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Theorem lsatfixedN 29199
Description: Show equality with the span of the sum of two vectors, one of which ( X) is fixed in advance. Compare lspfixed 15881. (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 29181 . . . 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 3298 . . . . . 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 2288 . . . . . . 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 2464 . . . . . 6  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( N `  { w } )  =/=  ( N `  { X } ) )
243, 5, 4, 11, 12, 16, 23lspsnne1 15870 . . . . 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 2464 . . . . . 6  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( N `  { w } )  =/=  ( N `  { Y } ) )
283, 5, 4, 11, 12, 18, 27lspsnne1 15870 . . . . 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 3212 . . . . . 6  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( N `  { w } ) 
C_  ( N `  { X ,  Y }
) )
32 eqid 2283 . . . . . . 7  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
33 lveclmod 15859 . . . . . . . . 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 15735 . . . . . . . 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 15752 . . . . . 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 15881 . . . 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 3303 . . . . . . . . . . 11  |-  ( ( N `  { Y } )  \  {  .0.  } )  C_  ( N `  { Y } )
4717snssd 3760 . . . . . . . . . . . 12  |-  ( ph  ->  { Y }  C_  V )
483, 4lspssv 15740 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  { Y }  C_  V )  ->  ( N `  { Y } )  C_  V )
4934, 47, 48syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { Y } )  C_  V
)
5046, 49syl5ss 3190 . . . . . . . . . 10  |-  ( ph  ->  ( ( N `  { Y } )  \  {  .0.  } )  C_  V )
51503ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( ( N `
 { Y }
)  \  {  .0.  } )  C_  V )
5251sselda 3180 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  z  e.  V
)
533, 10lmodvacl 15641 . . . . . . . 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 15869 . . . . . 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 15734 . . . . . . . 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 15752 . . . . . 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 2291 . . . . . 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 2560 . . . 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 2669 . 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 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    \ cdif 3149    C_ wss 3152   {csn 3640   {cpr 3641   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728   LVecclvec 15855  LSAtomsclsa 29164
This theorem is referenced by:  hdmaprnlem3eN  32051
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-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-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-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  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-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  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-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 29166
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