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Theorem dihatexv2 31529
Description: There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 17-Aug-2014.)
Hypotheses
Ref Expression
dihatexv2.a  |-  A  =  ( Atoms `  K )
dihatexv2.h  |-  H  =  ( LHyp `  K
)
dihatexv2.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihatexv2.v  |-  V  =  ( Base `  U
)
dihatexv2.o  |-  .0.  =  ( 0g `  U )
dihatexv2.n  |-  N  =  ( LSpan `  U )
dihatexv2.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dihatexv2.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
Assertion
Ref Expression
dihatexv2  |-  ( ph  ->  ( Q  e.  A  <->  E. x  e.  ( V 
\  {  .0.  }
) Q  =  ( `' I `  ( N `
 { x }
) ) ) )
Distinct variable groups:    x, A    x, I    x, K    x, N    x, Q    x, V    x, W    ph, x
Allowed substitution hints:    U( x)    H( x)    .0. ( x)

Proof of Theorem dihatexv2
StepHypRef Expression
1 eqid 2283 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 dihatexv2.a . . . 4  |-  A  =  ( Atoms `  K )
31, 2atbase 29479 . . 3  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
43anim2i 552 . 2  |-  ( (
ph  /\  Q  e.  A )  ->  ( ph  /\  Q  e.  (
Base `  K )
) )
5 dihatexv2.k . . . . . . 7  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
65adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( V  \  {  .0.  } ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 eldifi 3298 . . . . . . 7  |-  ( x  e.  ( V  \  {  .0.  } )  ->  x  e.  V )
8 dihatexv2.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
9 dihatexv2.u . . . . . . . 8  |-  U  =  ( ( DVecH `  K
) `  W )
10 dihatexv2.v . . . . . . . 8  |-  V  =  ( Base `  U
)
11 dihatexv2.n . . . . . . . 8  |-  N  =  ( LSpan `  U )
12 dihatexv2.i . . . . . . . 8  |-  I  =  ( ( DIsoH `  K
) `  W )
138, 9, 10, 11, 12dihlsprn 31521 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  x  e.  V
)  ->  ( N `  { x } )  e.  ran  I )
145, 7, 13syl2an 463 . . . . . 6  |-  ( (
ph  /\  x  e.  ( V  \  {  .0.  } ) )  ->  ( N `  { x } )  e.  ran  I )
151, 8, 12dihcnvcl 31461 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { x } )  e.  ran  I )  ->  ( `' I `  ( N `  {
x } ) )  e.  ( Base `  K
) )
166, 14, 15syl2anc 642 . . . . 5  |-  ( (
ph  /\  x  e.  ( V  \  {  .0.  } ) )  ->  ( `' I `  ( N `
 { x }
) )  e.  (
Base `  K )
)
17 eleq1a 2352 . . . . 5  |-  ( ( `' I `  ( N `
 { x }
) )  e.  (
Base `  K )  ->  ( Q  =  ( `' I `  ( N `
 { x }
) )  ->  Q  e.  ( Base `  K
) ) )
1816, 17syl 15 . . . 4  |-  ( (
ph  /\  x  e.  ( V  \  {  .0.  } ) )  ->  ( Q  =  ( `' I `  ( N `  { x } ) )  ->  Q  e.  ( Base `  K )
) )
1918rexlimdva 2667 . . 3  |-  ( ph  ->  ( E. x  e.  ( V  \  {  .0.  } ) Q  =  ( `' I `  ( N `  { x } ) )  ->  Q  e.  ( Base `  K ) ) )
2019imdistani 671 . 2  |-  ( (
ph  /\  E. x  e.  ( V  \  {  .0.  } ) Q  =  ( `' I `  ( N `  { x } ) ) )  ->  ( ph  /\  Q  e.  ( Base `  K ) ) )
21 dihatexv2.o . . . 4  |-  .0.  =  ( 0g `  U )
225adantr 451 . . . 4  |-  ( (
ph  /\  Q  e.  ( Base `  K )
)  ->  ( K  e.  HL  /\  W  e.  H ) )
23 simpr 447 . . . 4  |-  ( (
ph  /\  Q  e.  ( Base `  K )
)  ->  Q  e.  ( Base `  K )
)
241, 2, 8, 9, 10, 21, 11, 12, 22, 23dihatexv 31528 . . 3  |-  ( (
ph  /\  Q  e.  ( Base `  K )
)  ->  ( Q  e.  A  <->  E. x  e.  ( V  \  {  .0.  } ) ( I `  Q )  =  ( N `  { x } ) ) )
2522adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  Q  e.  ( Base `  K
) )  /\  x  e.  ( V  \  {  .0.  } ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2622, 7, 13syl2an 463 . . . . . . 7  |-  ( ( ( ph  /\  Q  e.  ( Base `  K
) )  /\  x  e.  ( V  \  {  .0.  } ) )  -> 
( N `  {
x } )  e. 
ran  I )
278, 12dihcnvid2 31463 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { x } )  e.  ran  I )  ->  ( I `  ( `' I `  ( N `
 { x }
) ) )  =  ( N `  {
x } ) )
2825, 26, 27syl2anc 642 . . . . . 6  |-  ( ( ( ph  /\  Q  e.  ( Base `  K
) )  /\  x  e.  ( V  \  {  .0.  } ) )  -> 
( I `  ( `' I `  ( N `
 { x }
) ) )  =  ( N `  {
x } ) )
2928eqeq2d 2294 . . . . 5  |-  ( ( ( ph  /\  Q  e.  ( Base `  K
) )  /\  x  e.  ( V  \  {  .0.  } ) )  -> 
( ( I `  Q )  =  ( I `  ( `' I `  ( N `
 { x }
) ) )  <->  ( I `  Q )  =  ( N `  { x } ) ) )
30 simplr 731 . . . . . 6  |-  ( ( ( ph  /\  Q  e.  ( Base `  K
) )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  Q  e.  ( Base `  K ) )
3125, 26, 15syl2anc 642 . . . . . 6  |-  ( ( ( ph  /\  Q  e.  ( Base `  K
) )  /\  x  e.  ( V  \  {  .0.  } ) )  -> 
( `' I `  ( N `  { x } ) )  e.  ( Base `  K
) )
321, 8, 12dih11 31455 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Q  e.  (
Base `  K )  /\  ( `' I `  ( N `  { x } ) )  e.  ( Base `  K
) )  ->  (
( I `  Q
)  =  ( I `
 ( `' I `  ( N `  {
x } ) ) )  <->  Q  =  ( `' I `  ( N `
 { x }
) ) ) )
3325, 30, 31, 32syl3anc 1182 . . . . 5  |-  ( ( ( ph  /\  Q  e.  ( Base `  K
) )  /\  x  e.  ( V  \  {  .0.  } ) )  -> 
( ( I `  Q )  =  ( I `  ( `' I `  ( N `
 { x }
) ) )  <->  Q  =  ( `' I `  ( N `
 { x }
) ) ) )
3429, 33bitr3d 246 . . . 4  |-  ( ( ( ph  /\  Q  e.  ( Base `  K
) )  /\  x  e.  ( V  \  {  .0.  } ) )  -> 
( ( I `  Q )  =  ( N `  { x } )  <->  Q  =  ( `' I `  ( N `
 { x }
) ) ) )
3534rexbidva 2560 . . 3  |-  ( (
ph  /\  Q  e.  ( Base `  K )
)  ->  ( E. x  e.  ( V  \  {  .0.  } ) ( I `  Q
)  =  ( N `
 { x }
)  <->  E. x  e.  ( V  \  {  .0.  } ) Q  =  ( `' I `  ( N `
 { x }
) ) ) )
3624, 35bitrd 244 . 2  |-  ( (
ph  /\  Q  e.  ( Base `  K )
)  ->  ( Q  e.  A  <->  E. x  e.  ( V  \  {  .0.  } ) Q  =  ( `' I `  ( N `
 { x }
) ) ) )
374, 20, 36pm5.21nd 868 1  |-  ( ph  ->  ( Q  e.  A  <->  E. x  e.  ( V 
\  {  .0.  }
) Q  =  ( `' I `  ( N `
 { x }
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    \ cdif 3149   {csn 3640   `'ccnv 4688   ran crn 4690   ` cfv 5255   Basecbs 13148   0gc0g 13400   LSpanclspn 15728   Atomscatm 29453   HLchlt 29540   LHypclh 30173   DVecHcdvh 31268   DIsoHcdih 31418
This theorem is referenced by:  djhcvat42  31605
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  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  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-rmo 2551  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 29166  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-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tendo 30944  df-edring 30946  df-disoa 31219  df-dvech 31269  df-dib 31329  df-dic 31363  df-dih 31419
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