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Theorem ltrnval1 34807
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 2462 . . . 4  |-  ( (
LDil `  K ) `  W )  =  ( ( LDil `  K
) `  W )
3 ltrnval1.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
41, 2, 3ltrnldil 34795 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T )  ->  F  e.  ( ( LDil `  K
) `  W )
)
543adant3 1011 . 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 34786 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  ( ( LDil `  K
) `  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( F `  X )  =  X )
95, 8syld3an2 1270 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762   class class class wbr 4442   ` cfv 5581   Basecbs 14481   lecple 14553   LHypclh 34657   LDilcldil 34773   LTrncltrn 34774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-ldil 34777  df-ltrn 34778
This theorem is referenced by:  ltrnid  34808  ltrnatb  34810  ltrnel  34812  ltrncnvel  34815  ltrneq  34822  ltrnmw  34824  cdlemc2  34865  cdlemd2  34872  cdlemg7N  35299
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