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Theorem lautlt 30888
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 444 . . . 4  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  K  e.  A )
2 simpr1 963 . . . 4  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  F  e.  I )
3 simpr2 964 . . . 4  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  X  e.  B )
4 simpr3 965 . . . 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 2436 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
7 lautlt.i . . . . 5  |-  I  =  ( LAut `  K
)
85, 6, 7lautle 30881 . . . 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 1185 . . 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 30883 . . . . . 6  |-  ( ( ( K  e.  A  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( ( F `  X )  =  ( F `  Y )  <-> 
X  =  Y ) )
111, 2, 3, 4, 10syl22anc 1185 . . . . 5  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  (
( F `  X
)  =  ( F `
 Y )  <->  X  =  Y ) )
1211bicomd 193 . . . 4  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( X  =  Y  <->  ( F `  X )  =  ( F `  Y ) ) )
1312necon3bid 2636 . . 3  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( X  =/=  Y  <->  ( F `  X )  =/=  ( F `  Y )
) )
149, 13anbi12d 692 . 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 14417 . . 3  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X
( le `  K
) Y  /\  X  =/=  Y ) ) )
17163adant3r1 1162 . 2  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( X  .<  Y  <->  ( X
( le `  K
) Y  /\  X  =/=  Y ) ) )
185, 7lautcl 30884 . . . 4  |-  ( ( ( K  e.  A  /\  F  e.  I
)  /\  X  e.  B )  ->  ( F `  X )  e.  B )
191, 2, 3, 18syl21anc 1183 . . 3  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( F `  X )  e.  B )
205, 7lautcl 30884 . . . 4  |-  ( ( ( K  e.  A  /\  F  e.  I
)  /\  Y  e.  B )  ->  ( F `  Y )  e.  B )
211, 2, 4, 20syl21anc 1183 . . 3  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( F `  Y )  e.  B )
226, 15pltval 14417 . . 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 1184 . 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 277 1  |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( X  .<  Y  <->  ( F `  X )  .<  ( F `  Y )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   ltcplt 14398   LAutclaut 30782
This theorem is referenced by:  lautcvr  30889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-plt 14415  df-laut 30786
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