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Theorem lspprabs 15848
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 2283 . . . . . . . 8  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
32lsssssubg 15715 . . . . . . 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 15734 . . . . . . 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 3181 . . . . 5  |-  ( ph  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
11 lspprabs.y . . . . . . 7  |-  ( ph  ->  Y  e.  V )
126, 2, 7lspsncl 15734 . . . . . . 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 3181 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (SubGrp `  W ) )
15 eqid 2283 . . . . . 6  |-  ( LSSum `  W )  =  (
LSSum `  W )
1615lsmub1 14967 . . . . 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 15836 . . . . . 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 15750 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
211, 5, 20syl2anc 642 . . . . . 6  |-  ( ph  ->  X  e.  ( N `
 { X }
) )
226, 7lspsnid 15750 . . . . . . 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 14961 . . . . . 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 15753 . . . 4  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  C_  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) ) )
286, 24lmodvacl 15641 . . . . . . . 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 15734 . . . . . . 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 3181 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  e.  (SubGrp `  W )
)
334, 19sseldd 3181 . . . . 5  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) )  e.  (SubGrp `  W )
)
3415lsmlub 14974 . . . . 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 14967 . . . . 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 15836 . . . . . 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 15672 . . . . . . . . 9  |-  ( W  e.  LMod  ->  W  e. 
Abel )
421, 41syl 15 . . . . . . . 8  |-  ( ph  ->  W  e.  Abel )
43 eqid 2283 . . . . . . . . 9  |-  ( -g `  W )  =  (
-g `  W )
446, 24, 43ablpncan2 15117 . . . . . . . 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 15750 . . . . . . . . 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 14963 . . . . . . 7  |-  ( ph  ->  ( ( X  .+  Y ) ( -g `  W ) X )  e.  ( ( N `
 { ( X 
.+  Y ) } ) ( LSSum `  W
) ( N `  { X } ) ) )
4945, 48eqeltrrd 2358 . . . . . 6  |-  ( ph  ->  Y  e.  ( ( N `  { ( X  .+  Y ) } ) ( LSSum `  W ) ( N `
 { X }
) ) )
5015lsmcom 15150 . . . . . . 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 2359 . . . . 5  |-  ( ph  ->  Y  e.  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) ) )
532, 7, 1, 40, 52lspsnel5a 15753 . . . 4  |-  ( ph  ->  ( N `  { Y } )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) ) )
544, 40sseldd 3181 . . . . 5  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) )  e.  (SubGrp `  W )
)
5515lsmlub 14974 . . . . 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 3196 . 2  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) )  =  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) ) )
596, 7, 15, 1, 5, 29lsmpr 15842 . 2  |-  ( ph  ->  ( N `  { X ,  ( X  .+  Y ) } )  =  ( ( N `
 { X }
) ( LSSum `  W
) ( N `  { ( X  .+  Y ) } ) ) )
606, 7, 15, 1, 5, 11lsmpr 15842 . 2  |-  ( ph  ->  ( N `  { X ,  Y }
)  =  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) )
6158, 59, 603eqtr4d 2325 1  |-  ( ph  ->  ( N `  { X ,  ( X  .+  Y ) } )  =  ( N `  { X ,  Y }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   {csn 3640   {cpr 3641   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   -gcsg 14365  SubGrpcsubg 14615   LSSumclsm 14945   Abelcabel 15090   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728
This theorem is referenced by:  lspabs2  15873  lspindp4  15890  mapdindp4  31913
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-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-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  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-lmod 15629  df-lss 15690  df-lsp 15729
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