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Theorem dicvscacl 35175
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 988 . . . 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 1016 . . . 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 2454 . . . . . . . 8  |-  ( Base `  U )  =  (
Base `  U )
93, 4, 5, 6, 7, 8dicssdvh 35170 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  ( Base `  U ) )
10 eqid 2454 . . . . . . . . . 10  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
11 dicvscacl.e . . . . . . . . . 10  |-  E  =  ( ( TEndo `  K
) `  W )
125, 10, 11, 7, 8dvhvbase 35071 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( ( ( LTrn `  K
) `  W )  X.  E ) )
1312eqcomd 2462 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( LTrn `  K ) `  W
)  X.  E )  =  ( Base `  U
) )
1413adantr 465 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( LTrn `  K ) `  W
)  X.  E )  =  ( Base `  U
) )
159, 14sseqtr4d 3502 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  ( (
( LTrn `  K ) `  W )  X.  E
) )
16153adant3 1008 . . . . 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 1017 . . . . 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 3466 . . . 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 35085 . . . 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 1217 . . 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 5858 . . . . . 6  |-  ( X  e.  E  ->  (  _I  `  X )  =  X )
232, 22syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `  Q ) ) )  ->  (  _I  `  X )  =  X )
2423coeq1d 5110 . . . 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 4175 . . 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 2498 . 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 2454 . . . . . . . 8  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
283, 4, 5, 27, 10, 6dicelval1sta 35171 . . . . . . 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 1215 . . . . . 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 5804 . . . . 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 35173 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Y  e.  ( I `  Q
) )  ->  ( 2nd `  Y )  e.  E )
32313adant3l 1215 . . . . . . 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 34746 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 2nd `  Y
)  e.  E )  ->  ( 2nd `  Y
) : ( (
LTrn `  K ) `  W ) --> ( (
LTrn `  K ) `  W ) )
341, 32, 33syl2anc 661 . . . . . 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 2454 . . . . . . . . 9  |-  ( oc
`  K )  =  ( oc `  K
)
363, 35, 4, 5lhpocnel 34001 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  A  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
37363ad2ant1 1009 . . . . . . 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 989 . . . . . . 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 2454 . . . . . . . 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 34562 . . . . . . 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 1219 . . . . . 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 5878 . . . . . 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 661 . . . . 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 2498 . . . 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 5802 . . . 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 2498 . . 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 34755 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  E  /\  ( 2nd `  Y
)  e.  E )  ->  ( X  o.  ( 2nd `  Y ) )  e.  E )
481, 2, 32, 47syl3anc 1219 . . . 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 2542 . . 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 5810 . . . . 5  |-  ( X `
 ( 1st `  Y
) )  e.  _V
51 fvex 5810 . . . . . 6  |-  (  _I 
`  X )  e. 
_V
52 fvex 5810 . . . . . 6  |-  ( 2nd `  Y )  e.  _V
5351, 52coex 6640 . . . . 5  |-  ( (  _I  `  X )  o.  ( 2nd `  Y
) )  e.  _V
543, 4, 5, 27, 10, 11, 6, 50, 53dicopelval 35161 . . . 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 1008 . . 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 913 . 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 2542 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    C_ wss 3437   <.cop 3992   class class class wbr 4401    _I cid 4740    X. cxp 4947    o. ccom 4953   -->wf 5523   ` cfv 5527   iota_crio 6161  (class class class)co 6201   1stc1st 6686   2ndc2nd 6687   Basecbs 14293   .scvsca 14362   lecple 14365   occoc 14366   Atomscatm 33247   HLchlt 33334   LHypclh 33967   LTrncltrn 34084   TEndoctendo 34735   DVecHcdvh 35062   DIsoCcdic 35156
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 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-riotaBAD 32943
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-undef 6903  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-plusg 14371  df-sca 14374  df-vsca 14375  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-p1 15330  df-lat 15336  df-clat 15398  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-llines 33481  df-lplanes 33482  df-lvols 33483  df-lines 33484  df-psubsp 33486  df-pmap 33487  df-padd 33779  df-lhyp 33971  df-laut 33972  df-ldil 34087  df-ltrn 34088  df-trl 34142  df-tendo 34738  df-dvech 35063  df-dic 35157
This theorem is referenced by:  diclss  35177
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