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Theorem ltrnj 30943
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 979 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  K  e.  HL )
2 hllat 30175 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 15 . 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 2296 . . . 4  |-  ( LAut `  K )  =  (
LAut `  K )
6 ltrnj.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
74, 5, 6ltrnlaut 30934 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F  e.  ( LAut `  K )
)
873adant3 975 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  F  e.  ( LAut `  K
) )
9 simp3l 983 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  X  e.  B )
10 simp3r 984 . 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 30904 . 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 1184 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   joincjn 14094   Latclat 14167   HLchlt 30162   LHypclh 30795   LAutclaut 30796   LTrncltrn 30912
This theorem is referenced by:  cdlemc2  31003  cdlemd2  31010  cdlemg2l  31414  cdlemg17h  31479  cdlemg17  31488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-lub 14124  df-join 14126  df-lat 14168  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-laut 30800  df-ldil 30915  df-ltrn 30916
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