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Theorem ltrncnvleN 30390
Description: Less-than or equal property of lattice translation converse. (Contributed by NM, 10-May-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ltrnle.b  |-  B  =  ( Base `  K
)
ltrnle.l  |-  .<_  =  ( le `  K )
ltrnle.h  |-  H  =  ( LHyp `  K
)
ltrnle.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrncnvleN  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X  .<_  Y  <->  ( `' F `  X )  .<_  ( `' F `  Y ) ) )

Proof of Theorem ltrncnvleN
StepHypRef Expression
1 simp1l 980 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  K  e.  V )
2 ltrnle.h . . . 4  |-  H  =  ( LHyp `  K
)
3 eqid 2366 . . . 4  |-  ( LAut `  K )  =  (
LAut `  K )
4 ltrnle.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
52, 3, 4ltrnlaut 30383 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T )  ->  F  e.  ( LAut `  K
) )
653adant3 976 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  F  e.  ( LAut `  K ) )
7 simp3 958 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X  e.  B  /\  Y  e.  B
) )
8 ltrnle.b . . 3  |-  B  =  ( Base `  K
)
9 ltrnle.l . . 3  |-  .<_  =  ( le `  K )
108, 9, 3lautcnvle 30349 . 2  |-  ( ( ( K  e.  V  /\  F  e.  ( LAut `  K ) )  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X  .<_  Y  <->  ( `' F `  X )  .<_  ( `' F `  Y ) ) )
111, 6, 7, 10syl21anc 1182 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X  .<_  Y  <->  ( `' F `  X )  .<_  ( `' F `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   class class class wbr 4125   `'ccnv 4791   ` cfv 5358   Basecbs 13356   lecple 13423   LHypclh 30244   LAutclaut 30245   LTrncltrn 30361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-map 6917  df-laut 30249  df-ldil 30364  df-ltrn 30365
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