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Theorem lauteq 30102
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 958 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  p  e.  A
)  ->  K  e.  HL )
2 simpl2 959 . . . . . . . 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 29297 . . . . . . . . 9  |-  ( p  e.  A  ->  p  e.  B )
65adantl 452 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  p  e.  A
)  ->  p  e.  B )
7 simpl3 960 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  p  e.  A
)  ->  X  e.  B )
8 eqid 2316 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
9 lauteq.i . . . . . . . . 9  |-  I  =  ( LAut `  K
)
103, 8, 9lautle 30091 . . . . . . . 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 1183 . . . . . . 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 4063 . . . . . . 7  |-  ( ( F `  p )  =  p  ->  (
( F `  p
) ( le `  K ) ( F `
 X )  <->  p ( le `  K ) ( F `  X ) ) )
1311, 12sylan9bb 680 . . . . . 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 192 . . . . 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 423 . . . 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 2655 . . 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 418 . 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 958 . . 3  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  A. p  e.  A  ( F `  p )  =  p )  ->  K  e.  HL )
19 simpl2 959 . . . 4  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  A. p  e.  A  ( F `  p )  =  p )  ->  F  e.  I )
20 simpl3 960 . . . 4  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  A. p  e.  A  ( F `  p )  =  p )  ->  X  e.  B )
213, 9lautcl 30094 . . . 4  |-  ( ( ( K  e.  HL  /\  F  e.  I )  /\  X  e.  B
)  ->  ( F `  X )  e.  B
)
2218, 19, 20, 21syl21anc 1181 . . 3  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  A. p  e.  A  ( F `  p )  =  p )  -> 
( F `  X
)  e.  B )
233, 8, 4hlateq 29406 . . 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 1182 . 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 201 1  |-  ( ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B )  /\  A. p  e.  A  ( F `  p )  =  p )  -> 
( F `  X
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   A.wral 2577   class class class wbr 4060   ` cfv 5292   Basecbs 13195   lecple 13262   Atomscatm 29271   HLchlt 29358   LAutclaut 29992
This theorem is referenced by:  ltrnid  30142
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-map 6817  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-lat 14201  df-clat 14263  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-laut 29996
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