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Theorem ltrnatb 35150
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 998 . . . 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 35138 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( F `  P )  e.  B
)
61, 52thd 240 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( P  e.  B  <->  ( F `  P )  e.  B
) )
7 simp1 996 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( K  e.  HL  /\  W  e.  H ) )
8 simp2 997 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  F  e.  T )
9 simp1l 1020 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  K  e.  HL )
10 hlop 34376 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
11 eqid 2467 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
122, 11op0cl 34198 . . . . . 6  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  B )
139, 10, 123syl 20 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( 0. `  K )  e.  B
)
14 eqid 2467 . . . . . 6  |-  (  <o  `  K )  =  ( 
<o  `  K )
152, 14, 3, 4ltrncvr 35146 . . . . 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 1232 . . . 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 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  K  e.  OP )
18 simp1r 1021 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  W  e.  H )
192, 3lhpbase 35011 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  B )
2018, 19syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  W  e.  B )
21 eqid 2467 . . . . . . . 8  |-  ( le
`  K )  =  ( le `  K
)
222, 21, 11op0le 34200 . . . . . . 7  |-  ( ( K  e.  OP  /\  W  e.  B )  ->  ( 0. `  K
) ( le `  K ) W )
2317, 20, 22syl2anc 661 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( 0. `  K ) ( le
`  K ) W )
242, 21, 3, 4ltrnval1 35147 . . . . . 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 1232 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( F `  ( 0. `  K
) )  =  ( 0. `  K ) )
2625breq1d 4457 . . . 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 253 . . 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 710 . 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 34300 . . 3  |-  ( K  e.  HL  ->  ( P  e.  A  <->  ( P  e.  B  /\  ( 0. `  K ) ( 
<o  `  K ) P ) ) )
319, 30syl 16 . 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 34300 . . 3  |-  ( K  e.  HL  ->  (
( F `  P
)  e.  A  <->  ( ( F `  P )  e.  B  /\  ( 0. `  K ) ( 
<o  `  K ) ( F `  P ) ) ) )
339, 32syl 16 . 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 285 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( P  e.  A  <->  ( F `  P )  e.  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5588   Basecbs 14493   lecple 14565   0.cp0 15527   OPcops 34186    <o ccvr 34276   Atomscatm 34277   HLchlt 34364   LHypclh 34997   LTrncltrn 35114
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-map 7423  df-plt 15448  df-glb 15465  df-p0 15529  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-hlat 34365  df-lhyp 35001  df-laut 35002  df-ldil 35117  df-ltrn 35118
This theorem is referenced by:  ltrncnvatb  35151  ltrnel  35152  ltrnat  35153
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