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Theorem djaclN 32008
Description: Closure of subspace join for  DVecA partial vector space. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
djacl.h  |-  H  =  ( LHyp `  K
)
djacl.t  |-  T  =  ( ( LTrn `  K
) `  W )
djacl.i  |-  I  =  ( ( DIsoA `  K
) `  W )
djacl.j  |-  J  =  ( ( vA `  K ) `  W
)
Assertion
Ref Expression
djaclN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  T  /\  Y  C_  T
) )  ->  ( X J Y )  e. 
ran  I )

Proof of Theorem djaclN
StepHypRef Expression
1 djacl.h . . 3  |-  H  =  ( LHyp `  K
)
2 djacl.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
3 djacl.i . . 3  |-  I  =  ( ( DIsoA `  K
) `  W )
4 eqid 2438 . . 3  |-  ( ( ocA `  K ) `
 W )  =  ( ( ocA `  K
) `  W )
5 djacl.j . . 3  |-  J  =  ( ( vA `  K ) `  W
)
61, 2, 3, 4, 5djavalN 32007 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  T  /\  Y  C_  T
) )  ->  ( X J Y )  =  ( ( ( ocA `  K ) `  W
) `  ( (
( ( ocA `  K
) `  W ) `  X )  i^i  (
( ( ocA `  K
) `  W ) `  Y ) ) ) )
7 inss1 3563 . . . 4  |-  ( ( ( ( ocA `  K
) `  W ) `  X )  i^i  (
( ( ocA `  K
) `  W ) `  Y ) )  C_  ( ( ( ocA `  K ) `  W
) `  X )
81, 2, 3, 4docaclN 31996 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( (
( ocA `  K
) `  W ) `  X )  e.  ran  I )
98adantrr 699 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  T  /\  Y  C_  T
) )  ->  (
( ( ocA `  K
) `  W ) `  X )  e.  ran  I )
101, 2, 3diaelrnN 31917 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( ( ocA `  K ) `
 W ) `  X )  e.  ran  I )  ->  (
( ( ocA `  K
) `  W ) `  X )  C_  T
)
119, 10syldan 458 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  T  /\  Y  C_  T
) )  ->  (
( ( ocA `  K
) `  W ) `  X )  C_  T
)
127, 11syl5ss 3361 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  T  /\  Y  C_  T
) )  ->  (
( ( ( ocA `  K ) `  W
) `  X )  i^i  ( ( ( ocA `  K ) `  W
) `  Y )
)  C_  T )
131, 2, 3, 4docaclN 31996 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( ( ( ocA `  K
) `  W ) `  X )  i^i  (
( ( ocA `  K
) `  W ) `  Y ) )  C_  T )  ->  (
( ( ocA `  K
) `  W ) `  ( ( ( ( ocA `  K ) `
 W ) `  X )  i^i  (
( ( ocA `  K
) `  W ) `  Y ) ) )  e.  ran  I )
1412, 13syldan 458 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  T  /\  Y  C_  T
) )  ->  (
( ( ocA `  K
) `  W ) `  ( ( ( ( ocA `  K ) `
 W ) `  X )  i^i  (
( ( ocA `  K
) `  W ) `  Y ) ) )  e.  ran  I )
156, 14eqeltrd 2512 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  T  /\  Y  C_  T
) )  ->  ( X J Y )  e. 
ran  I )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    i^i cin 3321    C_ wss 3322   ran crn 4882   ` cfv 5457  (class class class)co 6084   HLchlt 30222   LHypclh 30855   LTrncltrn 30972   DIsoAcdia 31900   ocAcocaN 31991   vAcdjaN 32003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-map 7023  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-p1 14474  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369  df-lplanes 30370  df-lvols 30371  df-lines 30372  df-psubsp 30374  df-pmap 30375  df-padd 30667  df-lhyp 30859  df-laut 30860  df-ldil 30975  df-ltrn 30976  df-trl 31030  df-disoa 31901  df-docaN 31992  df-djaN 32004
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