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Theorem lautlt 35288
Description: Less-than property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
lautlt.b  |-  B  =  ( Base `  K
)
lautlt.s  |-  .<  =  ( lt `  K )
lautlt.i  |-  I  =  ( LAut `  K
)
Assertion
Ref Expression
lautlt  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( X  .<  Y  <->  ( F `  X )  .<  ( F `  Y )
) )

Proof of Theorem lautlt
StepHypRef Expression
1 simpl 457 . . . 4  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  K  e.  A )
2 simpr1 1002 . . . 4  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  F  e.  I )
3 simpr2 1003 . . . 4  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  X  e.  B )
4 simpr3 1004 . . . 4  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  Y  e.  B )
5 lautlt.b . . . . 5  |-  B  =  ( Base `  K
)
6 eqid 2467 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
7 lautlt.i . . . . 5  |-  I  =  ( LAut `  K
)
85, 6, 7lautle 35281 . . . 4  |-  ( ( ( K  e.  A  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( le
`  K ) Y  <-> 
( F `  X
) ( le `  K ) ( F `
 Y ) ) )
91, 2, 3, 4, 8syl22anc 1229 . . 3  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( X ( le `  K ) Y  <->  ( F `  X ) ( le
`  K ) ( F `  Y ) ) )
105, 7laut11 35283 . . . . . 6  |-  ( ( ( K  e.  A  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( ( F `  X )  =  ( F `  Y )  <-> 
X  =  Y ) )
111, 2, 3, 4, 10syl22anc 1229 . . . . 5  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  (
( F `  X
)  =  ( F `
 Y )  <->  X  =  Y ) )
1211bicomd 201 . . . 4  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( X  =  Y  <->  ( F `  X )  =  ( F `  Y ) ) )
1312necon3bid 2725 . . 3  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( X  =/=  Y  <->  ( F `  X )  =/=  ( F `  Y )
) )
149, 13anbi12d 710 . 2  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  (
( X ( le
`  K ) Y  /\  X  =/=  Y
)  <->  ( ( F `
 X ) ( le `  K ) ( F `  Y
)  /\  ( F `  X )  =/=  ( F `  Y )
) ) )
15 lautlt.s . . . 4  |-  .<  =  ( lt `  K )
166, 15pltval 15464 . . 3  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X
( le `  K
) Y  /\  X  =/=  Y ) ) )
17163adant3r1 1205 . 2  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( X  .<  Y  <->  ( X
( le `  K
) Y  /\  X  =/=  Y ) ) )
185, 7lautcl 35284 . . . 4  |-  ( ( ( K  e.  A  /\  F  e.  I
)  /\  X  e.  B )  ->  ( F `  X )  e.  B )
191, 2, 3, 18syl21anc 1227 . . 3  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( F `  X )  e.  B )
205, 7lautcl 35284 . . . 4  |-  ( ( ( K  e.  A  /\  F  e.  I
)  /\  Y  e.  B )  ->  ( F `  Y )  e.  B )
211, 2, 4, 20syl21anc 1227 . . 3  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( F `  Y )  e.  B )
226, 15pltval 15464 . . 3  |-  ( ( K  e.  A  /\  ( F `  X )  e.  B  /\  ( F `  Y )  e.  B )  ->  (
( F `  X
)  .<  ( F `  Y )  <->  ( ( F `  X )
( le `  K
) ( F `  Y )  /\  ( F `  X )  =/=  ( F `  Y
) ) ) )
231, 19, 21, 22syl3anc 1228 . 2  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  (
( F `  X
)  .<  ( F `  Y )  <->  ( ( F `  X )
( le `  K
) ( F `  Y )  /\  ( F `  X )  =/=  ( F `  Y
) ) ) )
2414, 17, 233bitr4d 285 1  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( X  .<  Y  <->  ( F `  X )  .<  ( F `  Y )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594   Basecbs 14507   lecple 14579   ltcplt 15445   LAutclaut 35182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-plt 15462  df-laut 35186
This theorem is referenced by:  lautcvr  35289
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