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Theorem lautcnvle 30204
Description: Less-than or equal property of lattice automorphism converse. (Contributed by NM, 19-May-2012.)
Hypotheses
Ref Expression
lautcnvle.b  |-  B  =  ( Base `  K
)
lautcnvle.l  |-  .<_  =  ( le `  K )
lautcnvle.i  |-  I  =  ( LAut `  K
)
Assertion
Ref Expression
lautcnvle  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X  .<_  Y  <->  ( `' F `  X )  .<_  ( `' F `  Y ) ) )

Proof of Theorem lautcnvle
StepHypRef Expression
1 simpl 444 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( K  e.  V  /\  F  e.  I
) )
2 lautcnvle.b . . . . . 6  |-  B  =  ( Base `  K
)
3 lautcnvle.i . . . . . 6  |-  I  =  ( LAut `  K
)
42, 3laut1o 30200 . . . . 5  |-  ( ( K  e.  V  /\  F  e.  I )  ->  F : B -1-1-onto-> B )
54adantr 452 . . . 4  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  F : B -1-1-onto-> B )
6 simprl 733 . . . 4  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  X  e.  B )
7 f1ocnvdm 5958 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  X  e.  B )  ->  ( `' F `  X )  e.  B
)
85, 6, 7syl2anc 643 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( `' F `  X )  e.  B
)
9 simprr 734 . . . 4  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  B )
10 f1ocnvdm 5958 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  Y  e.  B )  ->  ( `' F `  Y )  e.  B
)
115, 9, 10syl2anc 643 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( `' F `  Y )  e.  B
)
12 lautcnvle.l . . . 4  |-  .<_  =  ( le `  K )
132, 12, 3lautle 30199 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( ( `' F `  X )  e.  B  /\  ( `' F `  Y )  e.  B ) )  ->  ( ( `' F `  X ) 
.<_  ( `' F `  Y )  <->  ( F `  ( `' F `  X ) )  .<_  ( F `  ( `' F `  Y ) ) ) )
141, 8, 11, 13syl12anc 1182 . 2  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( ( `' F `  X )  .<_  ( `' F `  Y )  <-> 
( F `  ( `' F `  X ) )  .<_  ( F `  ( `' F `  Y ) ) ) )
15 f1ocnvfv2 5955 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  X  e.  B )  ->  ( F `  ( `' F `  X ) )  =  X )
165, 6, 15syl2anc 643 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( F `  ( `' F `  X ) )  =  X )
17 f1ocnvfv2 5955 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  Y  e.  B )  ->  ( F `  ( `' F `  Y ) )  =  Y )
185, 9, 17syl2anc 643 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( F `  ( `' F `  Y ) )  =  Y )
1916, 18breq12d 4167 . 2  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( ( F `  ( `' F `  X ) )  .<_  ( F `  ( `' F `  Y ) )  <->  X  .<_  Y ) )
2014, 19bitr2d 246 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 1717   class class class wbr 4154   `'ccnv 4818   -1-1-onto->wf1o 5394   ` cfv 5395   Basecbs 13397   lecple 13464   LAutclaut 30100
This theorem is referenced by:  lautcnv  30205  lautj  30208  lautm  30209  ltrncnvleN  30245  ltrneq2  30263
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-map 6957  df-laut 30104
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