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Theorem lsmspsn 16156
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 16056 . . . 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 16056 . . . 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 15283 . . 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 16079 . . . . . . . . 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 16079 . . . . . . . . 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 2861 . . . . . . 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 2730 . . . . . 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 2861 . . . . . 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 2875 . . . . . . 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 2746 . . 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 2871 . . 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 16077 . . . 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 16077 . . . 4  |-  ( (
ph  /\  ( j  e.  K  /\  k  e.  K ) )  -> 
( k  .x.  Y
)  e.  ( N `
 { Y }
) )
41 oveq1 6088 . . . . . 6  |-  ( v  =  ( j  .x.  X )  ->  (
v  .+  w )  =  ( ( j 
.x.  X )  .+  w ) )
4241eqeq2d 2447 . . . . 5  |-  ( v  =  ( j  .x.  X )  ->  ( U  =  ( v  .+  w )  <->  U  =  ( ( j  .x.  X )  .+  w
) ) )
43 oveq2 6089 . . . . . 6  |-  ( w  =  ( k  .x.  Y )  ->  (
( j  .x.  X
)  .+  w )  =  ( ( j 
.x.  X )  .+  ( k  .x.  Y
) ) )
4443eqeq2d 2447 . . . . 5  |-  ( w  =  ( k  .x.  Y )  ->  ( U  =  ( (
j  .x.  X )  .+  w )  <->  U  =  ( ( j  .x.  X )  .+  (
k  .x.  Y )
) ) )
4542, 44ceqsrex2v 3071 . . . 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 2746 . 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 1652    e. wcel 1725   E.wrex 2706   {csn 3814   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529  Scalarcsca 13532   .scvsca 13533  SubGrpcsubg 14938   LSSumclsm 15268   LModclmod 15950   LSpanclspn 16047
This theorem is referenced by:  lsppr  16165  baerlem3lem2  32508  baerlem5alem2  32509  baerlem5blem2  32510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-lsm 15270  df-mgp 15649  df-rng 15663  df-ur 15665  df-lmod 15952  df-lss 16009  df-lsp 16048
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