Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ltrnatb Unicode version

Theorem ltrnatb 30326
Description: The lattice translation of an atom is an atom. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
ltrnatb.b  |-  B  =  ( Base `  K
)
ltrnatb.a  |-  A  =  ( Atoms `  K )
ltrnatb.h  |-  H  =  ( LHyp `  K
)
ltrnatb.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrnatb  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( P  e.  A  <->  ( F `  P )  e.  A
) )

Proof of Theorem ltrnatb
StepHypRef Expression
1 simp3 957 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  P  e.  B )
2 ltrnatb.b . . . . 5  |-  B  =  ( Base `  K
)
3 ltrnatb.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 ltrnatb.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
52, 3, 4ltrncl 30314 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( F `  P )  e.  B
)
61, 52thd 231 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( P  e.  B  <->  ( F `  P )  e.  B
) )
7 simp1 955 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( K  e.  HL  /\  W  e.  H ) )
8 simp2 956 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  F  e.  T )
9 simp1l 979 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  K  e.  HL )
10 hlop 29552 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
11 eqid 2283 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
122, 11op0cl 29374 . . . . . 6  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  B )
139, 10, 123syl 18 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( 0. `  K )  e.  B
)
14 eqid 2283 . . . . . 6  |-  (  <o  `  K )  =  ( 
<o  `  K )
152, 14, 3, 4ltrncvr 30322 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( 0. `  K )  e.  B  /\  P  e.  B
) )  ->  (
( 0. `  K
) (  <o  `  K
) P  <->  ( F `  ( 0. `  K
) ) (  <o  `  K ) ( F `
 P ) ) )
167, 8, 13, 1, 15syl112anc 1186 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( ( 0. `  K ) ( 
<o  `  K ) P  <-> 
( F `  ( 0. `  K ) ) (  <o  `  K )
( F `  P
) ) )
179, 10syl 15 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  K  e.  OP )
18 simp1r 980 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  W  e.  H )
192, 3lhpbase 30187 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  B )
2018, 19syl 15 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  W  e.  B )
21 eqid 2283 . . . . . . . 8  |-  ( le
`  K )  =  ( le `  K
)
222, 21, 11op0le 29376 . . . . . . 7  |-  ( ( K  e.  OP  /\  W  e.  B )  ->  ( 0. `  K
) ( le `  K ) W )
2317, 20, 22syl2anc 642 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( 0. `  K ) ( le
`  K ) W )
242, 21, 3, 4ltrnval1 30323 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( 0. `  K )  e.  B  /\  ( 0. `  K
) ( le `  K ) W ) )  ->  ( F `  ( 0. `  K
) )  =  ( 0. `  K ) )
257, 8, 13, 23, 24syl112anc 1186 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( F `  ( 0. `  K
) )  =  ( 0. `  K ) )
2625breq1d 4033 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( ( F `  ( 0. `  K ) ) ( 
<o  `  K ) ( F `  P )  <-> 
( 0. `  K
) (  <o  `  K
) ( F `  P ) ) )
2716, 26bitrd 244 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( ( 0. `  K ) ( 
<o  `  K ) P  <-> 
( 0. `  K
) (  <o  `  K
) ( F `  P ) ) )
286, 27anbi12d 691 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( ( P  e.  B  /\  ( 0. `  K ) (  <o  `  K ) P )  <->  ( ( F `  P )  e.  B  /\  ( 0. `  K ) ( 
<o  `  K ) ( F `  P ) ) ) )
29 ltrnatb.a . . . 4  |-  A  =  ( Atoms `  K )
302, 11, 14, 29isat 29476 . . 3  |-  ( K  e.  HL  ->  ( P  e.  A  <->  ( P  e.  B  /\  ( 0. `  K ) ( 
<o  `  K ) P ) ) )
319, 30syl 15 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( P  e.  A  <->  ( P  e.  B  /\  ( 0.
`  K ) ( 
<o  `  K ) P ) ) )
322, 11, 14, 29isat 29476 . . 3  |-  ( K  e.  HL  ->  (
( F `  P
)  e.  A  <->  ( ( F `  P )  e.  B  /\  ( 0. `  K ) ( 
<o  `  K ) ( F `  P ) ) ) )
339, 32syl 15 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( ( F `  P )  e.  A  <->  ( ( F `
 P )  e.  B  /\  ( 0.
`  K ) ( 
<o  `  K ) ( F `  P ) ) ) )
3428, 31, 333bitr4d 276 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( P  e.  A  <->  ( F `  P )  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   0.cp0 14143   OPcops 29362    <o ccvr 29452   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290
This theorem is referenced by:  ltrncnvatb  30327  ltrnel  30328  ltrnat  30329
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-undef 6298  df-riota 6304  df-map 6774  df-plt 14092  df-glb 14109  df-p0 14145  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-hlat 29541  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294
  Copyright terms: Public domain W3C validator