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Theorem lsmspsn 15853
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 15753 . . . 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 15753 . . . 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 14976 . . 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 15776 . . . . . . . . 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 15776 . . . . . . . . 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 2706 . . . . . . 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 2581 . . . . . 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 2706 . . . . . 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 2720 . . . . . . 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 2597 . . 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 2716 . . 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 15774 . . . 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 15774 . . . 4  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  -> 
( k  .x.  Y
)  e.  ( N `
 { Y }
) )
41 oveq1 5881 . . . . . 6  |-  ( v  =  ( j  .x.  X )  ->  (
v  .+  w )  =  ( ( j 
.x.  X )  .+  w ) )
4241eqeq2d 2307 . . . . 5  |-  ( v  =  ( j  .x.  X )  ->  ( U  =  ( v  .+  w )  <->  U  =  ( ( j  .x.  X )  .+  w
) ) )
43 oveq2 5882 . . . . . 6  |-  ( w  =  ( k  .x.  Y )  ->  (
( j  .x.  X
)  .+  w )  =  ( ( j 
.x.  X )  .+  ( k  .x.  Y
) ) )
4443eqeq2d 2307 . . . . 5  |-  ( w  =  ( k  .x.  Y )  ->  ( U  =  ( (
j  .x.  X )  .+  w )  <->  U  =  ( ( j  .x.  X )  .+  (
k  .x.  Y )
) ) )
4542, 44ceqsrex2v 2916 . . . 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 2597 . 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 1632    e. wcel 1696   E.wrex 2557   {csn 3653   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   .scvsca 13228  SubGrpcsubg 14631   LSSumclsm 14961   LModclmod 15643   LSpanclspn 15744
This theorem is referenced by:  lsppr  15862  baerlem3lem2  32522  baerlem5alem2  32523  baerlem5blem2  32524
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-lsm 14963  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lss 15706  df-lsp 15745
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