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Theorem ltrnval1 33115
Description: Value of a lattice translation under its co-atom. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
ltrnval1.b  |-  B  =  ( Base `  K
)
ltrnval1.l  |-  .<_  =  ( le `  K )
ltrnval1.h  |-  H  =  ( LHyp `  K
)
ltrnval1.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrnval1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( F `  X )  =  X )

Proof of Theorem ltrnval1
StepHypRef Expression
1 ltrnval1.h . . . 4  |-  H  =  ( LHyp `  K
)
2 eqid 2400 . . . 4  |-  ( (
LDil `  K ) `  W )  =  ( ( LDil `  K
) `  W )
3 ltrnval1.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
41, 2, 3ltrnldil 33103 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T )  ->  F  e.  ( ( LDil `  K
) `  W )
)
543adant3 1015 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  F  e.  ( ( LDil `  K
) `  W )
)
6 ltrnval1.b . . 3  |-  B  =  ( Base `  K
)
7 ltrnval1.l . . 3  |-  .<_  =  ( le `  K )
86, 7, 1, 2ldilval 33094 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  ( ( LDil `  K
) `  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( F `  X )  =  X )
95, 8syld3an2 1275 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( F `  X )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   class class class wbr 4392   ` cfv 5523   Basecbs 14731   lecple 14806   LHypclh 32965   LDilcldil 33081   LTrncltrn 33082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-ldil 33085  df-ltrn 33086
This theorem is referenced by:  ltrnid  33116  ltrnatb  33118  ltrnel  33120  ltrncnvel  33123  ltrneq  33130  ltrnmwOLD  33133  cdlemc2  33174  cdlemd2  33181  cdlemg7N  33609
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