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Theorem ltrncnvel 30953
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 30952 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( `' F `  P )  e.  A )
653adant3r 1179 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( `' F `  P )  e.  A )
7 simp3r 984 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
8 simp1 955 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
9 simp2 956 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  e.  T )
10 eqid 2296 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1110, 2atbase 30101 . . . . . 6  |-  ( ( `' F `  P )  e.  A  ->  ( `' F `  P )  e.  ( Base `  K
) )
126, 11syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( `' F `  P )  e.  ( Base `  K
) )
13 simp1r 980 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  H )
1410, 3lhpbase 30809 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1513, 14syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  ( Base `  K )
)
1610, 1, 3, 4ltrnle 30940 . . . . 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 1186 . . . 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 30935 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
19183adant3 975 . . . . . 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 983 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
2110, 2atbase 30101 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2220, 21syl 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  ( Base `  K )
)
23 f1ocnvfv2 5809 . . . . . 6  |-  ( ( F : ( Base `  K ) -1-1-onto-> ( Base `  K
)  /\  P  e.  ( Base `  K )
)  ->  ( F `  ( `' F `  P ) )  =  P )
2419, 22, 23syl2anc 642 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  ( `' F `  P ) )  =  P )
25 simp1l 979 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
26 hllat 30175 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
2725, 26syl 15 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  Lat )
2810, 1latref 14175 . . . . . . 7  |-  ( ( K  e.  Lat  /\  W  e.  ( Base `  K ) )  ->  W  .<_  W )
2927, 15, 28syl2anc 642 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  .<_  W )
3010, 1, 3, 4ltrnval1 30945 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( W  e.  (
Base `  K )  /\  W  .<_  W ) )  ->  ( F `  W )  =  W )
318, 9, 15, 29, 30syl112anc 1186 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  W )  =  W )
3224, 31breq12d 4052 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  ( `' F `  P )
)  .<_  ( F `  W )  <->  P  .<_  W ) )
3317, 32bitrd 244 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( `' F `  P ) 
.<_  W  <->  P  .<_  W ) )
347, 33mtbird 292 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  ( `' F `  P ) 
.<_  W )
356, 34jca 518 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   `'ccnv 4704   -1-1-onto->wf1o 5270   ` cfv 5271   Basecbs 13164   lecple 13231   Latclat 14167   Atomscatm 30075   HLchlt 30162   LHypclh 30795   LTrncltrn 30912
This theorem is referenced by:  ltrncnv  30957  cdlemg17h  31479
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-glb 14125  df-p0 14161  df-lat 14168  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916
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