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Theorem lsatfixedN 29821
Description: Show equality with the span of the sum of two vectors, one of which ( X) is fixed in advance. Compare lspfixed 15897. (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 29803 . . . 4  |-  ( W  e.  LVec  ->  ( Q  e.  A  <->  E. w  e.  ( V  \  {  .0.  } ) Q  =  ( N `  {
w } ) ) )
82, 7syl 15 . . 3  |-  ( ph  ->  ( Q  e.  A  <->  E. w  e.  ( V 
\  {  .0.  }
) Q  =  ( N `  { w } ) ) )
91, 8mpbid 201 . 2  |-  ( ph  ->  E. w  e.  ( V  \  {  .0.  } ) Q  =  ( N `  { w } ) )
10 lsatfixed.p . . . . 5  |-  .+  =  ( +g  `  W )
1123ad2ant1 976 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  W  e.  LVec )
12 simp2 956 . . . . . 6  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  w  e.  ( V  \  {  .0.  } ) )
13 eldifi 3311 . . . . . 6  |-  ( w  e.  ( V  \  {  .0.  } )  ->  w  e.  V )
1412, 13syl 15 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  w  e.  V
)
15 lsatfixed.x . . . . . 6  |-  ( ph  ->  X  e.  V )
16153ad2ant1 976 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  X  e.  V
)
17 lsatfixed.y . . . . . 6  |-  ( ph  ->  Y  e.  V )
18173ad2ant1 976 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  Y  e.  V
)
19 simp3 957 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  Q  =  ( N `  { w } ) )
2019eqcomd 2301 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( N `  { w } )  =  Q )
21 lsatfixed.e . . . . . . . 8  |-  ( ph  ->  Q  =/=  ( N `
 { X }
) )
22213ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  Q  =/=  ( N `  { X } ) )
2320, 22eqnetrd 2477 . . . . . 6  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( N `  { w } )  =/=  ( N `  { X } ) )
243, 5, 4, 11, 12, 16, 23lspsnne1 15886 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  -.  w  e.  ( N `  { X } ) )
25 lsatfixed.f . . . . . . . 8  |-  ( ph  ->  Q  =/=  ( N `
 { Y }
) )
26253ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  Q  =/=  ( N `  { Y } ) )
2720, 26eqnetrd 2477 . . . . . 6  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( N `  { w } )  =/=  ( N `  { Y } ) )
283, 5, 4, 11, 12, 18, 27lspsnne1 15886 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  -.  w  e.  ( N `  { Y } ) )
29 lsatfixed.g . . . . . . . 8  |-  ( ph  ->  Q  C_  ( N `  { X ,  Y } ) )
30293ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  Q  C_  ( N `  { X ,  Y } ) )
3120, 30eqsstrd 3225 . . . . . 6  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( N `  { w } ) 
C_  ( N `  { X ,  Y }
) )
32 eqid 2296 . . . . . . 7  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
33 lveclmod 15875 . . . . . . . . 9  |-  ( W  e.  LVec  ->  W  e. 
LMod )
342, 33syl 15 . . . . . . . 8  |-  ( ph  ->  W  e.  LMod )
35343ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  W  e.  LMod )
363, 32, 4, 34, 15, 17lspprcl 15751 . . . . . . . 8  |-  ( ph  ->  ( N `  { X ,  Y }
)  e.  ( LSubSp `  W ) )
37363ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( N `  { X ,  Y }
)  e.  ( LSubSp `  W ) )
383, 32, 4, 35, 37, 14lspsnel5 15768 . . . . . 6  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( w  e.  ( N `  { X ,  Y }
)  <->  ( N `  { w } ) 
C_  ( N `  { X ,  Y }
) ) )
3931, 38mpbird 223 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  w  e.  ( N `  { X ,  Y } ) )
403, 10, 5, 4, 11, 14, 16, 18, 24, 28, 39lspfixed 15897 . . . 4  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  E. z  e.  ( ( N `  { Y } )  \  {  .0.  } ) w  e.  ( N `  {
( X  .+  z
) } ) )
41 simpl1 958 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  ph )
4241, 2syl 15 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  W  e.  LVec )
43 simpl2 959 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  w  e.  ( V  \  {  .0.  } ) )
4441, 34syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  W  e.  LMod )
4541, 15syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  X  e.  V
)
46 difss 3316 . . . . . . . . . . 11  |-  ( ( N `  { Y } )  \  {  .0.  } )  C_  ( N `  { Y } )
4717snssd 3776 . . . . . . . . . . . 12  |-  ( ph  ->  { Y }  C_  V )
483, 4lspssv 15756 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  { Y }  C_  V )  ->  ( N `  { Y } )  C_  V )
4934, 47, 48syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { Y } )  C_  V
)
5046, 49syl5ss 3203 . . . . . . . . . 10  |-  ( ph  ->  ( ( N `  { Y } )  \  {  .0.  } )  C_  V )
51503ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( ( N `
 { Y }
)  \  {  .0.  } )  C_  V )
5251sselda 3193 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  z  e.  V
)
533, 10lmodvacl 15657 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  z  e.  V )  ->  ( X  .+  z )  e.  V )
5444, 45, 52, 53syl3anc 1182 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  ( X  .+  z )  e.  V
)
553, 5, 4, 42, 43, 54lspsncmp 15885 . . . . . 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
) } ) ) )
563, 32, 4lspsncl 15750 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  ( X  .+  z )  e.  V )  ->  ( N `  { ( X  .+  z ) } )  e.  ( LSubSp `  W ) )
5744, 54, 56syl2anc 642 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  ( N `  { ( X  .+  z ) } )  e.  ( LSubSp `  W
) )
5843, 13syl 15 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  w  e.  V
)
593, 32, 4, 44, 57, 58lspsnel5 15768 . . . . . 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 ) } ) ) )
60 simpl3 960 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  Q  =  ( N `  { w } ) )
6160eqeq1d 2304 . . . . . 6  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  ( Q  =  ( N `  {
( X  .+  z
) } )  <->  ( N `  { w } )  =  ( N `  { ( X  .+  z ) } ) ) )
6255, 59, 613bitr4rd 277 . . . . 5  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  ( Q  =  ( N `  {
( X  .+  z
) } )  <->  w  e.  ( N `  { ( X  .+  z ) } ) ) )
6362rexbidva 2573 . . . 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
) } ) ) )
6440, 63mpbird 223 . . 3  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  E. z  e.  ( ( N `  { Y } )  \  {  .0.  } ) Q  =  ( N `  {
( X  .+  z
) } ) )
6564rexlimdv3a 2682 . 2  |-  ( ph  ->  ( E. w  e.  ( V  \  {  .0.  } ) Q  =  ( N `  {
w } )  ->  E. z  e.  (
( N `  { Y } )  \  {  .0.  } ) Q  =  ( N `  {
( X  .+  z
) } ) ) )
669, 65mpd 14 1  |-  ( ph  ->  E. z  e.  ( ( N `  { Y } )  \  {  .0.  } ) Q  =  ( N `  {
( X  .+  z
) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    \ cdif 3162    C_ wss 3165   {csn 3653   {cpr 3654   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744   LVecclvec 15871  LSAtomsclsa 29786
This theorem is referenced by:  hdmaprnlem3eN  32673
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  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-rmo 2564  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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872  df-lsatoms 29788
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