Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  djaclN Structured version   Unicode version

Theorem djaclN 36964
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 2457 . . 3  |-  ( ( ocA `  K ) `
 W )  =  ( ( ocA `  K
) `  W )
5 djacl.j . . 3  |-  J  =  ( ( vA `  K ) `  W
)
61, 2, 3, 4, 5djavalN 36963 . 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 3714 . . . 4  |-  ( ( ( ( ocA `  K
) `  W ) `  X )  i^i  (
( ( ocA `  K
) `  W ) `  Y ) )  C_  ( ( ( ocA `  K ) `  W
) `  X )
81, 2, 3, 4docaclN 36952 . . . . . 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 36873 . . . . 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 3510 . . 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 36952 . . 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 2545 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 1395    e. wcel 1819    i^i cin 3470    C_ wss 3471   ran crn 5009   ` cfv 5594  (class class class)co 6296   HLchlt 35176   LHypclh 35809   LTrncltrn 35926   DIsoAcdia 36856   ocAcocaN 36947   vAcdjaN 36959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34785
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-undef 7020  df-map 7440  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-p1 15796  df-lat 15802  df-clat 15864  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-llines 35323  df-lplanes 35324  df-lvols 35325  df-lines 35326  df-psubsp 35328  df-pmap 35329  df-padd 35621  df-lhyp 35813  df-laut 35814  df-ldil 35929  df-ltrn 35930  df-trl 35985  df-disoa 36857  df-docaN 36948  df-djaN 36960
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator