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Theorem dibelval1st2N 35135
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 2454 . . 3  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
5 dibelval1st2.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
61, 2, 3, 4, 5dibelval1st 35133 . 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 35028 . 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 1264 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4401   ` cfv 5527   1stc1st 6686   Basecbs 14293   lecple 14365   LHypclh 33967   LTrncltrn 34084   trLctrl 34141   DIsoAcdia 35012   DIsoBcdib 35122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-1st 6688  df-disoa 35013  df-dib 35123
This theorem is referenced by: (None)
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