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Theorem lauteq 33739
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 991 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  p  e.  A
)  ->  K  e.  HL )
2 simpl2 992 . . . . . . . 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 32934 . . . . . . . . 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 993 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  p  e.  A
)  ->  X  e.  B )
8 eqid 2443 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
9 lauteq.i . . . . . . . . 9  |-  I  =  ( LAut `  K
)
103, 8, 9lautle 33728 . . . . . . . 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 1219 . . . . . . 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 4295 . . . . . . 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 2794 . . 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 991 . . 3  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  A. p  e.  A  ( F `  p )  =  p )  ->  K  e.  HL )
19 simpl2 992 . . . 4  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  A. p  e.  A  ( F `  p )  =  p )  ->  F  e.  I )
20 simpl3 993 . . . 4  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  A. p  e.  A  ( F `  p )  =  p )  ->  X  e.  B )
213, 9lautcl 33731 . . . 4  |-  ( ( ( K  e.  HL  /\  F  e.  I )  /\  X  e.  B
)  ->  ( F `  X )  e.  B
)
2218, 19, 20, 21syl21anc 1217 . . 3  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  A. p  e.  A  ( F `  p )  =  p )  -> 
( F `  X
)  e.  B )
233, 8, 4hlateq 33043 . . 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 1218 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2715   class class class wbr 4292   ` cfv 5418   Basecbs 14174   lecple 14245   Atomscatm 32908   HLchlt 32995   LAutclaut 33629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-map 7216  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-lat 15216  df-clat 15278  df-oposet 32821  df-ol 32823  df-oml 32824  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-laut 33633
This theorem is referenced by:  ltrnid  33779
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