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Theorem lsatfixedN 32621
Description: Show equality with the span of the sum of two vectors, one of which ( X) is fixed in advance. Compare lspfixed 18406. (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 32603 . . . 4  |-  ( W  e.  LVec  ->  ( Q  e.  A  <->  E. w  e.  ( V  \  {  .0.  } ) Q  =  ( N `  {
w } ) ) )
82, 7syl 17 . . 3  |-  ( ph  ->  ( Q  e.  A  <->  E. w  e.  ( V 
\  {  .0.  }
) Q  =  ( N `  { w } ) ) )
91, 8mpbid 215 . 2  |-  ( ph  ->  E. w  e.  ( V  \  {  .0.  } ) Q  =  ( N `  { w } ) )
10 lsatfixed.p . . . . 5  |-  .+  =  ( +g  `  W )
1123ad2ant1 1035 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  W  e.  LVec )
12 simp2 1015 . . . . . 6  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  w  e.  ( V  \  {  .0.  } ) )
1312eldifad 3428 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  w  e.  V
)
14 lsatfixed.x . . . . . 6  |-  ( ph  ->  X  e.  V )
15143ad2ant1 1035 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  X  e.  V
)
16 lsatfixed.y . . . . . 6  |-  ( ph  ->  Y  e.  V )
17163ad2ant1 1035 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  Y  e.  V
)
18 simp3 1016 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  Q  =  ( N `  { w } ) )
1918eqcomd 2468 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( N `  { w } )  =  Q )
20 lsatfixed.e . . . . . . . 8  |-  ( ph  ->  Q  =/=  ( N `
 { X }
) )
21203ad2ant1 1035 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  Q  =/=  ( N `  { X } ) )
2219, 21eqnetrd 2703 . . . . . 6  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( N `  { w } )  =/=  ( N `  { X } ) )
233, 5, 4, 11, 12, 15, 22lspsnne1 18395 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  -.  w  e.  ( N `  { X } ) )
24 lsatfixed.f . . . . . . . 8  |-  ( ph  ->  Q  =/=  ( N `
 { Y }
) )
25243ad2ant1 1035 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  Q  =/=  ( N `  { Y } ) )
2619, 25eqnetrd 2703 . . . . . 6  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( N `  { w } )  =/=  ( N `  { Y } ) )
273, 5, 4, 11, 12, 17, 26lspsnne1 18395 . . . . 5  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  -.  w  e.  ( N `  { Y } ) )
28 lsatfixed.g . . . . . . . 8  |-  ( ph  ->  Q  C_  ( N `  { X ,  Y } ) )
29283ad2ant1 1035 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  Q  C_  ( N `  { X ,  Y } ) )
3019, 29eqsstrd 3478 . . . . . 6  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( N `  { w } ) 
C_  ( N `  { X ,  Y }
) )
31 eqid 2462 . . . . . . 7  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
32 lveclmod 18384 . . . . . . . . 9  |-  ( W  e.  LVec  ->  W  e. 
LMod )
332, 32syl 17 . . . . . . . 8  |-  ( ph  ->  W  e.  LMod )
34333ad2ant1 1035 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  W  e.  LMod )
353, 31, 4, 33, 14, 16lspprcl 18256 . . . . . . . 8  |-  ( ph  ->  ( N `  { X ,  Y }
)  e.  ( LSubSp `  W ) )
36353ad2ant1 1035 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( N `  { X ,  Y }
)  e.  ( LSubSp `  W ) )
373, 31, 4, 34, 36, 13lspsnel5 18273 . . . . . 6  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( w  e.  ( N `  { X ,  Y }
)  <->  ( N `  { w } ) 
C_  ( N `  { X ,  Y }
) ) )
3830, 37mpbird 240 . . . . 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 18406 . . . 4  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  E. z  e.  ( ( N `  { Y } )  \  {  .0.  } ) w  e.  ( N `  {
( X  .+  z
) } ) )
40 simpl1 1017 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  ph )
4140, 2syl 17 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  W  e.  LVec )
42 simpl2 1018 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  w  e.  ( V  \  {  .0.  } ) )
4340, 33syl 17 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  W  e.  LMod )
4440, 14syl 17 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  X  e.  V
)
4516snssd 4130 . . . . . . . . . . . 12  |-  ( ph  ->  { Y }  C_  V )
463, 4lspssv 18261 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  { Y }  C_  V )  ->  ( N `  { Y } )  C_  V )
4733, 45, 46syl2anc 671 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { Y } )  C_  V
)
4847ssdifssd 3583 . . . . . . . . . 10  |-  ( ph  ->  ( ( N `  { Y } )  \  {  .0.  } )  C_  V )
49483ad2ant1 1035 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  ( ( N `
 { Y }
)  \  {  .0.  } )  C_  V )
5049sselda 3444 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  z  e.  V
)
513, 10lmodvacl 18160 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  z  e.  V )  ->  ( X  .+  z )  e.  V )
5243, 44, 50, 51syl3anc 1276 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  ( X  .+  z )  e.  V
)
533, 5, 4, 41, 42, 52lspsncmp 18394 . . . . . 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 18255 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  ( X  .+  z )  e.  V )  ->  ( N `  { ( X  .+  z ) } )  e.  ( LSubSp `  W ) )
5543, 52, 54syl2anc 671 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  ( N `  { ( X  .+  z ) } )  e.  ( LSubSp `  W
) )
5642eldifad 3428 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  w  e.  V
)
573, 31, 4, 43, 55, 56lspsnel5 18273 . . . . . 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 1019 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  Q  =  ( N `  { w } ) )
5958eqeq1d 2464 . . . . . 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 294 . . . . 5  |-  ( ( ( ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  { w } ) )  /\  z  e.  ( ( N `  { Y } )  \  {  .0.  } ) )  ->  ( Q  =  ( N `  {
( X  .+  z
) } )  <->  w  e.  ( N `  { ( X  .+  z ) } ) ) )
6160rexbidva 2910 . . . 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 240 . . 3  |-  ( (
ph  /\  w  e.  ( V  \  {  .0.  } )  /\  Q  =  ( N `  {
w } ) )  ->  E. z  e.  ( ( N `  { Y } )  \  {  .0.  } ) Q  =  ( N `  {
( X  .+  z
) } ) )
6362rexlimdv3a 2893 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   E.wrex 2750    \ cdif 3413    C_ wss 3416   {csn 3980   {cpr 3982   ` cfv 5605  (class class class)co 6320   Basecbs 15176   +g cplusg 15245   0gc0g 15393   LModclmod 18146   LSubSpclss 18210   LSpanclspn 18249   LVecclvec 18380  LSAtomsclsa 32586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-tpos 7004  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-er 7394  df-en 7601  df-dom 7602  df-sdom 7603  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-nn 10643  df-2 10701  df-3 10702  df-ndx 15179  df-slot 15180  df-base 15181  df-sets 15182  df-ress 15183  df-plusg 15258  df-mulr 15259  df-0g 15395  df-mgm 16543  df-sgrp 16582  df-mnd 16592  df-submnd 16638  df-grp 16728  df-minusg 16729  df-sbg 16730  df-subg 16869  df-cntz 17026  df-lsm 17343  df-cmn 17487  df-abl 17488  df-mgp 17779  df-ur 17791  df-ring 17837  df-oppr 17906  df-dvdsr 17924  df-unit 17925  df-invr 17955  df-drng 18032  df-lmod 18148  df-lss 18211  df-lsp 18250  df-lvec 18381  df-lsatoms 32588
This theorem is referenced by:  hdmaprnlem3eN  35475
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