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Theorem ltrncnvel 30939
Description: The converse of the lattice translation of an atom not under the fiducial co-atom. (Contributed by NM, 10-May-2013.)
Hypotheses
Ref Expression
ltrnel.l  |-  .<_  =  ( le `  K )
ltrnel.a  |-  A  =  ( Atoms `  K )
ltrnel.h  |-  H  =  ( LHyp `  K
)
ltrnel.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrncnvel  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( `' F `  P )  e.  A  /\  -.  ( `' F `  P ) 
.<_  W ) )

Proof of Theorem ltrncnvel
StepHypRef Expression
1 ltrnel.l . . . 4  |-  .<_  =  ( le `  K )
2 ltrnel.a . . . 4  |-  A  =  ( Atoms `  K )
3 ltrnel.h . . . 4  |-  H  =  ( LHyp `  K
)
4 ltrnel.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
51, 2, 3, 4ltrncnvat 30938 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( `' F `  P )  e.  A )
653adant3r 1181 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( `' F `  P )  e.  A )
7 simp3r 986 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
8 simp1 957 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
9 simp2 958 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  e.  T )
10 eqid 2436 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1110, 2atbase 30087 . . . . . 6  |-  ( ( `' F `  P )  e.  A  ->  ( `' F `  P )  e.  ( Base `  K
) )
126, 11syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( `' F `  P )  e.  ( Base `  K
) )
13 simp1r 982 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  H )
1410, 3lhpbase 30795 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1513, 14syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  ( Base `  K )
)
1610, 1, 3, 4ltrnle 30926 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( `' F `  P )  e.  (
Base `  K )  /\  W  e.  ( Base `  K ) ) )  ->  ( ( `' F `  P ) 
.<_  W  <->  ( F `  ( `' F `  P ) )  .<_  ( F `  W ) ) )
178, 9, 12, 15, 16syl112anc 1188 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( `' F `  P ) 
.<_  W  <->  ( F `  ( `' F `  P ) )  .<_  ( F `  W ) ) )
1810, 3, 4ltrn1o 30921 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
19183adant3 977 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
20 simp3l 985 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
2110, 2atbase 30087 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2220, 21syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  ( Base `  K )
)
23 f1ocnvfv2 6015 . . . . . 6  |-  ( ( F : ( Base `  K ) -1-1-onto-> ( Base `  K
)  /\  P  e.  ( Base `  K )
)  ->  ( F `  ( `' F `  P ) )  =  P )
2419, 22, 23syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  ( `' F `  P ) )  =  P )
25 simp1l 981 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
26 hllat 30161 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
2725, 26syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  Lat )
2810, 1latref 14482 . . . . . . 7  |-  ( ( K  e.  Lat  /\  W  e.  ( Base `  K ) )  ->  W  .<_  W )
2927, 15, 28syl2anc 643 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  .<_  W )
3010, 1, 3, 4ltrnval1 30931 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( W  e.  (
Base `  K )  /\  W  .<_  W ) )  ->  ( F `  W )  =  W )
318, 9, 15, 29, 30syl112anc 1188 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  W )  =  W )
3224, 31breq12d 4225 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  ( `' F `  P )
)  .<_  ( F `  W )  <->  P  .<_  W ) )
3317, 32bitrd 245 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( `' F `  P ) 
.<_  W  <->  P  .<_  W ) )
347, 33mtbird 293 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  ( `' F `  P ) 
.<_  W )
356, 34jca 519 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( `' F `  P )  e.  A  /\  -.  ( `' F `  P ) 
.<_  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   `'ccnv 4877   -1-1-onto->wf1o 5453   ` cfv 5454   Basecbs 13469   lecple 13536   Latclat 14474   Atomscatm 30061   HLchlt 30148   LHypclh 30781   LTrncltrn 30898
This theorem is referenced by:  ltrncnv  30943  cdlemg17h  31465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-undef 6543  df-riota 6549  df-map 7020  df-poset 14403  df-plt 14415  df-glb 14432  df-p0 14468  df-lat 14475  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902
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