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Theorem lautcnvle 34041
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 457 . . 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 34037 . . . . 5  |-  ( ( K  e.  V  /\  F  e.  I )  ->  F : B -1-1-onto-> B )
54adantr 465 . . . 4  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  F : B -1-1-onto-> B )
6 simprl 755 . . . 4  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  X  e.  B )
7 f1ocnvdm 6090 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  X  e.  B )  ->  ( `' F `  X )  e.  B
)
85, 6, 7syl2anc 661 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( `' F `  X )  e.  B
)
9 simprr 756 . . . 4  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  B )
10 f1ocnvdm 6090 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  Y  e.  B )  ->  ( `' F `  Y )  e.  B
)
115, 9, 10syl2anc 661 . . 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 34036 . . 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 1217 . 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 6085 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  X  e.  B )  ->  ( F `  ( `' F `  X ) )  =  X )
165, 6, 15syl2anc 661 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( F `  ( `' F `  X ) )  =  X )
17 f1ocnvfv2 6085 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  Y  e.  B )  ->  ( F `  ( `' F `  Y ) )  =  Y )
185, 9, 17syl2anc 661 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( F `  ( `' F `  Y ) )  =  Y )
1916, 18breq12d 4405 . 2  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( ( F `  ( `' F `  X ) )  .<_  ( F `  ( `' F `  Y ) )  <->  X  .<_  Y ) )
2014, 19bitr2d 254 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 369    = wceq 1370    e. wcel 1758   class class class wbr 4392   `'ccnv 4939   -1-1-onto->wf1o 5517   ` cfv 5518   Basecbs 14278   lecple 14349   LAutclaut 33937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-map 7318  df-laut 33941
This theorem is referenced by:  lautcnv  34042  lautj  34045  lautm  34046  ltrncnvleN  34082  ltrneq2  34100
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