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Theorem lspprabs 16167
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 2436 . . . . . . . 8  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
32lsssssubg 16034 . . . . . . 7  |-  ( W  e.  LMod  ->  ( LSubSp `  W )  C_  (SubGrp `  W ) )
41, 3syl 16 . . . . . 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 16053 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
91, 5, 8syl2anc 643 . . . . . 6  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
104, 9sseldd 3349 . . . . 5  |-  ( ph  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
11 lspprabs.y . . . . . . 7  |-  ( ph  ->  Y  e.  V )
126, 2, 7lspsncl 16053 . . . . . . 7  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
131, 11, 12syl2anc 643 . . . . . 6  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
144, 13sseldd 3349 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (SubGrp `  W ) )
15 eqid 2436 . . . . . 6  |-  ( LSSum `  W )  =  (
LSSum `  W )
1615lsmub1 15290 . . . . 5  |-  ( ( ( N `  { X } )  e.  (SubGrp `  W )  /\  ( N `  { Y } )  e.  (SubGrp `  W ) )  -> 
( N `  { X } )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) )
1710, 14, 16syl2anc 643 . . . 4  |-  ( ph  ->  ( N `  { X } )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) )
182, 15lsmcl 16155 . . . . . 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 1184 . . . . 5  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) )  e.  ( LSubSp `  W )
)
206, 7lspsnid 16069 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
211, 5, 20syl2anc 643 . . . . . 6  |-  ( ph  ->  X  e.  ( N `
 { X }
) )
226, 7lspsnid 16069 . . . . . . 7  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  Y  e.  ( N `  { Y } ) )
231, 11, 22syl2anc 643 . . . . . 6  |-  ( ph  ->  Y  e.  ( N `
 { Y }
) )
24 lspprabs.p . . . . . . 7  |-  .+  =  ( +g  `  W )
2524, 15lsmelvali 15284 . . . . . 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 1185 . . . . 5  |-  ( ph  ->  ( X  .+  Y
)  e.  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) )
272, 7, 1, 19, 26lspsnel5a 16072 . . . 4  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  C_  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) ) )
286, 24lmodvacl 15964 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .+  Y )  e.  V )
291, 5, 11, 28syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( X  .+  Y
)  e.  V )
306, 2, 7lspsncl 16053 . . . . . . 7  |-  ( ( W  e.  LMod  /\  ( X  .+  Y )  e.  V )  ->  ( N `  { ( X  .+  Y ) } )  e.  ( LSubSp `  W ) )
311, 29, 30syl2anc 643 . . . . . 6  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  e.  ( LSubSp `  W )
)
324, 31sseldd 3349 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  e.  (SubGrp `  W )
)
334, 19sseldd 3349 . . . . 5  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) )  e.  (SubGrp `  W )
)
3415lsmlub 15297 . . . . 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 1184 . . . 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 888 . . 3  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) )  C_  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) ) )
3715lsmub1 15290 . . . . 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 643 . . . 4  |-  ( ph  ->  ( N `  { X } )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) ) )
392, 15lsmcl 16155 . . . . . 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 1184 . . . . 5  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) )  e.  ( LSubSp `  W )
)
41 eqid 2436 . . . . . . 7  |-  ( -g `  W )  =  (
-g `  W )
426, 7lspsnid 16069 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  ( X  .+  Y )  e.  V )  ->  ( X  .+  Y )  e.  ( N `  {
( X  .+  Y
) } ) )
431, 29, 42syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( X  .+  Y
)  e.  ( N `
 { ( X 
.+  Y ) } ) )
4441, 15, 32, 10, 43, 21lsmelvalmi 15286 . . . . . 6  |-  ( ph  ->  ( ( X  .+  Y ) ( -g `  W ) X )  e.  ( ( N `
 { ( X 
.+  Y ) } ) ( LSSum `  W
) ( N `  { X } ) ) )
45 lmodabl 15991 . . . . . . . 8  |-  ( W  e.  LMod  ->  W  e. 
Abel )
461, 45syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  Abel )
476, 24, 41ablpncan2 15440 . . . . . . 7  |-  ( ( W  e.  Abel  /\  X  e.  V  /\  Y  e.  V )  ->  (
( X  .+  Y
) ( -g `  W
) X )  =  Y )
4846, 5, 11, 47syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( ( X  .+  Y ) ( -g `  W ) X )  =  Y )
4915lsmcom 15473 . . . . . . 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 ) } ) ) )
5046, 32, 10, 49syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( ( N `  { ( X  .+  Y ) } ) ( LSSum `  W )
( N `  { X } ) )  =  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) ) )
5144, 48, 503eltr3d 2516 . . . . 5  |-  ( ph  ->  Y  e.  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) ) )
522, 7, 1, 40, 51lspsnel5a 16072 . . . 4  |-  ( ph  ->  ( N `  { Y } )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) ) )
534, 40sseldd 3349 . . . . 5  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) )  e.  (SubGrp `  W )
)
5415lsmlub 15297 . . . . 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 ) } ) ) ) )
5510, 14, 53, 54syl3anc 1184 . . . 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 ) } ) ) ) )
5638, 52, 55mpbi2and 888 . . 3  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) )  C_  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) ) )
5736, 56eqssd 3365 . 2  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) )  =  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) ) )
586, 7, 15, 1, 5, 29lsmpr 16161 . 2  |-  ( ph  ->  ( N `  { X ,  ( X  .+  Y ) } )  =  ( ( N `
 { X }
) ( LSSum `  W
) ( N `  { ( X  .+  Y ) } ) ) )
596, 7, 15, 1, 5, 11lsmpr 16161 . 2  |-  ( ph  ->  ( N `  { X ,  Y }
)  =  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) )
6057, 58, 593eqtr4d 2478 1  |-  ( ph  ->  ( N `  { X ,  ( X  .+  Y ) } )  =  ( N `  { X ,  Y }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3320   {csn 3814   {cpr 3815   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   -gcsg 14688  SubGrpcsubg 14938   LSSumclsm 15268   Abelcabel 15413   LModclmod 15950   LSubSpclss 16008   LSpanclspn 16047
This theorem is referenced by:  lspabs2  16192  lspindp4  16209  mapdindp4  32521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-cntz 15116  df-lsm 15270  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-lmod 15952  df-lss 16009  df-lsp 16048
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