MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lspprabs Unicode version

Theorem lspprabs 15864
Description: Absorption of vector sum into span of pair. (Contributed by NM, 27-Apr-2015.)
Hypotheses
Ref Expression
lspprabs.v  |-  V  =  ( Base `  W
)
lspprabs.p  |-  .+  =  ( +g  `  W )
lspprabs.n  |-  N  =  ( LSpan `  W )
lspprabs.w  |-  ( ph  ->  W  e.  LMod )
lspprabs.x  |-  ( ph  ->  X  e.  V )
lspprabs.y  |-  ( ph  ->  Y  e.  V )
Assertion
Ref Expression
lspprabs  |-  ( ph  ->  ( N `  { X ,  ( X  .+  Y ) } )  =  ( N `  { X ,  Y }
) )

Proof of Theorem lspprabs
StepHypRef Expression
1 lspprabs.w . . . . . . 7  |-  ( ph  ->  W  e.  LMod )
2 eqid 2296 . . . . . . . 8  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
32lsssssubg 15731 . . . . . . 7  |-  ( W  e.  LMod  ->  ( LSubSp `  W )  C_  (SubGrp `  W ) )
41, 3syl 15 . . . . . 6  |-  ( ph  ->  ( LSubSp `  W )  C_  (SubGrp `  W )
)
5 lspprabs.x . . . . . . 7  |-  ( ph  ->  X  e.  V )
6 lspprabs.v . . . . . . . 8  |-  V  =  ( Base `  W
)
7 lspprabs.n . . . . . . . 8  |-  N  =  ( LSpan `  W )
86, 2, 7lspsncl 15750 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
91, 5, 8syl2anc 642 . . . . . 6  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
104, 9sseldd 3194 . . . . 5  |-  ( ph  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
11 lspprabs.y . . . . . . 7  |-  ( ph  ->  Y  e.  V )
126, 2, 7lspsncl 15750 . . . . . . 7  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
131, 11, 12syl2anc 642 . . . . . 6  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
144, 13sseldd 3194 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (SubGrp `  W ) )
15 eqid 2296 . . . . . 6  |-  ( LSSum `  W )  =  (
LSSum `  W )
1615lsmub1 14983 . . . . 5  |-  ( ( ( N `  { X } )  e.  (SubGrp `  W )  /\  ( N `  { Y } )  e.  (SubGrp `  W ) )  -> 
( N `  { X } )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) )
1710, 14, 16syl2anc 642 . . . 4  |-  ( ph  ->  ( N `  { X } )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) )
182, 15lsmcl 15852 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( N `  { X } )  e.  (
LSubSp `  W )  /\  ( N `  { Y } )  e.  (
LSubSp `  W ) )  ->  ( ( N `
 { X }
) ( LSSum `  W
) ( N `  { Y } ) )  e.  ( LSubSp `  W
) )
191, 9, 13, 18syl3anc 1182 . . . . 5  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) )  e.  ( LSubSp `  W )
)
206, 7lspsnid 15766 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
211, 5, 20syl2anc 642 . . . . . 6  |-  ( ph  ->  X  e.  ( N `
 { X }
) )
226, 7lspsnid 15766 . . . . . . 7  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  Y  e.  ( N `  { Y } ) )
231, 11, 22syl2anc 642 . . . . . 6  |-  ( ph  ->  Y  e.  ( N `
 { Y }
) )
24 lspprabs.p . . . . . . 7  |-  .+  =  ( +g  `  W )
2524, 15lsmelvali 14977 . . . . . 6  |-  ( ( ( ( N `  { X } )  e.  (SubGrp `  W )  /\  ( N `  { Y } )  e.  (SubGrp `  W ) )  /\  ( X  e.  ( N `  { X } )  /\  Y  e.  ( N `  { Y } ) ) )  ->  ( X  .+  Y )  e.  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) )
2610, 14, 21, 23, 25syl22anc 1183 . . . . 5  |-  ( ph  ->  ( X  .+  Y
)  e.  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) )
272, 7, 1, 19, 26lspsnel5a 15769 . . . 4  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  C_  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) ) )
286, 24lmodvacl 15657 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .+  Y )  e.  V )
291, 5, 11, 28syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( X  .+  Y
)  e.  V )
306, 2, 7lspsncl 15750 . . . . . . 7  |-  ( ( W  e.  LMod  /\  ( X  .+  Y )  e.  V )  ->  ( N `  { ( X  .+  Y ) } )  e.  ( LSubSp `  W ) )
311, 29, 30syl2anc 642 . . . . . 6  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  e.  ( LSubSp `  W )
)
324, 31sseldd 3194 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  e.  (SubGrp `  W )
)
334, 19sseldd 3194 . . . . 5  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) )  e.  (SubGrp `  W )
)
3415lsmlub 14990 . . . . 5  |-  ( ( ( N `  { X } )  e.  (SubGrp `  W )  /\  ( N `  { ( X  .+  Y ) } )  e.  (SubGrp `  W )  /\  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) )  e.  (SubGrp `  W ) )  -> 
( ( ( N `
 { X }
)  C_  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) )  /\  ( N `  { ( X  .+  Y ) } )  C_  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) )  <->  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) ) )
3510, 32, 33, 34syl3anc 1182 . . . 4  |-  ( ph  ->  ( ( ( N `
 { X }
)  C_  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) )  /\  ( N `  { ( X  .+  Y ) } )  C_  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) )  <->  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) ) )
3617, 27, 35mpbi2and 887 . . 3  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) )  C_  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) ) )
3715lsmub1 14983 . . . . 5  |-  ( ( ( N `  { X } )  e.  (SubGrp `  W )  /\  ( N `  { ( X  .+  Y ) } )  e.  (SubGrp `  W ) )  -> 
( N `  { X } )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) ) )
3810, 32, 37syl2anc 642 . . . 4  |-  ( ph  ->  ( N `  { X } )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) ) )
392, 15lsmcl 15852 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( N `  { X } )  e.  (
LSubSp `  W )  /\  ( N `  { ( X  .+  Y ) } )  e.  (
LSubSp `  W ) )  ->  ( ( N `
 { X }
) ( LSSum `  W
) ( N `  { ( X  .+  Y ) } ) )  e.  ( LSubSp `  W ) )
401, 9, 31, 39syl3anc 1182 . . . . 5  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) )  e.  ( LSubSp `  W )
)
41 lmodabl 15688 . . . . . . . . 9  |-  ( W  e.  LMod  ->  W  e. 
Abel )
421, 41syl 15 . . . . . . . 8  |-  ( ph  ->  W  e.  Abel )
43 eqid 2296 . . . . . . . . 9  |-  ( -g `  W )  =  (
-g `  W )
446, 24, 43ablpncan2 15133 . . . . . . . 8  |-  ( ( W  e.  Abel  /\  X  e.  V  /\  Y  e.  V )  ->  (
( X  .+  Y
) ( -g `  W
) X )  =  Y )
4542, 5, 11, 44syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( X  .+  Y ) ( -g `  W ) X )  =  Y )
466, 7lspsnid 15766 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  ( X  .+  Y )  e.  V )  ->  ( X  .+  Y )  e.  ( N `  {
( X  .+  Y
) } ) )
471, 29, 46syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( X  .+  Y
)  e.  ( N `
 { ( X 
.+  Y ) } ) )
4843, 15, 32, 10, 47, 21lsmelvalmi 14979 . . . . . . 7  |-  ( ph  ->  ( ( X  .+  Y ) ( -g `  W ) X )  e.  ( ( N `
 { ( X 
.+  Y ) } ) ( LSSum `  W
) ( N `  { X } ) ) )
4945, 48eqeltrrd 2371 . . . . . 6  |-  ( ph  ->  Y  e.  ( ( N `  { ( X  .+  Y ) } ) ( LSSum `  W ) ( N `
 { X }
) ) )
5015lsmcom 15166 . . . . . . 7  |-  ( ( W  e.  Abel  /\  ( N `  { ( X  .+  Y ) } )  e.  (SubGrp `  W )  /\  ( N `  { X } )  e.  (SubGrp `  W ) )  -> 
( ( N `  { ( X  .+  Y ) } ) ( LSSum `  W )
( N `  { X } ) )  =  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) ) )
5142, 32, 10, 50syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( ( N `  { ( X  .+  Y ) } ) ( LSSum `  W )
( N `  { X } ) )  =  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) ) )
5249, 51eleqtrd 2372 . . . . 5  |-  ( ph  ->  Y  e.  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) ) )
532, 7, 1, 40, 52lspsnel5a 15769 . . . 4  |-  ( ph  ->  ( N `  { Y } )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) ) )
544, 40sseldd 3194 . . . . 5  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) )  e.  (SubGrp `  W )
)
5515lsmlub 14990 . . . . 5  |-  ( ( ( N `  { X } )  e.  (SubGrp `  W )  /\  ( N `  { Y } )  e.  (SubGrp `  W )  /\  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) )  e.  (SubGrp `  W ) )  -> 
( ( ( N `
 { X }
)  C_  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) )  /\  ( N `  { Y } )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) ) )  <->  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) ) ) )
5610, 14, 54, 55syl3anc 1182 . . . 4  |-  ( ph  ->  ( ( ( N `
 { X }
)  C_  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) )  /\  ( N `  { Y } )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) ) )  <->  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) ) ) )
5738, 53, 56mpbi2and 887 . . 3  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) )  C_  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) ) )
5836, 57eqssd 3209 . 2  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) )  =  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) ) )
596, 7, 15, 1, 5, 29lsmpr 15858 . 2  |-  ( ph  ->  ( N `  { X ,  ( X  .+  Y ) } )  =  ( ( N `
 { X }
) ( LSSum `  W
) ( N `  { ( X  .+  Y ) } ) ) )
606, 7, 15, 1, 5, 11lsmpr 15858 . 2  |-  ( ph  ->  ( N `  { X ,  Y }
)  =  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) )
6158, 59, 603eqtr4d 2338 1  |-  ( ph  ->  ( N `  { X ,  ( X  .+  Y ) } )  =  ( N `  { X ,  Y }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   {csn 3653   {cpr 3654   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   -gcsg 14381  SubGrpcsubg 14631   LSSumclsm 14961   Abelcabel 15106   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744
This theorem is referenced by:  lspabs2  15889  lspindp4  15906  mapdindp4  32535
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-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-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  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-lmod 15645  df-lss 15706  df-lsp 15745
  Copyright terms: Public domain W3C validator