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Theorem lauteq 30284
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 29479 . . . . . . . . 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 2283 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
9 lauteq.i . . . . . . . . 9  |-  I  =  ( LAut `  K
)
103, 8, 9lautle 30273 . . . . . . . 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 4026 . . . . . . 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 2621 . . 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 30276 . . . 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 29588 . . 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 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Atomscatm 29453   HLchlt 29540   LAutclaut 30174
This theorem is referenced by:  ltrnid  30324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-laut 30178
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