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Theorem djaclN 31631
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 2412 . . 3  |-  ( ( ocA `  K ) `
 W )  =  ( ( ocA `  K
) `  W )
5 djacl.j . . 3  |-  J  =  ( ( vA `  K ) `  W
)
61, 2, 3, 4, 5djavalN 31630 . 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 3529 . . . 4  |-  ( ( ( ( ocA `  K
) `  W ) `  X )  i^i  (
( ( ocA `  K
) `  W ) `  Y ) )  C_  ( ( ( ocA `  K ) `  W
) `  X )
81, 2, 3, 4docaclN 31619 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( (
( ocA `  K
) `  W ) `  X )  e.  ran  I )
98adantrr 698 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  T  /\  Y  C_  T
) )  ->  (
( ( ocA `  K
) `  W ) `  X )  e.  ran  I )
101, 2, 3diaelrnN 31540 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( ( ocA `  K ) `
 W ) `  X )  e.  ran  I )  ->  (
( ( ocA `  K
) `  W ) `  X )  C_  T
)
119, 10syldan 457 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  T  /\  Y  C_  T
) )  ->  (
( ( ocA `  K
) `  W ) `  X )  C_  T
)
127, 11syl5ss 3327 . . 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 31619 . . 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 457 . 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 2486 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 359    = wceq 1649    e. wcel 1721    i^i cin 3287    C_ wss 3288   ran crn 4846   ` cfv 5421  (class class class)co 6048   HLchlt 29845   LHypclh 30478   LTrncltrn 30595   DIsoAcdia 31523   ocAcocaN 31614   vAcdjaN 31626
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-map 6987  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-llines 29992  df-lplanes 29993  df-lvols 29994  df-lines 29995  df-psubsp 29997  df-pmap 29998  df-padd 30290  df-lhyp 30482  df-laut 30483  df-ldil 30598  df-ltrn 30599  df-trl 30653  df-disoa 31524  df-docaN 31615  df-djaN 31627
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