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Theorem dihatexv2 36537
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 2467 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 dihatexv2.a . . . 4  |-  A  =  ( Atoms `  K )
31, 2atbase 34487 . . 3  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
43anim2i 569 . 2  |-  ( (
ph  /\  Q  e.  A )  ->  ( ph  /\  Q  e.  (
Base `  K )
) )
5 dihatexv2.k . . . . . . 7  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
65adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( V  \  {  .0.  } ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 eldifi 3631 . . . . . . 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 36529 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  x  e.  V
)  ->  ( N `  { x } )  e.  ran  I )
145, 7, 13syl2an 477 . . . . . 6  |-  ( (
ph  /\  x  e.  ( V  \  {  .0.  } ) )  ->  ( N `  { x } )  e.  ran  I )
151, 8, 12dihcnvcl 36469 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { x } )  e.  ran  I )  ->  ( `' I `  ( N `  {
x } ) )  e.  ( Base `  K
) )
166, 14, 15syl2anc 661 . . . . 5  |-  ( (
ph  /\  x  e.  ( V  \  {  .0.  } ) )  ->  ( `' I `  ( N `
 { x }
) )  e.  (
Base `  K )
)
17 eleq1a 2550 . . . . 5  |-  ( ( `' I `  ( N `
 { x }
) )  e.  (
Base `  K )  ->  ( Q  =  ( `' I `  ( N `
 { x }
) )  ->  Q  e.  ( Base `  K
) ) )
1816, 17syl 16 . . . 4  |-  ( (
ph  /\  x  e.  ( V  \  {  .0.  } ) )  ->  ( Q  =  ( `' I `  ( N `  { x } ) )  ->  Q  e.  ( Base `  K )
) )
1918rexlimdva 2959 . . 3  |-  ( ph  ->  ( E. x  e.  ( V  \  {  .0.  } ) Q  =  ( `' I `  ( N `  { x } ) )  ->  Q  e.  ( Base `  K ) ) )
2019imdistani 690 . 2  |-  ( (
ph  /\  E. x  e.  ( V  \  {  .0.  } ) Q  =  ( `' I `  ( N `  { x } ) ) )  ->  ( ph  /\  Q  e.  ( Base `  K ) ) )
21 dihatexv2.o . . . 4  |-  .0.  =  ( 0g `  U )
225adantr 465 . . . 4  |-  ( (
ph  /\  Q  e.  ( Base `  K )
)  ->  ( K  e.  HL  /\  W  e.  H ) )
23 simpr 461 . . . 4  |-  ( (
ph  /\  Q  e.  ( Base `  K )
)  ->  Q  e.  ( Base `  K )
)
241, 2, 8, 9, 10, 21, 11, 12, 22, 23dihatexv 36536 . . 3  |-  ( (
ph  /\  Q  e.  ( Base `  K )
)  ->  ( Q  e.  A  <->  E. x  e.  ( V  \  {  .0.  } ) ( I `  Q )  =  ( N `  { x } ) ) )
2522adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  Q  e.  ( Base `  K
) )  /\  x  e.  ( V  \  {  .0.  } ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2622, 7, 13syl2an 477 . . . . . . 7  |-  ( ( ( ph  /\  Q  e.  ( Base `  K
) )  /\  x  e.  ( V  \  {  .0.  } ) )  -> 
( N `  {
x } )  e. 
ran  I )
278, 12dihcnvid2 36471 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { x } )  e.  ran  I )  ->  ( I `  ( `' I `  ( N `
 { x }
) ) )  =  ( N `  {
x } ) )
2825, 26, 27syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  Q  e.  ( Base `  K
) )  /\  x  e.  ( V  \  {  .0.  } ) )  -> 
( I `  ( `' I `  ( N `
 { x }
) ) )  =  ( N `  {
x } ) )
2928eqeq2d 2481 . . . . 5  |-  ( ( ( ph  /\  Q  e.  ( Base `  K
) )  /\  x  e.  ( V  \  {  .0.  } ) )  -> 
( ( I `  Q )  =  ( I `  ( `' I `  ( N `
 { x }
) ) )  <->  ( I `  Q )  =  ( N `  { x } ) ) )
30 simplr 754 . . . . . 6  |-  ( ( ( ph  /\  Q  e.  ( Base `  K
) )  /\  x  e.  ( V  \  {  .0.  } ) )  ->  Q  e.  ( Base `  K ) )
3125, 26, 15syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  Q  e.  ( Base `  K
) )  /\  x  e.  ( V  \  {  .0.  } ) )  -> 
( `' I `  ( N `  { x } ) )  e.  ( Base `  K
) )
321, 8, 12dih11 36463 . . . . . 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 1228 . . . . 5  |-  ( ( ( ph  /\  Q  e.  ( Base `  K
) )  /\  x  e.  ( V  \  {  .0.  } ) )  -> 
( ( I `  Q )  =  ( I `  ( `' I `  ( N `
 { x }
) ) )  <->  Q  =  ( `' I `  ( N `
 { x }
) ) ) )
3429, 33bitr3d 255 . . . 4  |-  ( ( ( ph  /\  Q  e.  ( Base `  K
) )  /\  x  e.  ( V  \  {  .0.  } ) )  -> 
( ( I `  Q )  =  ( N `  { x } )  <->  Q  =  ( `' I `  ( N `
 { x }
) ) ) )
3534rexbidva 2975 . . 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 253 . 2  |-  ( (
ph  /\  Q  e.  ( Base `  K )
)  ->  ( Q  e.  A  <->  E. x  e.  ( V  \  {  .0.  } ) Q  =  ( `' I `  ( N `
 { x }
) ) ) )
374, 20, 36pm5.21nd 898 1  |-  ( ph  ->  ( Q  e.  A  <->  E. x  e.  ( V 
\  {  .0.  }
) Q  =  ( `' I `  ( N `
 { x }
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2818    \ cdif 3478   {csn 4033   `'ccnv 5004   ran crn 5006   ` cfv 5594   Basecbs 14507   0gc0g 14712   LSpanclspn 17488   Atomscatm 34461   HLchlt 34548   LHypclh 35181   DVecHcdvh 36276   DIsoHcdih 36426
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-riotaBAD 34157
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-undef 7014  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-0g 14714  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-cntz 16227  df-lsm 16529  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-dvr 17204  df-drng 17269  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lvec 17620  df-lsatoms 34174  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-llines 34695  df-lplanes 34696  df-lvols 34697  df-lines 34698  df-psubsp 34700  df-pmap 34701  df-padd 34993  df-lhyp 35185  df-laut 35186  df-ldil 35301  df-ltrn 35302  df-trl 35356  df-tendo 35952  df-edring 35954  df-disoa 36227  df-dvech 36277  df-dib 36337  df-dic 36371  df-dih 36427
This theorem is referenced by:  djhcvat42  36613
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