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Theorem ltrnm 30617
Description: Lattice translation of a meet. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
ltrnm.b  |-  B  =  ( Base `  K
)
ltrnm.m  |-  ./\  =  ( meet `  K )
ltrnm.h  |-  H  =  ( LHyp `  K
)
ltrnm.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrnm  |-  ( ( ( 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 ltrnm
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 29850 . . 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 ltrnm.h . . . 4  |-  H  =  ( LHyp `  K
)
5 eqid 2408 . . . 4  |-  ( LAut `  K )  =  (
LAut `  K )
6 ltrnm.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
74, 5, 6ltrnlaut 30609 . . 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 ltrnm.b . . 3  |-  B  =  ( Base `  K
)
12 ltrnm.m . . 3  |-  ./\  =  ( meet `  K )
1311, 12, 5lautm 30580 . 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 1649    e. wcel 1721   ` cfv 5417  (class class class)co 6044   Basecbs 13428   meetcmee 14361   Latclat 14433   HLchlt 29837   LHypclh 30470   LAutclaut 30471   LTrncltrn 30587
This theorem is referenced by:  ltrnmw  30637  cdlemd2  30685  cdlemg17  31163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-map 6983  df-poset 14362  df-glb 14391  df-meet 14393  df-lat 14434  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-laut 30475  df-ldil 30590  df-ltrn 30591
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