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Theorem lsatfixedN 29808
Description: Show equality with the span of the sum of two vectors, one of which ( X) is fixed in advance. Compare lspfixed 16201. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
lsatfixed.v  |-  V  =  ( Base `  W
)
lsatfixed.p  |-  .+  =  ( +g  `  W )
lsatfixed.o  |-  .0.  =  ( 0g `  W )
lsatfixed.n  |-  N  =  ( LSpan `  W )
lsatfixed.a  |-  A  =  (LSAtoms `  W )
lsatfixed.w  |-  ( ph  ->  W  e.  LVec )
lsatfixed.q  |-  ( ph  ->  Q  e.  A )
lsatfixed.x  |-  ( ph  ->  X  e.  V )
lsatfixed.y  |-  ( ph  ->  Y  e.  V )
lsatfixed.e  |-  ( ph  ->  Q  =/=  ( N `
 { X }
) )
lsatfixed.f  |-  ( ph  ->  Q  =/=  ( N `
 { Y }
) )
lsatfixed.g  |-  ( ph  ->  Q  C_  ( N `  { X ,  Y } ) )
Assertion
Ref Expression
lsatfixedN  |-  ( ph  ->  E. z  e.  ( ( N `  { Y } )  \  {  .0.  } ) Q  =  ( N `  {
( X  .+  z
) } ) )
Distinct variable groups:    z, N    z,  .0.    z,  .+    ph, z    z, Q    z, V    z, W    z, X    z, Y
Allowed substitution hint:    A( z)

Proof of Theorem lsatfixedN
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 lsatfixed.q . . 3  |-  ( ph  ->  Q  e.  A )
2 lsatfixed.w . . . 4  |-  ( ph  ->  W  e.  LVec )
3 lsatfixed.v . . . . 5  |-  V  =  ( Base `  W
)
4 lsatfixed.n . . . . 5  |-  N  =  ( LSpan `  W )
5 lsatfixed.o . . . . 5  |-  .0.  =  ( 0g `  W )
6 lsatfixed.a . . . . 5  |-  A  =  (LSAtoms `  W )
73, 4, 5, 6islsat 29790 . . . 4  |-  ( W  e.  LVec  ->  ( Q  e.  A  <->  E. w  e.  ( V  \  {  .0.  } ) Q  =  ( N `  {
w } ) ) )
82, 7syl 16 . . 3  |-  ( ph  ->  ( Q  e.  A  <->  E. w  e.  ( V 
\  {  .0.  }
) Q  =  ( N `  { w } ) ) )
91, 8mpbid 203 . 2  |-  ( ph  ->  E. w  e.  ( V  \  {  .0.  } ) Q  =  ( N `  { w } ) )
10 lsatfixed.p . . . . 5  |-  .+  =  ( +g  `  W )
1123ad2ant1 979 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  W  e.  LVec )
12 simp2 959 . . . . . 6  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  w  e.  ( V  \  {  .0.  } ) )
1312eldifad 3333 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  w  e.  V
)
14 lsatfixed.x . . . . . 6  |-  ( ph  ->  X  e.  V )
15143ad2ant1 979 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  X  e.  V
)
16 lsatfixed.y . . . . . 6  |-  ( ph  ->  Y  e.  V )
17163ad2ant1 979 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  Y  e.  V
)
18 simp3 960 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  Q  =  ( N `  { w } ) )
1918eqcomd 2442 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( N `  { w } )  =  Q )
20 lsatfixed.e . . . . . . . 8  |-  ( ph  ->  Q  =/=  ( N `
 { X }
) )
21203ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  Q  =/=  ( N `  { X } ) )
2219, 21eqnetrd 2620 . . . . . 6  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( N `  { w } )  =/=  ( N `  { X } ) )
233, 5, 4, 11, 12, 15, 22lspsnne1 16190 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  -.  w  e.  ( N `  { X } ) )
24 lsatfixed.f . . . . . . . 8  |-  ( ph  ->  Q  =/=  ( N `
 { Y }
) )
25243ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  Q  =/=  ( N `  { Y } ) )
2619, 25eqnetrd 2620 . . . . . 6  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( N `  { w } )  =/=  ( N `  { Y } ) )
273, 5, 4, 11, 12, 17, 26lspsnne1 16190 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  -.  w  e.  ( N `  { Y } ) )
28 lsatfixed.g . . . . . . . 8  |-  ( ph  ->  Q  C_  ( N `  { X ,  Y } ) )
29283ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  Q  C_  ( N `  { X ,  Y } ) )
3019, 29eqsstrd 3383 . . . . . 6  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( N `  { w } ) 
C_  ( N `  { X ,  Y }
) )
31 eqid 2437 . . . . . . 7  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
32 lveclmod 16179 . . . . . . . . 9  |-  ( W  e.  LVec  ->  W  e. 
LMod )
332, 32syl 16 . . . . . . . 8  |-  ( ph  ->  W  e.  LMod )
34333ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  W  e.  LMod )
353, 31, 4, 33, 14, 16lspprcl 16055 . . . . . . . 8  |-  ( ph  ->  ( N `  { X ,  Y }
)  e.  ( LSubSp `  W ) )
36353ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( N `  { X ,  Y }
)  e.  ( LSubSp `  W ) )
373, 31, 4, 34, 36, 13lspsnel5 16072 . . . . . 6  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( w  e.  ( N `  { X ,  Y }
)  <->  ( N `  { w } ) 
C_  ( N `  { X ,  Y }
) ) )
3830, 37mpbird 225 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  w  e.  ( N `  { X ,  Y } ) )
393, 10, 5, 4, 11, 13, 15, 17, 23, 27, 38lspfixed 16201 . . . 4  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  E. z  e.  ( ( N `  { Y } )  \  {  .0.  } ) w  e.  ( N `  {
( X  .+  z
) } ) )
40 simpl1 961 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  ph )
4140, 2syl 16 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  W  e.  LVec )
42 simpl2 962 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  w  e.  ( V  \  {  .0.  } ) )
4340, 33syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  W  e.  LMod )
4440, 14syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  X  e.  V
)
4516snssd 3944 . . . . . . . . . . . 12  |-  ( ph  ->  { Y }  C_  V )
463, 4lspssv 16060 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  { Y }  C_  V )  ->  ( N `  { Y } )  C_  V )
4733, 45, 46syl2anc 644 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { Y } )  C_  V
)
4847ssdifssd 3486 . . . . . . . . . 10  |-  ( ph  ->  ( ( N `  { Y } )  \  {  .0.  } )  C_  V )
49483ad2ant1 979 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( ( N `
 { Y }
)  \  {  .0.  } )  C_  V )
5049sselda 3349 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  z  e.  V
)
513, 10lmodvacl 15965 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  z  e.  V )  ->  ( X  .+  z )  e.  V )
5243, 44, 50, 51syl3anc 1185 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  ( X  .+  z )  e.  V
)
533, 5, 4, 41, 42, 52lspsncmp 16189 . . . . . 6  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  ( ( N `
 { w }
)  C_  ( N `  { ( X  .+  z ) } )  <-> 
( N `  {
w } )  =  ( N `  {
( X  .+  z
) } ) ) )
543, 31, 4lspsncl 16054 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  ( X  .+  z )  e.  V )  ->  ( N `  { ( X  .+  z ) } )  e.  ( LSubSp `  W ) )
5543, 52, 54syl2anc 644 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  ( N `  { ( X  .+  z ) } )  e.  ( LSubSp `  W
) )
5642eldifad 3333 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  w  e.  V
)
573, 31, 4, 43, 55, 56lspsnel5 16072 . . . . . 6  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  ( w  e.  ( N `  {
( X  .+  z
) } )  <->  ( N `  { w } ) 
C_  ( N `  { ( X  .+  z ) } ) ) )
58 simpl3 963 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  Q  =  ( N `  { w } ) )
5958eqeq1d 2445 . . . . . 6  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  ( Q  =  ( N `  {
( X  .+  z
) } )  <->  ( N `  { w } )  =  ( N `  { ( X  .+  z ) } ) ) )
6053, 57, 593bitr4rd 279 . . . . 5  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  ( Q  =  ( N `  {
( X  .+  z
) } )  <->  w  e.  ( N `  { ( X  .+  z ) } ) ) )
6160rexbidva 2723 . . . 4  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( E. z  e.  ( ( N `  { Y } )  \  {  .0.  } ) Q  =  ( N `  { ( X  .+  z ) } )  <->  E. z  e.  (
( N `  { Y } )  \  {  .0.  } ) w  e.  ( N `  {
( X  .+  z
) } ) ) )
6239, 61mpbird 225 . . 3  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  E. z  e.  ( ( N `  { Y } )  \  {  .0.  } ) Q  =  ( N `  {
( X  .+  z
) } ) )
6362rexlimdv3a 2833 . 2  |-  ( ph  ->  ( E. w  e.  ( V  \  {  .0.  } ) Q  =  ( N `  {
w } )  ->  E. z  e.  (
( N `  { Y } )  \  {  .0.  } ) Q  =  ( N `  {
( X  .+  z
) } ) ) )
649, 63mpd 15 1  |-  ( ph  ->  E. z  e.  ( ( N `  { Y } )  \  {  .0.  } ) Q  =  ( N `  {
( X  .+  z
) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   E.wrex 2707    \ cdif 3318    C_ wss 3321   {csn 3815   {cpr 3816   ` cfv 5455  (class class class)co 6082   Basecbs 13470   +g cplusg 13530   0gc0g 13724   LModclmod 15951   LSubSpclss 16009   LSpanclspn 16048   LVecclvec 16175  LSAtomsclsa 29773
This theorem is referenced by:  hdmaprnlem3eN  32660
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-tpos 6480  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-3 10060  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-mulr 13544  df-0g 13728  df-mnd 14691  df-submnd 14740  df-grp 14813  df-minusg 14814  df-sbg 14815  df-subg 14942  df-cntz 15117  df-lsm 15271  df-cmn 15415  df-abl 15416  df-mgp 15650  df-rng 15664  df-ur 15666  df-oppr 15729  df-dvdsr 15747  df-unit 15748  df-invr 15778  df-drng 15838  df-lmod 15953  df-lss 16010  df-lsp 16049  df-lvec 16176  df-lsatoms 29775
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