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Theorem diadmleN 31934
Description: A member of domain of the partial isomorphism A is under the fiducial hyperplane. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
diadmle.l  |-  .<_  =  ( le `  K )
diadmle.h  |-  H  =  ( LHyp `  K
)
diadmle.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
diadmleN  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  X  .<_  W )

Proof of Theorem diadmleN
StepHypRef Expression
1 eqid 2442 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 diadmle.l . . 3  |-  .<_  =  ( le `  K )
3 diadmle.h . . 3  |-  H  =  ( LHyp `  K
)
4 diadmle.i . . 3  |-  I  =  ( ( DIsoA `  K
) `  W )
51, 2, 3, 4diaeldm 31932 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  ( X  e.  (
Base `  K )  /\  X  .<_  W ) ) )
65simplbda 609 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  X  .<_  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727   class class class wbr 4237   dom cdm 4907   ` cfv 5483   Basecbs 13500   lecple 13567   LHypclh 30879   DIsoAcdia 31924
This theorem is referenced by:  diaocN  32021  doca2N  32022  djajN  32033
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 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-disoa 31925
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