Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dibelval1st2N Structured version   Unicode version

Theorem dibelval1st2N 37330
Description: Membership in value of the partial isomorphism B for a lattice  K. (Contributed by NM, 13-Feb-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dibelval1st2.b  |-  B  =  ( Base `  K
)
dibelval1st2.l  |-  .<_  =  ( le `  K )
dibelval1st2.h  |-  H  =  ( LHyp `  K
)
dibelval1st2.t  |-  T  =  ( ( LTrn `  K
) `  W )
dibelval1st2.r  |-  R  =  ( ( trL `  K
) `  W )
dibelval1st2.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dibelval1st2N  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  Y  e.  ( I `  X
) )  ->  ( R `  ( 1st `  Y ) )  .<_  X )

Proof of Theorem dibelval1st2N
StepHypRef Expression
1 dibelval1st2.b . . 3  |-  B  =  ( Base `  K
)
2 dibelval1st2.l . . 3  |-  .<_  =  ( le `  K )
3 dibelval1st2.h . . 3  |-  H  =  ( LHyp `  K
)
4 eqid 2396 . . 3  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
5 dibelval1st2.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
61, 2, 3, 4, 5dibelval1st 37328 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  Y  e.  ( I `  X
) )  ->  ( 1st `  Y )  e.  ( ( ( DIsoA `  K ) `  W
) `  X )
)
7 dibelval1st2.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
8 dibelval1st2.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
91, 2, 3, 7, 8, 4diatrl 37223 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( 1st `  Y )  e.  ( ( ( DIsoA `  K
) `  W ) `  X ) )  -> 
( R `  ( 1st `  Y ) ) 
.<_  X )
106, 9syld3an3 1271 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  Y  e.  ( I `  X
) )  ->  ( R `  ( 1st `  Y ) )  .<_  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836   class class class wbr 4384   ` cfv 5513   1stc1st 6719   Basecbs 14657   lecple 14732   LHypclh 36160   LTrncltrn 36277   trLctrl 36335   DIsoAcdia 37207   DIsoBcdib 37317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-1st 6721  df-disoa 37208  df-dib 37318
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator