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Theorem ltrnj 30929
Description: Lattice translation of a meet. TODO: change antecedent to 
K  e.  HL (Contributed by NM, 25-May-2012.)
Hypotheses
Ref Expression
ltrnj.b  |-  B  =  ( Base `  K
)
ltrnj.j  |-  .\/  =  ( join `  K )
ltrnj.h  |-  H  =  ( LHyp `  K
)
ltrnj.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrnj  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( F `  ( X  .\/  Y ) )  =  ( ( F `  X )  .\/  ( F `  Y )
) )

Proof of Theorem ltrnj
StepHypRef Expression
1 simp1l 981 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  K  e.  HL )
2 hllat 30161 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  K  e.  Lat )
4 ltrnj.h . . . 4  |-  H  =  ( LHyp `  K
)
5 eqid 2436 . . . 4  |-  ( LAut `  K )  =  (
LAut `  K )
6 ltrnj.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
74, 5, 6ltrnlaut 30920 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F  e.  ( LAut `  K )
)
873adant3 977 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  F  e.  ( LAut `  K
) )
9 simp3l 985 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  X  e.  B )
10 simp3r 986 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  Y  e.  B )
11 ltrnj.b . . 3  |-  B  =  ( Base `  K
)
12 ltrnj.j . . 3  |-  .\/  =  ( join `  K )
1311, 12, 5lautj 30890 . 2  |-  ( ( K  e.  Lat  /\  ( F  e.  ( LAut `  K )  /\  X  e.  B  /\  Y  e.  B )
)  ->  ( F `  ( X  .\/  Y
) )  =  ( ( F `  X
)  .\/  ( F `  Y ) ) )
143, 8, 9, 10, 13syl13anc 1186 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( F `  ( X  .\/  Y ) )  =  ( ( F `  X )  .\/  ( F `  Y )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   joincjn 14401   Latclat 14474   HLchlt 30148   LHypclh 30781   LAutclaut 30782   LTrncltrn 30898
This theorem is referenced by:  cdlemc2  30989  cdlemd2  30996  cdlemg2l  31400  cdlemg17h  31465  cdlemg17  31474
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-map 7020  df-poset 14403  df-lub 14431  df-join 14433  df-lat 14475  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-laut 30786  df-ldil 30901  df-ltrn 30902
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