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Theorem dicvscacl 34211
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 997 . . . 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 1025 . . . 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 2402 . . . . . . . 8  |-  ( Base `  U )  =  (
Base `  U )
93, 4, 5, 6, 7, 8dicssdvh 34206 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  ( Base `  U ) )
10 eqid 2402 . . . . . . . . . 10  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
11 dicvscacl.e . . . . . . . . . 10  |-  E  =  ( ( TEndo `  K
) `  W )
125, 10, 11, 7, 8dvhvbase 34107 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( ( ( LTrn `  K
) `  W )  X.  E ) )
1312eqcomd 2410 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( LTrn `  K ) `  W
)  X.  E )  =  ( Base `  U
) )
1413adantr 463 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( LTrn `  K ) `  W
)  X.  E )  =  ( Base `  U
) )
159, 14sseqtr4d 3479 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  ( (
( LTrn `  K ) `  W )  X.  E
) )
16153adant3 1017 . . . . 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 1026 . . . . 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 3443 . . . 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 34121 . . . 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 1228 . . 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 5906 . . . . . 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 4985 . . . 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 4166 . . 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 2446 . 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 2402 . . . . . . . 8  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
283, 4, 5, 27, 10, 6dicelval1sta 34207 . . . . . . 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 1226 . . . . . 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 5853 . . . . 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 34209 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Y  e.  ( I `  Q
) )  ->  ( 2nd `  Y )  e.  E )
32313adant3l 1226 . . . . . . 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 33782 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 2nd `  Y
)  e.  E )  ->  ( 2nd `  Y
) : ( (
LTrn `  K ) `  W ) --> ( (
LTrn `  K ) `  W ) )
341, 32, 33syl2anc 659 . . . . . 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 2402 . . . . . . . . 9  |-  ( oc
`  K )  =  ( oc `  K
)
363, 35, 4, 5lhpocnel 33035 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  A  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
37363ad2ant1 1018 . . . . . . 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 998 . . . . . . 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 2402 . . . . . . . 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 33598 . . . . . . 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 1230 . . . . . 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 5926 . . . . . 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 659 . . . . 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 2446 . . . 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 5851 . . . 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 2446 . . 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 33791 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  E  /\  ( 2nd `  Y
)  e.  E )  ->  ( X  o.  ( 2nd `  Y ) )  e.  E )
481, 2, 32, 47syl3anc 1230 . . . 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 2490 . . 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 5859 . . . . 5  |-  ( X `
 ( 1st `  Y
) )  e.  _V
51 fvex 5859 . . . . . 6  |-  (  _I 
`  X )  e. 
_V
52 fvex 5859 . . . . . 6  |-  ( 2nd `  Y )  e.  _V
5351, 52coex 6736 . . . . 5  |-  ( (  _I  `  X )  o.  ( 2nd `  Y
) )  e.  _V
543, 4, 5, 27, 10, 11, 6, 50, 53dicopelval 34197 . . . 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 1017 . . 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 923 . 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 2490 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 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    C_ wss 3414   <.cop 3978   class class class wbr 4395    _I cid 4733    X. cxp 4821    o. ccom 4827   -->wf 5565   ` cfv 5569   iota_crio 6239  (class class class)co 6278   1stc1st 6782   2ndc2nd 6783   Basecbs 14841   .scvsca 14913   lecple 14916   occoc 14917   Atomscatm 32281   HLchlt 32368   LHypclh 33001   LTrncltrn 33118   TEndoctendo 33771   DVecHcdvh 34098   DIsoCcdic 34192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-riotaBAD 31977
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-undef 7005  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-plusg 14922  df-sca 14925  df-vsca 14926  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-p1 15994  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-llines 32515  df-lplanes 32516  df-lvols 32517  df-lines 32518  df-psubsp 32520  df-pmap 32521  df-padd 32813  df-lhyp 33005  df-laut 33006  df-ldil 33121  df-ltrn 33122  df-trl 33177  df-tendo 33774  df-dvech 34099  df-dic 34193
This theorem is referenced by:  diclss  34213
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