Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lautcnv Structured version   Unicode version

Theorem lautcnv 34097
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 2454 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 lautcnv.i . . . 4  |-  I  =  ( LAut `  K
)
31, 2laut1o 34092 . . 3  |-  ( ( K  e.  V  /\  F  e.  I )  ->  F : ( Base `  K ) -1-1-onto-> ( Base `  K
) )
4 f1ocnv 5764 . . 3  |-  ( F : ( Base `  K
)
-1-1-onto-> ( Base `  K )  ->  `' F : ( Base `  K ) -1-1-onto-> ( Base `  K
) )
53, 4syl 16 . 2  |-  ( ( K  e.  V  /\  F  e.  I )  ->  `' F : ( Base `  K ) -1-1-onto-> ( Base `  K
) )
6 eqid 2454 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
71, 6, 2lautcnvle 34096 . . 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 2914 . 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 34090 . . 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 465 . 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 913 1  |-  ( ( K  e.  V  /\  F  e.  I )  ->  `' F  e.  I
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   class class class wbr 4403   `'ccnv 4950   -1-1-onto->wf1o 5528   ` cfv 5529   Basecbs 14296   lecple 14368   LAutclaut 33992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-map 7329  df-laut 33996
This theorem is referenced by:  ldilcnv  34122
  Copyright terms: Public domain W3C validator