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Theorem lautcnvle 30900
Description: Less-than or equal property of lattice automorphism converse. (Contributed by NM, 19-May-2012.)
Hypotheses
Ref Expression
lautcnvle.b  |-  B  =  ( Base `  K
)
lautcnvle.l  |-  .<_  =  ( le `  K )
lautcnvle.i  |-  I  =  ( LAut `  K
)
Assertion
Ref Expression
lautcnvle  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X  .<_  Y  <->  ( `' F `  X )  .<_  ( `' F `  Y ) ) )

Proof of Theorem lautcnvle
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( K  e.  V  /\  F  e.  I
) )
2 lautcnvle.b . . . . . 6  |-  B  =  ( Base `  K
)
3 lautcnvle.i . . . . . 6  |-  I  =  ( LAut `  K
)
42, 3laut1o 30896 . . . . 5  |-  ( ( K  e.  V  /\  F  e.  I )  ->  F : B -1-1-onto-> B )
54adantr 451 . . . 4  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  F : B -1-1-onto-> B )
6 simprl 732 . . . 4  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  X  e.  B )
7 f1ocnvdm 5812 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  X  e.  B )  ->  ( `' F `  X )  e.  B
)
85, 6, 7syl2anc 642 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( `' F `  X )  e.  B
)
9 simprr 733 . . . 4  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  B )
10 f1ocnvdm 5812 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  Y  e.  B )  ->  ( `' F `  Y )  e.  B
)
115, 9, 10syl2anc 642 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( `' F `  Y )  e.  B
)
12 lautcnvle.l . . . 4  |-  .<_  =  ( le `  K )
132, 12, 3lautle 30895 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( ( `' F `  X )  e.  B  /\  ( `' F `  Y )  e.  B ) )  ->  ( ( `' F `  X ) 
.<_  ( `' F `  Y )  <->  ( F `  ( `' F `  X ) )  .<_  ( F `  ( `' F `  Y ) ) ) )
141, 8, 11, 13syl12anc 1180 . 2  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( ( `' F `  X )  .<_  ( `' F `  Y )  <-> 
( F `  ( `' F `  X ) )  .<_  ( F `  ( `' F `  Y ) ) ) )
15 f1ocnvfv2 5809 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  X  e.  B )  ->  ( F `  ( `' F `  X ) )  =  X )
165, 6, 15syl2anc 642 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( F `  ( `' F `  X ) )  =  X )
17 f1ocnvfv2 5809 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  Y  e.  B )  ->  ( F `  ( `' F `  Y ) )  =  Y )
185, 9, 17syl2anc 642 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( F `  ( `' F `  Y ) )  =  Y )
1916, 18breq12d 4052 . 2  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( ( F `  ( `' F `  X ) )  .<_  ( F `  ( `' F `  Y ) )  <->  X  .<_  Y ) )
2014, 19bitr2d 245 1  |-  ( ( ( K  e.  V  /\  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 176    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   `'ccnv 4704   -1-1-onto->wf1o 5270   ` cfv 5271   Basecbs 13164   lecple 13231   LAutclaut 30796
This theorem is referenced by:  lautcnv  30901  lautj  30904  lautm  30905  ltrncnvleN  30941  ltrneq2  30959
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-laut 30800
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