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Theorem lautcnvle 30787
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 30783 . . . . 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 6010 . . . 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 6010 . . . 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 30782 . . 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 6007 . . . 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 6007 . . . 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 4217 . 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 1652    e. wcel 1725   class class class wbr 4204   `'ccnv 4869   -1-1-onto->wf1o 5445   ` cfv 5446   Basecbs 13459   lecple 13526   LAutclaut 30683
This theorem is referenced by:  lautcnv  30788  lautj  30791  lautm  30792  ltrncnvleN  30828  ltrneq2  30846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-laut 30687
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