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Theorem djaclN 34877
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 2443 . . 3  |-  ( ( ocA `  K ) `
 W )  =  ( ( ocA `  K
) `  W )
5 djacl.j . . 3  |-  J  =  ( ( vA `  K ) `  W
)
61, 2, 3, 4, 5djavalN 34876 . 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 3591 . . . 4  |-  ( ( ( ( ocA `  K
) `  W ) `  X )  i^i  (
( ( ocA `  K
) `  W ) `  Y ) )  C_  ( ( ( ocA `  K ) `  W
) `  X )
81, 2, 3, 4docaclN 34865 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( (
( ocA `  K
) `  W ) `  X )  e.  ran  I )
98adantrr 716 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  T  /\  Y  C_  T
) )  ->  (
( ( ocA `  K
) `  W ) `  X )  e.  ran  I )
101, 2, 3diaelrnN 34786 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( ( ocA `  K ) `
 W ) `  X )  e.  ran  I )  ->  (
( ( ocA `  K
) `  W ) `  X )  C_  T
)
119, 10syldan 470 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  T  /\  Y  C_  T
) )  ->  (
( ( ocA `  K
) `  W ) `  X )  C_  T
)
127, 11syl5ss 3388 . . 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 34865 . . 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 470 . 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 2517 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  T  /\  Y  C_  T
) )  ->  ( X J Y )  e. 
ran  I )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3348    C_ wss 3349   ran crn 4862   ` cfv 5439  (class class class)co 6112   HLchlt 33091   LHypclh 33724   LTrncltrn 33841   DIsoAcdia 34769   ocAcocaN 34860   vAcdjaN 34872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-riotaBAD 32700
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-undef 6813  df-map 7237  df-poset 15137  df-plt 15149  df-lub 15165  df-glb 15166  df-join 15167  df-meet 15168  df-p0 15230  df-p1 15231  df-lat 15237  df-clat 15299  df-oposet 32917  df-ol 32919  df-oml 32920  df-covers 33007  df-ats 33008  df-atl 33039  df-cvlat 33063  df-hlat 33092  df-llines 33238  df-lplanes 33239  df-lvols 33240  df-lines 33241  df-psubsp 33243  df-pmap 33244  df-padd 33536  df-lhyp 33728  df-laut 33729  df-ldil 33844  df-ltrn 33845  df-trl 33899  df-disoa 34770  df-docaN 34861  df-djaN 34873
This theorem is referenced by: (None)
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