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Theorem dihatexv2 36789
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 2441 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 dihatexv2.a . . . 4  |-  A  =  ( Atoms `  K )
31, 2atbase 34737 . . 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 3609 . . . . . . 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 36781 . . . . . . 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 36721 . . . . . 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 2524 . . . . 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 2933 . . 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 36788 . . 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 36723 . . . . . . 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 2455 . . . . 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 36715 . . . . . 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 1227 . . . . 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 2949 . . 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 1381    e. wcel 1802   E.wrex 2792    \ cdif 3456   {csn 4011   `'ccnv 4985   ran crn 4987   ` cfv 5575   Basecbs 14506   0gc0g 14711   LSpanclspn 17488   Atomscatm 34711   HLchlt 34798   LHypclh 35431   DVecHcdvh 36528   DIsoHcdih 36678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-riotaBAD 34407
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-int 4269  df-iun 4314  df-iin 4315  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-1st 6782  df-2nd 6783  df-tpos 6954  df-undef 7001  df-recs 7041  df-rdg 7075  df-1o 7129  df-oadd 7133  df-er 7310  df-map 7421  df-en 7516  df-dom 7517  df-sdom 7518  df-fin 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-nn 10540  df-2 10597  df-3 10598  df-4 10599  df-5 10600  df-6 10601  df-n0 10799  df-z 10868  df-uz 11088  df-fz 11679  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-sca 14587  df-vsca 14588  df-0g 14713  df-preset 15428  df-poset 15446  df-plt 15459  df-lub 15475  df-glb 15476  df-join 15477  df-meet 15478  df-p0 15540  df-p1 15541  df-lat 15547  df-clat 15609  df-mgm 15743  df-sgrp 15782  df-mnd 15792  df-submnd 15838  df-grp 15928  df-minusg 15929  df-sbg 15930  df-subg 16069  df-cntz 16226  df-lsm 16527  df-cmn 16671  df-abl 16672  df-mgp 17013  df-ur 17025  df-ring 17071  df-oppr 17143  df-dvdsr 17161  df-unit 17162  df-invr 17192  df-dvr 17203  df-drng 17269  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lvec 17620  df-lsatoms 34424  df-oposet 34624  df-ol 34626  df-oml 34627  df-covers 34714  df-ats 34715  df-atl 34746  df-cvlat 34770  df-hlat 34799  df-llines 34945  df-lplanes 34946  df-lvols 34947  df-lines 34948  df-psubsp 34950  df-pmap 34951  df-padd 35243  df-lhyp 35435  df-laut 35436  df-ldil 35551  df-ltrn 35552  df-trl 35607  df-tendo 36204  df-edring 36206  df-disoa 36479  df-dvech 36529  df-dib 36589  df-dic 36623  df-dih 36679
This theorem is referenced by:  djhcvat42  36865
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