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Theorem lautle 36205
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 36204 . . 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 622 . 2  |-  ( ( K  e.  V  /\  F  e.  I )  ->  A. x  e.  B  A. y  e.  B  ( x  .<_  y  <->  ( F `  x )  .<_  ( F `
 y ) ) )
6 breq1 4442 . . . 4  |-  ( x  =  X  ->  (
x  .<_  y  <->  X  .<_  y ) )
7 fveq2 5848 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
87breq1d 4449 . . . 4  |-  ( x  =  X  ->  (
( F `  x
)  .<_  ( F `  y )  <->  ( F `  X )  .<_  ( F `
 y ) ) )
96, 8bibi12d 319 . . 3  |-  ( x  =  X  ->  (
( x  .<_  y  <->  ( F `  x )  .<_  ( F `
 y ) )  <-> 
( X  .<_  y  <->  ( F `  X )  .<_  ( F `
 y ) ) ) )
10 breq2 4443 . . . 4  |-  ( y  =  Y  ->  ( X  .<_  y  <->  X  .<_  Y ) )
11 fveq2 5848 . . . . 5  |-  ( y  =  Y  ->  ( F `  y )  =  ( F `  Y ) )
1211breq2d 4451 . . . 4  |-  ( y  =  Y  ->  (
( F `  X
)  .<_  ( F `  y )  <->  ( F `  X )  .<_  ( F `
 Y ) ) )
1310, 12bibi12d 319 . . 3  |-  ( y  =  Y  ->  (
( X  .<_  y  <->  ( F `  X )  .<_  ( F `
 y ) )  <-> 
( X  .<_  Y  <->  ( F `  X )  .<_  ( F `
 Y ) ) ) )
149, 13rspc2v 3216 . 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 467 1  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X  .<_  Y  <->  ( F `  X )  .<_  ( F `
 Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   class class class wbr 4439   -1-1-onto->wf1o 5569   ` cfv 5570   Basecbs 14716   lecple 14791   LAutclaut 36106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-laut 36110
This theorem is referenced by:  lautcnvle  36210  lautlt  36212  lautj  36214  lautm  36215  lauteq  36216  lautco  36218  ltrnle  36250
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