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Theorem lspprabs 17181
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 2443 . . . . . . . 8  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
32lsssssubg 17044 . . . . . . 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 17063 . . . . . . 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 3362 . . . . 5  |-  ( ph  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
11 lspprabs.y . . . . . . 7  |-  ( ph  ->  Y  e.  V )
126, 2, 7lspsncl 17063 . . . . . . 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 3362 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (SubGrp `  W ) )
15 eqid 2443 . . . . . 6  |-  ( LSSum `  W )  =  (
LSSum `  W )
1615lsmub1 16160 . . . . 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 17169 . . . . . 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 1218 . . . . 5  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) )  e.  ( LSubSp `  W )
)
206, 7lspsnid 17079 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
211, 5, 20syl2anc 661 . . . . . 6  |-  ( ph  ->  X  e.  ( N `
 { X }
) )
226, 7lspsnid 17079 . . . . . . 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 16154 . . . . . 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 1219 . . . . 5  |-  ( ph  ->  ( X  .+  Y
)  e.  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) )
272, 7, 1, 19, 26lspsnel5a 17082 . . . 4  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  C_  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) ) )
286, 24lmodvacl 16967 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .+  Y )  e.  V )
291, 5, 11, 28syl3anc 1218 . . . . . . 7  |-  ( ph  ->  ( X  .+  Y
)  e.  V )
306, 2, 7lspsncl 17063 . . . . . . 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 3362 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  e.  (SubGrp `  W )
)
334, 19sseldd 3362 . . . . 5  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) )  e.  (SubGrp `  W )
)
3415lsmlub 16167 . . . . 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 1218 . . . 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 912 . . 3  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) )  C_  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) ) )
3715lsmub1 16160 . . . . 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 17169 . . . . . 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 1218 . . . . 5  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) )  e.  ( LSubSp `  W )
)
41 eqid 2443 . . . . . . 7  |-  ( -g `  W )  =  (
-g `  W )
426, 7lspsnid 17079 . . . . . . . 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 16156 . . . . . 6  |-  ( ph  ->  ( ( X  .+  Y ) ( -g `  W ) X )  e.  ( ( N `
 { ( X 
.+  Y ) } ) ( LSSum `  W
) ( N `  { X } ) ) )
45 lmodabl 16997 . . . . . . . 8  |-  ( W  e.  LMod  ->  W  e. 
Abel )
461, 45syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  Abel )
476, 24, 41ablpncan2 16310 . . . . . . 7  |-  ( ( W  e.  Abel  /\  X  e.  V  /\  Y  e.  V )  ->  (
( X  .+  Y
) ( -g `  W
) X )  =  Y )
4846, 5, 11, 47syl3anc 1218 . . . . . 6  |-  ( ph  ->  ( ( X  .+  Y ) ( -g `  W ) X )  =  Y )
4915lsmcom 16345 . . . . . . 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 1218 . . . . . 6  |-  ( ph  ->  ( ( N `  { ( X  .+  Y ) } ) ( LSSum `  W )
( N `  { X } ) )  =  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) ) )
5144, 48, 503eltr3d 2523 . . . . 5  |-  ( ph  ->  Y  e.  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) ) )
522, 7, 1, 40, 51lspsnel5a 17082 . . . 4  |-  ( ph  ->  ( N `  { Y } )  C_  (
( N `  { X } ) ( LSSum `  W ) ( N `
 { ( X 
.+  Y ) } ) ) )
534, 40sseldd 3362 . . . . 5  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) )  e.  (SubGrp `  W )
)
5415lsmlub 16167 . . . . 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 1218 . . . 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 912 . . 3  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) )  C_  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) ) )
5736, 56eqssd 3378 . 2  |-  ( ph  ->  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { ( X  .+  Y ) } ) )  =  ( ( N `  { X } ) (
LSSum `  W ) ( N `  { Y } ) ) )
586, 7, 15, 1, 5, 29lsmpr 17175 . 2  |-  ( ph  ->  ( N `  { X ,  ( X  .+  Y ) } )  =  ( ( N `
 { X }
) ( LSSum `  W
) ( N `  { ( X  .+  Y ) } ) ) )
596, 7, 15, 1, 5, 11lsmpr 17175 . 2  |-  ( ph  ->  ( N `  { X ,  Y }
)  =  ( ( N `  { X } ) ( LSSum `  W ) ( N `
 { Y }
) ) )
6057, 58, 593eqtr4d 2485 1  |-  ( ph  ->  ( N `  { X ,  ( X  .+  Y ) } )  =  ( N `  { X ,  Y }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3333   {csn 3882   {cpr 3884   ` cfv 5423  (class class class)co 6096   Basecbs 14179   +g cplusg 14243   -gcsg 15418  SubGrpcsubg 15680   LSSumclsm 16138   Abelcabel 16283   LModclmod 16953   LSubSpclss 17018   LSpanclspn 17057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-0g 14385  df-mnd 15420  df-submnd 15470  df-grp 15550  df-minusg 15551  df-sbg 15552  df-subg 15683  df-cntz 15840  df-lsm 16140  df-cmn 16284  df-abl 16285  df-mgp 16597  df-ur 16609  df-rng 16652  df-lmod 16955  df-lss 17019  df-lsp 17058
This theorem is referenced by:  lspabs2  17206  lspindp4  17223  mapdindp4  35373
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