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Theorem lautcnv 30901
Description: The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)
Hypothesis
Ref Expression
lautcnv.i  |-  I  =  ( LAut `  K
)
Assertion
Ref Expression
lautcnv  |-  ( ( K  e.  V  /\  F  e.  I )  ->  `' F  e.  I
)

Proof of Theorem lautcnv
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 lautcnv.i . . . 4  |-  I  =  ( LAut `  K
)
31, 2laut1o 30896 . . 3  |-  ( ( K  e.  V  /\  F  e.  I )  ->  F : ( Base `  K ) -1-1-onto-> ( Base `  K
) )
4 f1ocnv 5501 . . 3  |-  ( F : ( Base `  K
)
-1-1-onto-> ( Base `  K )  ->  `' F : ( Base `  K ) -1-1-onto-> ( Base `  K
) )
53, 4syl 15 . 2  |-  ( ( K  e.  V  /\  F  e.  I )  ->  `' F : ( Base `  K ) -1-1-onto-> ( Base `  K
) )
6 eqid 2296 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
71, 6, 2lautcnvle 30900 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
) )  ->  (
x ( le `  K ) y  <->  ( `' F `  x )
( le `  K
) ( `' F `  y ) ) )
87ralrimivva 2648 . 2  |-  ( ( K  e.  V  /\  F  e.  I )  ->  A. x  e.  (
Base `  K ) A. y  e.  ( Base `  K ) ( x ( le `  K ) y  <->  ( `' F `  x )
( le `  K
) ( `' F `  y ) ) )
91, 6, 2islaut 30894 . . 3  |-  ( K  e.  V  ->  ( `' F  e.  I  <->  ( `' F : ( Base `  K ) -1-1-onto-> ( Base `  K
)  /\  A. x  e.  ( Base `  K
) A. y  e.  ( Base `  K
) ( x ( le `  K ) y  <->  ( `' F `  x ) ( le
`  K ) ( `' F `  y ) ) ) ) )
109adantr 451 . 2  |-  ( ( K  e.  V  /\  F  e.  I )  ->  ( `' F  e.  I  <->  ( `' F : ( Base `  K
)
-1-1-onto-> ( Base `  K )  /\  A. x  e.  (
Base `  K ) A. y  e.  ( Base `  K ) ( x ( le `  K ) y  <->  ( `' F `  x )
( le `  K
) ( `' F `  y ) ) ) ) )
115, 8, 10mpbir2and 888 1  |-  ( ( K  e.  V  /\  F  e.  I )  ->  `' F  e.  I
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   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:  ldilcnv  30926
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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