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Theorem lautcnv 33107
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 2402 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 lautcnv.i . . . 4  |-  I  =  ( LAut `  K
)
31, 2laut1o 33102 . . 3  |-  ( ( K  e.  V  /\  F  e.  I )  ->  F : ( Base `  K ) -1-1-onto-> ( Base `  K
) )
4 f1ocnv 5811 . . 3  |-  ( F : ( Base `  K
)
-1-1-onto-> ( Base `  K )  ->  `' F : ( Base `  K ) -1-1-onto-> ( Base `  K
) )
53, 4syl 17 . 2  |-  ( ( K  e.  V  /\  F  e.  I )  ->  `' F : ( Base `  K ) -1-1-onto-> ( Base `  K
) )
6 eqid 2402 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
71, 6, 2lautcnvle 33106 . . 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 2825 . 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 33100 . . 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 463 . 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 923 1  |-  ( ( K  e.  V  /\  F  e.  I )  ->  `' F  e.  I
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   class class class wbr 4395   `'ccnv 4822   -1-1-onto->wf1o 5568   ` cfv 5569   Basecbs 14841   lecple 14916   LAutclaut 33002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-map 7459  df-laut 33006
This theorem is referenced by:  ldilcnv  33132
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