MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lsmspsn Unicode version

Theorem lsmspsn 16111
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 16011 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
61, 2, 5syl2anc 643 . . 3  |-  ( ph  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
7 lsmspsn.y . . . 4  |-  ( ph  ->  Y  e.  V )
83, 4lspsnsubg 16011 . . . 4  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (SubGrp `  W ) )
91, 7, 8syl2anc 643 . . 3  |-  ( ph  ->  ( N `  { Y } )  e.  (SubGrp `  W ) )
10 lsmspsn.a . . . 4  |-  .+  =  ( +g  `  W )
11 lsmspsn.p . . . 4  |-  .(+)  =  (
LSSum `  W )
1210, 11lsmelval 15238 . . 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 643 . 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 16034 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
v  e.  ( N `
 { X }
)  <->  E. j  e.  K  v  =  ( j  .x.  X ) ) )
181, 2, 17syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( v  e.  ( N `  { X } )  <->  E. j  e.  K  v  =  ( j  .x.  X
) ) )
1914, 15, 3, 16, 4lspsnel 16034 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  (
w  e.  ( N `
 { Y }
)  <->  E. k  e.  K  w  =  ( k  .x.  Y ) ) )
201, 7, 19syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( w  e.  ( N `  { Y } )  <->  E. k  e.  K  w  =  ( k  .x.  Y
) ) )
2118, 20anbi12d 692 . . . . . . 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 471 . . . . . 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 495 . . . . 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 2821 . . . . . . 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 2691 . . . . . 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 2821 . . . . . 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 2835 . . . . . . 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 677 . . . . . 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 264 . . . . 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 253 . . . 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 2707 . . 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 2831 . . 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 253 . 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 452 . . . . 5  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  ->  W  e.  LMod )
35 simprl 733 . . . . 5  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  -> 
j  e.  K )
362adantr 452 . . . . 5  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  ->  X  e.  V )
373, 16, 14, 15, 4, 34, 35, 36lspsneli 16032 . . . 4  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  -> 
( j  .x.  X
)  e.  ( N `
 { X }
) )
38 simprr 734 . . . . 5  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  -> 
k  e.  K )
397adantr 452 . . . . 5  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  ->  Y  e.  V )
403, 16, 14, 15, 4, 34, 38, 39lspsneli 16032 . . . 4  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  -> 
( k  .x.  Y
)  e.  ( N `
 { Y }
) )
41 oveq1 6047 . . . . . 6  |-  ( v  =  ( j  .x.  X )  ->  (
v  .+  w )  =  ( ( j 
.x.  X )  .+  w ) )
4241eqeq2d 2415 . . . . 5  |-  ( v  =  ( j  .x.  X )  ->  ( U  =  ( v  .+  w )  <->  U  =  ( ( j  .x.  X )  .+  w
) ) )
43 oveq2 6048 . . . . . 6  |-  ( w  =  ( k  .x.  Y )  ->  (
( j  .x.  X
)  .+  w )  =  ( ( j 
.x.  X )  .+  ( k  .x.  Y
) ) )
4443eqeq2d 2415 . . . . 5  |-  ( w  =  ( k  .x.  Y )  ->  ( U  =  ( (
j  .x.  X )  .+  w )  <->  U  =  ( ( j  .x.  X )  .+  (
k  .x.  Y )
) ) )
4542, 44ceqsrex2v 3031 . . . 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 643 . . 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 2707 . 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 271 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667   {csn 3774   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484  Scalarcsca 13487   .scvsca 13488  SubGrpcsubg 14893   LSSumclsm 15223   LModclmod 15905   LSpanclspn 16002
This theorem is referenced by:  lsppr  16120  baerlem3lem2  32193  baerlem5alem2  32194  baerlem5blem2  32195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-lsm 15225  df-mgp 15604  df-rng 15618  df-ur 15620  df-lmod 15907  df-lss 15964  df-lsp 16003
  Copyright terms: Public domain W3C validator