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Theorem lautcnvle 36226
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 455 . . 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 36222 . . . . 5  |-  ( ( K  e.  V  /\  F  e.  I )  ->  F : B -1-1-onto-> B )
54adantr 463 . . . 4  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  F : B -1-1-onto-> B )
6 simprl 754 . . . 4  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  X  e.  B )
7 f1ocnvdm 6089 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  X  e.  B )  ->  ( `' F `  X )  e.  B
)
85, 6, 7syl2anc 659 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( `' F `  X )  e.  B
)
9 simprr 755 . . . 4  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  B )
10 f1ocnvdm 6089 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  Y  e.  B )  ->  ( `' F `  Y )  e.  B
)
115, 9, 10syl2anc 659 . . 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 36221 . . 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 1224 . 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 6084 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  X  e.  B )  ->  ( F `  ( `' F `  X ) )  =  X )
165, 6, 15syl2anc 659 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( F `  ( `' F `  X ) )  =  X )
17 f1ocnvfv2 6084 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  Y  e.  B )  ->  ( F `  ( `' F `  Y ) )  =  Y )
185, 9, 17syl2anc 659 . . 3  |-  ( ( ( K  e.  V  /\  F  e.  I
)  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( F `  ( `' F `  Y ) )  =  Y )
1916, 18breq12d 4380 . 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 367    = wceq 1399    e. wcel 1826   class class class wbr 4367   `'ccnv 4912   -1-1-onto->wf1o 5495   ` cfv 5496   Basecbs 14634   lecple 14709   LAutclaut 36122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-map 7340  df-laut 36126
This theorem is referenced by:  lautcnv  36227  lautj  36230  lautm  36231  ltrncnvleN  36267  ltrneq2  36285
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