscacl.e  |-  E  =  ( ( TEndo `  K
) `  W )
dicvscacl.u  |-  U  =  ( ( DVecH `  K
) `  W )
dicvscacl.i  |-  I  =  ( ( DIsoC `  K
) `  W )
dicvscacl.s  |-  .x.  =  ( .s `  U )
Assertion
Ref Expression
dicvscacl  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( X  .x.  Y )  e.  ( I `  Q ) )

Proof of Theorem dicvscacl
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 simp1 1005 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp3l 1033 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  X  e.  E
)
3 dicvscacl.l . . . . . . . 8  |-  .<_  =  ( le `  K )
4 dicvscacl.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
5 dicvscacl.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
6 dicvscacl.i . . . . . . . 8  |-  I  =  ( ( DIsoC `  K
) `  W )
7 dicvscacl.u . . . . . . . 8  |-  U  =  ( ( DVecH `  K
) `  W )
8 eqid 2422 . . . . . . . 8  |-  ( Base `  U )  =  (
Base `  U )
93, 4, 5, 6, 7, 8dicssdvh 34723 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  ( Base `  U ) )
10 eqid 2422 . . . . . . . . . 10  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
11 dicvscacl.e . . . . . . . . . 10  |-  E  =  ( ( TEndo `  K
) `  W )
125, 10, 11, 7, 8dvhvbase 34624 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( ( ( LTrn `  K
) `  W )  X.  E ) )
1312eqcomd 2430 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( LTrn `  K ) `  W
)  X.  E )  =  ( Base `  U
) )
1413adantr 466 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( LTrn `  K ) `  W
)  X.  E )  =  ( Base `  U
) )
159, 14sseqtr4d 3501 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  ( (
( LTrn `  K ) `  W )  X.  E
) )
16153adant3 1025 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( I `  Q )  C_  (
( ( LTrn `  K
) `  W )  X.  E ) )
17 simp3r 1034 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  Y  e.  ( I `  Q ) )
1816, 17sseldd 3465 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  Y  e.  ( ( ( LTrn `  K
) `  W )  X.  E ) )
19 dicvscacl.s . . . . 5  |-  .x.  =  ( .s `  U )
205, 10, 11, 7, 19dvhvsca 34638 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  E  /\  Y  e.  ( ( ( LTrn `  K ) `  W
)  X.  E ) ) )  ->  ( X  .x.  Y )  = 
<. ( X `  ( 1st `  Y ) ) ,  ( X  o.  ( 2nd `  Y ) ) >. )
211, 2, 18, 20syl12anc 1262 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( X  .x.  Y )  =  <. ( X `  ( 1st `  Y ) ) ,  ( X  o.  ( 2nd `  Y ) )
>. )
22 fvi 5938 . . . . . 6  |-  ( X  e.  E  ->  (  _I  `  X )  =  X )
232, 22syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  (  _I  `  X )  =  X )
2423coeq1d 5015 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( (  _I 
`  X )  o.  ( 2nd `  Y
) )  =  ( X  o.  ( 2nd `  Y ) ) )
2524opeq2d 4194 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  <. ( X `  ( 1st `  Y ) ) ,  ( (  _I  `  X )  o.  ( 2nd `  Y
) ) >.  =  <. ( X `  ( 1st `  Y ) ) ,  ( X  o.  ( 2nd `  Y ) )
>. )
2621, 25eqtr4d 2466 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( X  .x.  Y )  =  <. ( X `  ( 1st `  Y ) ) ,  ( (  _I  `  X )  o.  ( 2nd `  Y ) )
>. )
27 eqid 2422 . . . . . . . 8  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
283, 4, 5, 27, 10, 6dicelval1sta 34724 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Y  e.  ( I `  Q
) )  ->  ( 1st `  Y )  =  ( ( 2nd `  Y
) `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) ) )
29283adant3l 1260 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) ) )
3029fveq2d 5885 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( X `  ( 1st `  Y ) )  =  ( X `
 ( ( 2nd `  Y ) `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  Q ) ) ) )
313, 4, 5, 11, 6dicelval2nd 34726 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Y  e.  ( I `  Q
) )  ->  ( 2nd `  Y )  e.  E )
32313adant3l 1260 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( 2nd `  Y
)  e.  E )
335, 10, 11tendof 34299 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 2nd `  Y
)  e.  E )  ->  ( 2nd `  Y
) : ( (
LTrn `  K ) `  W ) --> ( (
LTrn `  K ) `  W ) )
341, 32, 33syl2anc 665 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( 2nd `  Y
) : ( (
LTrn `  K ) `  W ) --> ( (
LTrn `  K ) `  W ) )
35 eqid 2422 . . . . . . . . 9  |-  ( oc
`  K )  =  ( oc `  K
)
363, 35, 4, 5lhpocnel 33552 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  A  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
37363ad2ant1 1026 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( ( ( oc `  K ) `
 W )  e.  A  /\  -.  (
( oc `  K
) `  W )  .<_  W ) )
38 simp2 1006 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
39 eqid 2422 . . . . . . . 8  |-  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q )  =  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q )
403, 4, 5, 10, 39ltrniotacl 34115 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( ( oc `  K ) `
 W )  e.  A  /\  -.  (
( oc `  K
) `  W )  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q )  e.  ( (
LTrn `  K ) `  W ) )
411, 37, 38, 40syl3anc 1264 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q )  e.  ( ( LTrn `  K ) `  W
) )
42 fvco3 5958 . . . . . 6  |-  ( ( ( 2nd `  Y
) : ( (
LTrn `  K ) `  W ) --> ( (
LTrn `  K ) `  W )  /\  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  Q )  e.  ( ( LTrn `  K
) `  W )
)  ->  ( ( X  o.  ( 2nd `  Y ) ) `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) )  =  ( X `
 ( ( 2nd `  Y ) `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  Q ) ) ) )
4334, 41, 42syl2anc 665 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( ( X  o.  ( 2nd `  Y
) ) `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  Q ) )  =  ( X `  (
( 2nd `  Y
) `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) ) ) )
4430, 43eqtr4d 2466 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( X `  ( 1st `  Y ) )  =  ( ( X  o.  ( 2nd `  Y ) ) `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) ) )
4524fveq1d 5883 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( ( (  _I  `  X )  o.  ( 2nd `  Y
) ) `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  Q ) )  =  ( ( X  o.  ( 2nd `  Y ) ) `  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q ) ) )
4644, 45eqtr4d 2466 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( X `  ( 1st `  Y ) )  =  ( ( (  _I  `  X
)  o.  ( 2nd `  Y ) ) `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) ) )
475, 11tendococl 34308 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  E  /\  ( 2nd `  Y
)  e.  E )  ->  ( X  o.  ( 2nd `  Y ) )  e.  E )
481, 2, 32, 47syl3anc 1264 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( X  o.  ( 2nd `  Y ) )  e.  E )
4924, 48eqeltrd 2507 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( (  _I 
`  X )  o.  ( 2nd `  Y
) )  e.  E
)
50 fvex 5891 . . . . 5  |-  ( X `
 ( 1st `  Y
) )  e.  _V
51 fvex 5891 . . . . . 6  |-  (  _I 
`  X )  e. 
_V
52 fvex 5891 . . . . . 6  |-  ( 2nd `  Y )  e.  _V
5351, 52coex 6759 . . . . 5  |-  ( (  _I  `  X )  o.  ( 2nd `  Y
) )  e.  _V
543, 4, 5, 27, 10, 11, 6, 50, 53dicopelval 34714 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( <. ( X `  ( 1st `  Y ) ) ,  ( (  _I  `  X )  o.  ( 2nd `  Y
) ) >.  e.  ( I `  Q )  <-> 
( ( X `  ( 1st `  Y ) )  =  ( ( (  _I  `  X
)  o.  ( 2nd `  Y ) ) `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) )  /\  ( (  _I  `  X )  o.  ( 2nd `  Y
) )  e.  E
) ) )
55543adant3 1025 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( <. ( X `  ( 1st `  Y ) ) ,  ( (  _I  `  X )  o.  ( 2nd `  Y ) )
>.  e.  ( I `  Q )  <->  ( ( X `  ( 1st `  Y ) )  =  ( ( (  _I 
`  X )  o.  ( 2nd `  Y
) ) `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  Q ) )  /\  ( (  _I  `  X )  o.  ( 2nd `  Y ) )  e.  E ) ) )
5646, 49, 55mpbir2and 930 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  <. ( X `  ( 1st `  Y ) ) ,  ( (  _I  `  X )  o.  ( 2nd `  Y
) ) >.  e.  ( I `  Q ) )
5726, 56eqeltrd 2507 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  ( X  .x.  Y )  e.  ( I `  Q ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    C_ wss 3436   <.cop 4004   class class class wbr 4423    _I cid 4763    X. cxp 4851    o. ccom 4857   -->wf 5597   ` cfv 5601   iota_crio 6266  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806   Basecbs 15120   .scvsca 15193   lecple 15196   occoc 15197   Atomscatm 32798   HLchlt 32885   LHypclh 33518   LTrncltrn 33635   TEndoctendo 34288   DVecHcdvh 34615   DIsoCcdic 34709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-riotaBAD 32494
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-undef 7031  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-struct 15122  df-ndx 15123  df-slot 15124  df-base 15125  df-plusg 15202  df-sca 15205  df-vsca 15206  df-preset 16172  df-poset 16190  df-plt 16203  df-lub 16219  df-glb 16220  df-join 16221  df-meet 16222  df-p0 16284  df-p1 16285  df-lat 16291  df-clat 16353  df-oposet 32711  df-ol 32713  df-oml 32714  df-covers 32801  df-ats 32802  df-atl 32833  df-cvlat 32857  df-hlat 32886  df-llines 33032  df-lplanes 33033  df-lvols 33034  df-lines 33035  df-psubsp 33037  df-pmap 33038  df-padd 33330  df-lhyp 33522  df-laut 33523  df-ldil 33638  df-ltrn 33639  df-trl 33694  df-tendo 34291  df-dvech 34616  df-dic 34710
This theorem is referenced by:  diclss  34730
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