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Theorem lautcnv 30279
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 2283 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 lautcnv.i . . . 4  |-  I  =  ( LAut `  K
)
31, 2laut1o 30274 . . 3  |-  ( ( K  e.  V  /\  F  e.  I )  ->  F : ( Base `  K ) -1-1-onto-> ( Base `  K
) )
4 f1ocnv 5485 . . 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 2283 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
71, 6, 2lautcnvle 30278 . . 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 2635 . 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 30272 . . 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 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   `'ccnv 4688   -1-1-onto->wf1o 5254   ` cfv 5255   Basecbs 13148   lecple 13215   LAutclaut 30174
This theorem is referenced by:  ldilcnv  30304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-laut 30178
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