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Theorem lautle 30273
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 30272 . . 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 4026 . . . 4  |-  ( x  =  X  ->  (
x  .<_  y  <->  X  .<_  y ) )
7 fveq2 5525 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
87breq1d 4033 . . . 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 4027 . . . 4  |-  ( y  =  Y  ->  ( X  .<_  y  <->  X  .<_  Y ) )
11 fveq2 5525 . . . . 5  |-  ( y  =  Y  ->  ( F `  y )  =  ( F `  Y ) )
1211breq2d 4035 . . . 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 2890 . 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 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   -1-1-onto->wf1o 5254   ` cfv 5255   Basecbs 13148   lecple 13215   LAutclaut 30174
This theorem is referenced by:  lautcnvle  30278  lautlt  30280  lautj  30282  lautm  30283  lauteq  30284  lautco  30286  ltrnle  30318
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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