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Theorem idlaut 34046
Description: The identity function is a lattice automorphism. (Contributed by NM, 18-May-2012.)
Hypotheses
Ref Expression
idlaut.b  |-  B  =  ( Base `  K
)
idlaut.i  |-  I  =  ( LAut `  K
)
Assertion
Ref Expression
idlaut  |-  ( K  e.  A  ->  (  _I  |`  B )  e.  I )

Proof of Theorem idlaut
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oi 5774 . . 3  |-  (  _I  |`  B ) : B -1-1-onto-> B
21a1i 11 . 2  |-  ( K  e.  A  ->  (  _I  |`  B ) : B -1-1-onto-> B )
3 fvresi 6003 . . . . . 6  |-  ( x  e.  B  ->  (
(  _I  |`  B ) `
 x )  =  x )
4 fvresi 6003 . . . . . 6  |-  ( y  e.  B  ->  (
(  _I  |`  B ) `
 y )  =  y )
53, 4breqan12d 4405 . . . . 5  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( ( (  _I  |`  B ) `  x
) ( le `  K ) ( (  _I  |`  B ) `  y )  <->  x ( le `  K ) y ) )
65bicomd 201 . . . 4  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x ( le
`  K ) y  <-> 
( (  _I  |`  B ) `
 x ) ( le `  K ) ( (  _I  |`  B ) `
 y ) ) )
76rgen2a 2890 . . 3  |-  A. x  e.  B  A. y  e.  B  ( x
( le `  K
) y  <->  ( (  _I  |`  B ) `  x ) ( le
`  K ) ( (  _I  |`  B ) `
 y ) )
87a1i 11 . 2  |-  ( K  e.  A  ->  A. x  e.  B  A. y  e.  B  ( x
( le `  K
) y  <->  ( (  _I  |`  B ) `  x ) ( le
`  K ) ( (  _I  |`  B ) `
 y ) ) )
9 idlaut.b . . 3  |-  B  =  ( Base `  K
)
10 eqid 2451 . . 3  |-  ( le
`  K )  =  ( le `  K
)
11 idlaut.i . . 3  |-  I  =  ( LAut `  K
)
129, 10, 11islaut 34033 . 2  |-  ( K  e.  A  ->  (
(  _I  |`  B )  e.  I  <->  ( (  _I  |`  B ) : B -1-1-onto-> B  /\  A. x  e.  B  A. y  e.  B  ( x
( le `  K
) y  <->  ( (  _I  |`  B ) `  x ) ( le
`  K ) ( (  _I  |`  B ) `
 y ) ) ) ) )
132, 8, 12mpbir2and 913 1  |-  ( K  e.  A  ->  (  _I  |`  B )  e.  I )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   class class class wbr 4390    _I cid 4729    |` cres 4940   -1-1-onto->wf1o 5515   ` cfv 5516   Basecbs 14276   lecple 14347   LAutclaut 33935
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 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-map 7316  df-laut 33939
This theorem is referenced by:  idldil  34064
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