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Theorem lauteq 35292
Description: A lattice automorphism argument is equal to its value if all atoms are equal to their values. (Contributed by NM, 24-May-2012.)
Hypotheses
Ref Expression
lauteq.b  |-  B  =  ( Base `  K
)
lauteq.a  |-  A  =  ( Atoms `  K )
lauteq.i  |-  I  =  ( LAut `  K
)
Assertion
Ref Expression
lauteq  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  A. p  e.  A  ( F `  p )  =  p )  -> 
( F `  X
)  =  X )
Distinct variable groups:    A, p    B, p    F, p    I, p    K, p    X, p

Proof of Theorem lauteq
StepHypRef Expression
1 simpl1 999 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  p  e.  A
)  ->  K  e.  HL )
2 simpl2 1000 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  p  e.  A
)  ->  F  e.  I )
3 lauteq.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
4 lauteq.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
53, 4atbase 34487 . . . . . . . . 9  |-  ( p  e.  A  ->  p  e.  B )
65adantl 466 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  p  e.  A
)  ->  p  e.  B )
7 simpl3 1001 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  p  e.  A
)  ->  X  e.  B )
8 eqid 2467 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
9 lauteq.i . . . . . . . . 9  |-  I  =  ( LAut `  K
)
103, 8, 9lautle 35281 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  F  e.  I )  /\  ( p  e.  B  /\  X  e.  B ) )  -> 
( p ( le
`  K ) X  <-> 
( F `  p
) ( le `  K ) ( F `
 X ) ) )
111, 2, 6, 7, 10syl22anc 1229 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  p  e.  A
)  ->  ( p
( le `  K
) X  <->  ( F `  p ) ( le
`  K ) ( F `  X ) ) )
12 breq1 4456 . . . . . . 7  |-  ( ( F `  p )  =  p  ->  (
( F `  p
) ( le `  K ) ( F `
 X )  <->  p ( le `  K ) ( F `  X ) ) )
1311, 12sylan9bb 699 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  p  e.  A )  /\  ( F `  p )  =  p )  ->  (
p ( le `  K ) X  <->  p ( le `  K ) ( F `  X ) ) )
1413bicomd 201 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  p  e.  A )  /\  ( F `  p )  =  p )  ->  (
p ( le `  K ) ( F `
 X )  <->  p ( le `  K ) X ) )
1514ex 434 . . . 4  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  p  e.  A
)  ->  ( ( F `  p )  =  p  ->  ( p ( le `  K
) ( F `  X )  <->  p ( le `  K ) X ) ) )
1615ralimdva 2875 . . 3  |-  ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  ->  ( A. p  e.  A  ( F `  p )  =  p  ->  A. p  e.  A  ( p ( le
`  K ) ( F `  X )  <-> 
p ( le `  K ) X ) ) )
1716imp 429 . 2  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  A. p  e.  A  ( F `  p )  =  p )  ->  A. p  e.  A  ( p ( le
`  K ) ( F `  X )  <-> 
p ( le `  K ) X ) )
18 simpl1 999 . . 3  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  A. p  e.  A  ( F `  p )  =  p )  ->  K  e.  HL )
19 simpl2 1000 . . . 4  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  A. p  e.  A  ( F `  p )  =  p )  ->  F  e.  I )
20 simpl3 1001 . . . 4  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  A. p  e.  A  ( F `  p )  =  p )  ->  X  e.  B )
213, 9lautcl 35284 . . . 4  |-  ( ( ( K  e.  HL  /\  F  e.  I )  /\  X  e.  B
)  ->  ( F `  X )  e.  B
)
2218, 19, 20, 21syl21anc 1227 . . 3  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  A. p  e.  A  ( F `  p )  =  p )  -> 
( F `  X
)  e.  B )
233, 8, 4hlateq 34596 . . 3  |-  ( ( K  e.  HL  /\  ( F `  X )  e.  B  /\  X  e.  B )  ->  ( A. p  e.  A  ( p ( le
`  K ) ( F `  X )  <-> 
p ( le `  K ) X )  <-> 
( F `  X
)  =  X ) )
2418, 22, 20, 23syl3anc 1228 . 2  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  A. p  e.  A  ( F `  p )  =  p )  -> 
( A. p  e.  A  ( p ( le `  K ) ( F `  X
)  <->  p ( le
`  K ) X )  <->  ( F `  X )  =  X ) )
2517, 24mpbid 210 1  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  A. p  e.  A  ( F `  p )  =  p )  -> 
( F `  X
)  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   class class class wbr 4453   ` cfv 5594   Basecbs 14507   lecple 14579   Atomscatm 34461   HLchlt 34548   LAutclaut 35182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-clat 15612  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-laut 35186
This theorem is referenced by:  ltrnid  35332
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