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Theorem lsmspsn 17143
Description: Member of subspace sum of spans of singletons. (Contributed by NM, 8-Apr-2015.)
Hypotheses
Ref Expression
lsmspsn.v  |-  V  =  ( Base `  W
)
lsmspsn.a  |-  .+  =  ( +g  `  W )
lsmspsn.f  |-  F  =  (Scalar `  W )
lsmspsn.k  |-  K  =  ( Base `  F
)
lsmspsn.t  |-  .x.  =  ( .s `  W )
lsmspsn.p  |-  .(+)  =  (
LSSum `  W )
lsmspsn.n  |-  N  =  ( LSpan `  W )
lsmspsn.w  |-  ( ph  ->  W  e.  LMod )
lsmspsn.x  |-  ( ph  ->  X  e.  V )
lsmspsn.y  |-  ( ph  ->  Y  e.  V )
Assertion
Ref Expression
lsmspsn  |-  ( ph  ->  ( U  e.  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  <->  E. j  e.  K  E. k  e.  K  U  =  ( ( j  .x.  X )  .+  (
k  .x.  Y )
) ) )
Distinct variable groups:    j, k,  .+    j, F, k    j, K, k    j, N, k    .x. , j, k    U, j, k    j, V, k   
j, W, k    j, X, k    j, Y, k    ph, j, k
Allowed substitution hints:    .(+) ( j, k)

Proof of Theorem lsmspsn
Dummy variables  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lsmspsn.w . . . 4  |-  ( ph  ->  W  e.  LMod )
2 lsmspsn.x . . . 4  |-  ( ph  ->  X  e.  V )
3 lsmspsn.v . . . . 5  |-  V  =  ( Base `  W
)
4 lsmspsn.n . . . . 5  |-  N  =  ( LSpan `  W )
53, 4lspsnsubg 17039 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
61, 2, 5syl2anc 656 . . 3  |-  ( ph  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
7 lsmspsn.y . . . 4  |-  ( ph  ->  Y  e.  V )
83, 4lspsnsubg 17039 . . . 4  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (SubGrp `  W ) )
91, 7, 8syl2anc 656 . . 3  |-  ( ph  ->  ( N `  { Y } )  e.  (SubGrp `  W ) )
10 lsmspsn.a . . . 4  |-  .+  =  ( +g  `  W )
11 lsmspsn.p . . . 4  |-  .(+)  =  (
LSSum `  W )
1210, 11lsmelval 16141 . . 3  |-  ( ( ( N `  { X } )  e.  (SubGrp `  W )  /\  ( N `  { Y } )  e.  (SubGrp `  W ) )  -> 
( U  e.  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  <->  E. v  e.  ( N `  { X } ) E. w  e.  ( N `  { Y } ) U  =  ( v  .+  w
) ) )
136, 9, 12syl2anc 656 . 2  |-  ( ph  ->  ( U  e.  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  <->  E. v  e.  ( N `  { X } ) E. w  e.  ( N `  { Y } ) U  =  ( v  .+  w
) ) )
14 lsmspsn.f . . . . . . . . . 10  |-  F  =  (Scalar `  W )
15 lsmspsn.k . . . . . . . . . 10  |-  K  =  ( Base `  F
)
16 lsmspsn.t . . . . . . . . . 10  |-  .x.  =  ( .s `  W )
1714, 15, 3, 16, 4lspsnel 17062 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
v  e.  ( N `
 { X }
)  <->  E. j  e.  K  v  =  ( j  .x.  X ) ) )
181, 2, 17syl2anc 656 . . . . . . . 8  |-  ( ph  ->  ( v  e.  ( N `  { X } )  <->  E. j  e.  K  v  =  ( j  .x.  X
) ) )
1914, 15, 3, 16, 4lspsnel 17062 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  (
w  e.  ( N `
 { Y }
)  <->  E. k  e.  K  w  =  ( k  .x.  Y ) ) )
201, 7, 19syl2anc 656 . . . . . . . 8  |-  ( ph  ->  ( w  e.  ( N `  { Y } )  <->  E. k  e.  K  w  =  ( k  .x.  Y
) ) )
2118, 20anbi12d 705 . . . . . . 7  |-  ( ph  ->  ( ( v  e.  ( N `  { X } )  /\  w  e.  ( N `  { Y } ) )  <->  ( E. j  e.  K  v  =  ( j  .x.  X )  /\  E. k  e.  K  w  =  ( k  .x.  Y ) ) ) )
2221biimpa 481 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( N `  { X } )  /\  w  e.  ( N `  { Y } ) ) )  ->  ( E. j  e.  K  v  =  ( j  .x.  X
)  /\  E. k  e.  K  w  =  ( k  .x.  Y
) ) )
2322biantrurd 505 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( N `  { X } )  /\  w  e.  ( N `  { Y } ) ) )  ->  ( U  =  ( v  .+  w
)  <->  ( ( E. j  e.  K  v  =  ( j  .x.  X )  /\  E. k  e.  K  w  =  ( k  .x.  Y ) )  /\  U  =  ( v  .+  w ) ) ) )
24 r19.41v 2871 . . . . . . 7  |-  ( E. k  e.  K  ( ( v  =  ( j  .x.  X )  /\  w  =  ( k  .x.  Y ) )  /\  U  =  ( v  .+  w
) )  <->  ( E. k  e.  K  (
v  =  ( j 
.x.  X )  /\  w  =  ( k  .x.  Y ) )  /\  U  =  ( v  .+  w ) ) )
2524rexbii 2738 . . . . . 6  |-  ( E. j  e.  K  E. k  e.  K  (
( v  =  ( j  .x.  X )  /\  w  =  ( k  .x.  Y ) )  /\  U  =  ( v  .+  w
) )  <->  E. j  e.  K  ( E. k  e.  K  (
v  =  ( j 
.x.  X )  /\  w  =  ( k  .x.  Y ) )  /\  U  =  ( v  .+  w ) ) )
26 r19.41v 2871 . . . . . 6  |-  ( E. j  e.  K  ( E. k  e.  K  ( v  =  ( j  .x.  X )  /\  w  =  ( k  .x.  Y ) )  /\  U  =  ( v  .+  w
) )  <->  ( E. j  e.  K  E. k  e.  K  (
v  =  ( j 
.x.  X )  /\  w  =  ( k  .x.  Y ) )  /\  U  =  ( v  .+  w ) ) )
27 reeanv 2886 . . . . . . 7  |-  ( E. j  e.  K  E. k  e.  K  (
v  =  ( j 
.x.  X )  /\  w  =  ( k  .x.  Y ) )  <->  ( E. j  e.  K  v  =  ( j  .x.  X )  /\  E. k  e.  K  w  =  ( k  .x.  Y ) ) )
2827anbi1i 690 . . . . . 6  |-  ( ( E. j  e.  K  E. k  e.  K  ( v  =  ( j  .x.  X )  /\  w  =  ( k  .x.  Y ) )  /\  U  =  ( v  .+  w
) )  <->  ( ( E. j  e.  K  v  =  ( j  .x.  X )  /\  E. k  e.  K  w  =  ( k  .x.  Y ) )  /\  U  =  ( v  .+  w ) ) )
2925, 26, 283bitrri 272 . . . . 5  |-  ( ( ( E. j  e.  K  v  =  ( j  .x.  X )  /\  E. k  e.  K  w  =  ( k  .x.  Y ) )  /\  U  =  ( v  .+  w
) )  <->  E. j  e.  K  E. k  e.  K  ( (
v  =  ( j 
.x.  X )  /\  w  =  ( k  .x.  Y ) )  /\  U  =  ( v  .+  w ) ) )
3023, 29syl6bb 261 . . . 4  |-  ( (
ph  /\  ( v  e.  ( N `  { X } )  /\  w  e.  ( N `  { Y } ) ) )  ->  ( U  =  ( v  .+  w
)  <->  E. j  e.  K  E. k  e.  K  ( ( v  =  ( j  .x.  X
)  /\  w  =  ( k  .x.  Y
) )  /\  U  =  ( v  .+  w ) ) ) )
31302rexbidva 2754 . . 3  |-  ( ph  ->  ( E. v  e.  ( N `  { X } ) E. w  e.  ( N `  { Y } ) U  =  ( v  .+  w
)  <->  E. v  e.  ( N `  { X } ) E. w  e.  ( N `  { Y } ) E. j  e.  K  E. k  e.  K  ( (
v  =  ( j 
.x.  X )  /\  w  =  ( k  .x.  Y ) )  /\  U  =  ( v  .+  w ) ) ) )
32 rexrot4 2882 . . 3  |-  ( E. v  e.  ( N `
 { X }
) E. w  e.  ( N `  { Y } ) E. j  e.  K  E. k  e.  K  ( (
v  =  ( j 
.x.  X )  /\  w  =  ( k  .x.  Y ) )  /\  U  =  ( v  .+  w ) )  <->  E. j  e.  K  E. k  e.  K  E. v  e.  ( N `  { X } ) E. w  e.  ( N `  { Y } ) ( ( v  =  ( j 
.x.  X )  /\  w  =  ( k  .x.  Y ) )  /\  U  =  ( v  .+  w ) ) )
3331, 32syl6bb 261 . 2  |-  ( ph  ->  ( E. v  e.  ( N `  { X } ) E. w  e.  ( N `  { Y } ) U  =  ( v  .+  w
)  <->  E. j  e.  K  E. k  e.  K  E. v  e.  ( N `  { X } ) E. w  e.  ( N `  { Y } ) ( ( v  =  ( j 
.x.  X )  /\  w  =  ( k  .x.  Y ) )  /\  U  =  ( v  .+  w ) ) ) )
341adantr 462 . . . . 5  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  ->  W  e.  LMod )
35 simprl 750 . . . . 5  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  -> 
j  e.  K )
362adantr 462 . . . . 5  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  ->  X  e.  V )
373, 16, 14, 15, 4, 34, 35, 36lspsneli 17060 . . . 4  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  -> 
( j  .x.  X
)  e.  ( N `
 { X }
) )
38 simprr 751 . . . . 5  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  -> 
k  e.  K )
397adantr 462 . . . . 5  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  ->  Y  e.  V )
403, 16, 14, 15, 4, 34, 38, 39lspsneli 17060 . . . 4  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  -> 
( k  .x.  Y
)  e.  ( N `
 { Y }
) )
41 oveq1 6097 . . . . . 6  |-  ( v  =  ( j  .x.  X )  ->  (
v  .+  w )  =  ( ( j 
.x.  X )  .+  w ) )
4241eqeq2d 2452 . . . . 5  |-  ( v  =  ( j  .x.  X )  ->  ( U  =  ( v  .+  w )  <->  U  =  ( ( j  .x.  X )  .+  w
) ) )
43 oveq2 6098 . . . . . 6  |-  ( w  =  ( k  .x.  Y )  ->  (
( j  .x.  X
)  .+  w )  =  ( ( j 
.x.  X )  .+  ( k  .x.  Y
) ) )
4443eqeq2d 2452 . . . . 5  |-  ( w  =  ( k  .x.  Y )  ->  ( U  =  ( (
j  .x.  X )  .+  w )  <->  U  =  ( ( j  .x.  X )  .+  (
k  .x.  Y )
) ) )
4542, 44ceqsrex2v 3092 . . . 4  |-  ( ( ( j  .x.  X
)  e.  ( N `
 { X }
)  /\  ( k  .x.  Y )  e.  ( N `  { Y } ) )  -> 
( E. v  e.  ( N `  { X } ) E. w  e.  ( N `  { Y } ) ( ( v  =  ( j 
.x.  X )  /\  w  =  ( k  .x.  Y ) )  /\  U  =  ( v  .+  w ) )  <->  U  =  ( ( j  .x.  X )  .+  (
k  .x.  Y )
) ) )
4637, 40, 45syl2anc 656 . . 3  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  -> 
( E. v  e.  ( N `  { X } ) E. w  e.  ( N `  { Y } ) ( ( v  =  ( j 
.x.  X )  /\  w  =  ( k  .x.  Y ) )  /\  U  =  ( v  .+  w ) )  <->  U  =  ( ( j  .x.  X )  .+  (
k  .x.  Y )
) ) )
47462rexbidva 2754 . 2  |-  ( ph  ->  ( E. j  e.  K  E. k  e.  K  E. v  e.  ( N `  { X } ) E. w  e.  ( N `  { Y } ) ( ( v  =  ( j 
.x.  X )  /\  w  =  ( k  .x.  Y ) )  /\  U  =  ( v  .+  w ) )  <->  E. j  e.  K  E. k  e.  K  U  =  ( ( j  .x.  X )  .+  (
k  .x.  Y )
) ) )
4813, 33, 473bitrd 279 1  |-  ( ph  ->  ( U  e.  ( ( N `  { X } )  .(+)  ( N `
 { Y }
) )  <->  E. j  e.  K  E. k  e.  K  U  =  ( ( j  .x.  X )  .+  (
k  .x.  Y )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   E.wrex 2714   {csn 3874   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234  Scalarcsca 14237   .scvsca 14238  SubGrpcsubg 15668   LSSumclsm 16126   LModclmod 16928   LSpanclspn 17030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-sbg 15540  df-subg 15671  df-lsm 16128  df-mgp 16582  df-ur 16594  df-rng 16637  df-lmod 16930  df-lss 16992  df-lsp 17031
This theorem is referenced by:  lsppr  17152  baerlem3lem2  35077  baerlem5alem2  35078  baerlem5blem2  35079
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