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Theorem lautle 30342
Description: Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012.)
Hypotheses
Ref Expression
lautset.b  |-  B  =  ( Base `  K
)
lautset.l  |-  .<_  =  ( le `  K )
lautset.i  |-  I  =  ( LAut `  K
)
Assertion
Ref Expression
lautle  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X  .<_  Y  <->  ( F `  X )  .<_  ( F `
 Y ) ) )

Proof of Theorem lautle
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lautset.b . . . 4  |-  B  =  ( Base `  K
)
2 lautset.l . . . 4  |-  .<_  =  ( le `  K )
3 lautset.i . . . 4  |-  I  =  ( LAut `  K
)
41, 2, 3islaut 30341 . . 3  |-  ( K  e.  V  ->  ( F  e.  I  <->  ( F : B -1-1-onto-> B  /\  A. x  e.  B  A. y  e.  B  ( x  .<_  y  <->  ( F `  x )  .<_  ( F `
 y ) ) ) ) )
54simplbda 607 . 2  |-  ( ( K  e.  V  /\  F  e.  I )  ->  A. x  e.  B  A. y  e.  B  ( x  .<_  y  <->  ( F `  x )  .<_  ( F `
 y ) ) )
6 breq1 4107 . . . 4  |-  ( x  =  X  ->  (
x  .<_  y  <->  X  .<_  y ) )
7 fveq2 5608 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
87breq1d 4114 . . . 4  |-  ( x  =  X  ->  (
( F `  x
)  .<_  ( F `  y )  <->  ( F `  X )  .<_  ( F `
 y ) ) )
96, 8bibi12d 312 . . 3  |-  ( x  =  X  ->  (
( x  .<_  y  <->  ( F `  x )  .<_  ( F `
 y ) )  <-> 
( X  .<_  y  <->  ( F `  X )  .<_  ( F `
 y ) ) ) )
10 breq2 4108 . . . 4  |-  ( y  =  Y  ->  ( X  .<_  y  <->  X  .<_  Y ) )
11 fveq2 5608 . . . . 5  |-  ( y  =  Y  ->  ( F `  y )  =  ( F `  Y ) )
1211breq2d 4116 . . . 4  |-  ( y  =  Y  ->  (
( F `  X
)  .<_  ( F `  y )  <->  ( F `  X )  .<_  ( F `
 Y ) ) )
1310, 12bibi12d 312 . . 3  |-  ( y  =  Y  ->  (
( X  .<_  y  <->  ( F `  X )  .<_  ( F `
 y ) )  <-> 
( X  .<_  Y  <->  ( F `  X )  .<_  ( F `
 Y ) ) ) )
149, 13rspc2v 2966 . 2  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( A. x  e.  B  A. y  e.  B  ( x  .<_  y  <-> 
( F `  x
)  .<_  ( F `  y ) )  -> 
( X  .<_  Y  <->  ( F `  X )  .<_  ( F `
 Y ) ) ) )
155, 14mpan9 455 1  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X  .<_  Y  <->  ( F `  X )  .<_  ( F `
 Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   class class class wbr 4104   -1-1-onto->wf1o 5336   ` cfv 5337   Basecbs 13245   lecple 13312   LAutclaut 30243
This theorem is referenced by:  lautcnvle  30347  lautlt  30349  lautj  30351  lautm  30352  lauteq  30353  lautco  30355  ltrnle  30387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-map 6862  df-laut 30247
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