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

Theorem lsmspsn 15837
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 15737 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
61, 2, 5syl2anc 642 . . 3  |-  ( ph  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
7 lsmspsn.y . . . 4  |-  ( ph  ->  Y  e.  V )
83, 4lspsnsubg 15737 . . . 4  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (SubGrp `  W ) )
91, 7, 8syl2anc 642 . . 3  |-  ( ph  ->  ( N `  { Y } )  e.  (SubGrp `  W ) )
10 lsmspsn.a . . . 4  |-  .+  =  ( +g  `  W )
11 lsmspsn.p . . . 4  |-  .(+)  =  (
LSSum `  W )
1210, 11lsmelval 14960 . . 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 642 . 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 15760 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
v  e.  ( N `
 { X }
)  <->  E. j  e.  K  v  =  ( j  .x.  X ) ) )
181, 2, 17syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( v  e.  ( N `  { X } )  <->  E. j  e.  K  v  =  ( j  .x.  X
) ) )
1914, 15, 3, 16, 4lspsnel 15760 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  (
w  e.  ( N `
 { Y }
)  <->  E. k  e.  K  w  =  ( k  .x.  Y ) ) )
201, 7, 19syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( w  e.  ( N `  { Y } )  <->  E. k  e.  K  w  =  ( k  .x.  Y
) ) )
2118, 20anbi12d 691 . . . . . . 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 470 . . . . . 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 494 . . . . 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 2693 . . . . . . 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 2568 . . . . . 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 2693 . . . . . 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 2707 . . . . . . 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 676 . . . . . 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 263 . . . . 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 252 . . . 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 2584 . . 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 2703 . . 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 252 . 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 451 . . . . 5  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  ->  W  e.  LMod )
35 simprl 732 . . . . 5  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  -> 
j  e.  K )
362adantr 451 . . . . 5  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  ->  X  e.  V )
373, 16, 14, 15, 4, 34, 35, 36lspsneli 15758 . . . 4  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  -> 
( j  .x.  X
)  e.  ( N `
 { X }
) )
38 simprr 733 . . . . 5  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  -> 
k  e.  K )
397adantr 451 . . . . 5  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  ->  Y  e.  V )
403, 16, 14, 15, 4, 34, 38, 39lspsneli 15758 . . . 4  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  -> 
( k  .x.  Y
)  e.  ( N `
 { Y }
) )
41 oveq1 5865 . . . . . 6  |-  ( v  =  ( j  .x.  X )  ->  (
v  .+  w )  =  ( ( j 
.x.  X )  .+  w ) )
4241eqeq2d 2294 . . . . 5  |-  ( v  =  ( j  .x.  X )  ->  ( U  =  ( v  .+  w )  <->  U  =  ( ( j  .x.  X )  .+  w
) ) )
43 oveq2 5866 . . . . . 6  |-  ( w  =  ( k  .x.  Y )  ->  (
( j  .x.  X
)  .+  w )  =  ( ( j 
.x.  X )  .+  ( k  .x.  Y
) ) )
4443eqeq2d 2294 . . . . 5  |-  ( w  =  ( k  .x.  Y )  ->  ( U  =  ( (
j  .x.  X )  .+  w )  <->  U  =  ( ( j  .x.  X )  .+  (
k  .x.  Y )
) ) )
4542, 44ceqsrex2v 2903 . . . 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 642 . . 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 2584 . 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 270 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212  SubGrpcsubg 14615   LSSumclsm 14945   LModclmod 15627   LSpanclspn 15728
This theorem is referenced by:  lsppr  15846  baerlem3lem2  31900  baerlem5alem2  31901  baerlem5blem2  31902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-lsm 14947  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lss 15690  df-lsp 15729
  Copyright terms: Public domain W3C validator