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Theorem lautle 30578
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 30577 . . 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 608 . 2  |-  ( ( K  e.  V  /\  F  e.  I )  ->  A. x  e.  B  A. y  e.  B  ( x  .<_  y  <->  ( F `  x )  .<_  ( F `
 y ) ) )
6 breq1 4183 . . . 4  |-  ( x  =  X  ->  (
x  .<_  y  <->  X  .<_  y ) )
7 fveq2 5695 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
87breq1d 4190 . . . 4  |-  ( x  =  X  ->  (
( F `  x
)  .<_  ( F `  y )  <->  ( F `  X )  .<_  ( F `
 y ) ) )
96, 8bibi12d 313 . . 3  |-  ( x  =  X  ->  (
( x  .<_  y  <->  ( F `  x )  .<_  ( F `
 y ) )  <-> 
( X  .<_  y  <->  ( F `  X )  .<_  ( F `
 y ) ) ) )
10 breq2 4184 . . . 4  |-  ( y  =  Y  ->  ( X  .<_  y  <->  X  .<_  Y ) )
11 fveq2 5695 . . . . 5  |-  ( y  =  Y  ->  ( F `  y )  =  ( F `  Y ) )
1211breq2d 4192 . . . 4  |-  ( y  =  Y  ->  (
( F `  X
)  .<_  ( F `  y )  <->  ( F `  X )  .<_  ( F `
 Y ) ) )
1310, 12bibi12d 313 . . 3  |-  ( y  =  Y  ->  (
( X  .<_  y  <->  ( F `  X )  .<_  ( F `
 y ) )  <-> 
( X  .<_  Y  <->  ( F `  X )  .<_  ( F `
 Y ) ) ) )
149, 13rspc2v 3026 . 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 456 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   class class class wbr 4180   -1-1-onto->wf1o 5420   ` cfv 5421   Basecbs 13432   lecple 13499   LAutclaut 30479
This theorem is referenced by:  lautcnvle  30583  lautlt  30585  lautj  30587  lautm  30588  lauteq  30589  lautco  30591  ltrnle  30623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-map 6987  df-laut 30483
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