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Theorem lspprabs 17521
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 2467 . . . . . . . 8  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
32lsssssubg 17384 . . . . . . 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 17403 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
91, 5, 8syl2anc 661 . . . . . 6  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
104, 9sseldd 3505 . . . . 5  |-  ( ph  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
11 lspprabs.y . . . . . . 7  |-  ( ph  ->  Y  e.  V )
126, 2, 7lspsncl 17403 . . . . . . 7  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
131, 11, 12syl2anc 661 . . . . . 6  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
144, 13sseldd 3505 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (SubGrp `  W ) )
15 eqid 2467 . . . . . 6  |-  ( LSSum `  W )  =  (
LSSum `  W )
1615lsmub1 16469 . . . . 5  |-  ( ( ( N `  { X } )  e.  (SubGrp `  W )  /\  ( N `  { Y } )  e.  (SubGrp `  W ) )  -> 
( N `  { X } )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) )
1710, 14, 16syl2anc 661 . . . 4  |-  ( ph  ->  ( N `  { X } )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) )
182, 15lsmcl 17509 . . . . . 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 1228 . . . . 5  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) )  e.  ( LSubSp `  W )
)
206, 7lspsnid 17419 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
211, 5, 20syl2anc 661 . . . . . 6  |-  ( ph  ->  X  e.  ( N `
 { X }
) )
226, 7lspsnid 17419 . . . . . . 7  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  Y  e.  ( N `  { Y } ) )
231, 11, 22syl2anc 661 . . . . . 6  |-  ( ph  ->  Y  e.  ( N `
 { Y }
) )
24 lspprabs.p . . . . . . 7  |-  .+  =  ( +g  `  W )
2524, 15lsmelvali 16463 . . . . . 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 1229 . . . . 5  |-  ( ph  ->  ( X  .+  Y
)  e.  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) )
272, 7, 1, 19, 26lspsnel5a 17422 . . . 4  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  C_  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) ) )
286, 24lmodvacl 17306 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .+  Y )  e.  V )
291, 5, 11, 28syl3anc 1228 . . . . . . 7  |-  ( ph  ->  ( X  .+  Y
)  e.  V )
306, 2, 7lspsncl 17403 . . . . . . 7  |-  ( ( W  e.  LMod  /\  ( X  .+  Y )  e.  V )  ->  ( N `  { ( X  .+  Y ) } )  e.  ( LSubSp `  W ) )
311, 29, 30syl2anc 661 . . . . . 6  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  e.  ( LSubSp `  W )
)
324, 31sseldd 3505 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  e.  (SubGrp `  W )
)
334, 19sseldd 3505 . . . . 5  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) )  e.  (SubGrp `  W )
)
3415lsmlub 16476 . . . . 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 1228 . . . 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 919 . . 3  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) )  C_  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) ) )
3715lsmub1 16469 . . . . 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 661 . . . 4  |-  ( ph  ->  ( N `  { X } )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) ) )
392, 15lsmcl 17509 . . . . . 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 1228 . . . . 5  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) )  e.  ( LSubSp `  W )
)
41 eqid 2467 . . . . . . 7  |-  ( -g `  W )  =  (
-g `  W )
426, 7lspsnid 17419 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  ( X  .+  Y )  e.  V )  ->  ( X  .+  Y )  e.  ( N `  {
( X  .+  Y
) } ) )
431, 29, 42syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( X  .+  Y
)  e.  ( N `
 { ( X 
.+  Y ) } ) )
4441, 15, 32, 10, 43, 21lsmelvalmi 16465 . . . . . 6  |-  ( ph  ->  ( ( X  .+  Y ) ( -g `  W ) X )  e.  ( ( N `
 { ( X 
.+  Y ) } ) ( LSSum `  W
) ( N `  { X } ) ) )
45 lmodabl 17337 . . . . . . . 8  |-  ( W  e.  LMod  ->  W  e. 
Abel )
461, 45syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  Abel )
476, 24, 41ablpncan2 16619 . . . . . . 7  |-  ( ( W  e.  Abel  /\  X  e.  V  /\  Y  e.  V )  ->  (
( X  .+  Y
) ( -g `  W
) X )  =  Y )
4846, 5, 11, 47syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( ( X  .+  Y ) ( -g `  W ) X )  =  Y )
4915lsmcom 16654 . . . . . . 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 1228 . . . . . 6  |-  ( ph  ->  ( ( N `  { ( X  .+  Y ) } ) ( LSSum `  W )
( N `  { X } ) )  =  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) ) )
5144, 48, 503eltr3d 2569 . . . . 5  |-  ( ph  ->  Y  e.  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) ) )
522, 7, 1, 40, 51lspsnel5a 17422 . . . 4  |-  ( ph  ->  ( N `  { Y } )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) ) )
534, 40sseldd 3505 . . . . 5  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) )  e.  (SubGrp `  W )
)
5415lsmlub 16476 . . . . 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 1228 . . . 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 919 . . 3  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) )  C_  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) ) )
5736, 56eqssd 3521 . 2  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) )  =  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) ) )
586, 7, 15, 1, 5, 29lsmpr 17515 . 2  |-  ( ph  ->  ( N `  { X ,  ( X  .+  Y ) } )  =  ( ( N `
 { X }
) ( LSSum `  W
) ( N `  { ( X  .+  Y ) } ) ) )
596, 7, 15, 1, 5, 11lsmpr 17515 . 2  |-  ( ph  ->  ( N `  { X ,  Y }
)  =  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) )
6057, 58, 593eqtr4d 2518 1  |-  ( ph  ->  ( N `  { X ,  ( X  .+  Y ) } )  =  ( N `  { X ,  Y }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3476   {csn 4027   {cpr 4029   ` cfv 5586  (class class class)co 6282   Basecbs 14483   +g cplusg 14548   -gcsg 15723  SubGrpcsubg 15987   LSSumclsm 16447   Abelcabl 16592   LModclmod 17292   LSubSpclss 17358   LSpanclspn 17397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-0g 14690  df-mnd 15725  df-submnd 15775  df-grp 15855  df-minusg 15856  df-sbg 15857  df-subg 15990  df-cntz 16147  df-lsm 16449  df-cmn 16593  df-abl 16594  df-mgp 16929  df-ur 16941  df-rng 16985  df-lmod 17294  df-lss 17359  df-lsp 17398
This theorem is referenced by:  lspabs2  17546  lspindp4  17563  mapdindp4  36520
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