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Theorem lsmsatcv 29493
Description: Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 23107 analog.) Explicit atom version of lsmcv 16168. (Contributed by NM, 29-Oct-2014.)
Hypotheses
Ref Expression
lsmsatcv.s  |-  S  =  ( LSubSp `  W )
lsmsatcv.p  |-  .(+)  =  (
LSSum `  W )
lsmsatcv.a  |-  A  =  (LSAtoms `  W )
lsmsatcv.w  |-  ( ph  ->  W  e.  LVec )
lsmsatcv.t  |-  ( ph  ->  T  e.  S )
lsmsatcv.u  |-  ( ph  ->  U  e.  S )
lsmsatcv.x  |-  ( ph  ->  Q  e.  A )
Assertion
Ref Expression
lsmsatcv  |-  ( (
ph  /\  T  C.  U  /\  U  C_  ( T  .(+)  Q ) )  ->  U  =  ( T  .(+)  Q )
)

Proof of Theorem lsmsatcv
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 lsmsatcv.w . . . 4  |-  ( ph  ->  W  e.  LVec )
2 lsmsatcv.x . . . 4  |-  ( ph  ->  Q  e.  A )
3 eqid 2404 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2404 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 lsmsatcv.a . . . . 5  |-  A  =  (LSAtoms `  W )
63, 4, 5islsati 29477 . . . 4  |-  ( ( W  e.  LVec  /\  Q  e.  A )  ->  E. v  e.  ( Base `  W
) Q  =  ( ( LSpan `  W ) `  { v } ) )
71, 2, 6syl2anc 643 . . 3  |-  ( ph  ->  E. v  e.  (
Base `  W ) Q  =  ( ( LSpan `  W ) `  { v } ) )
8 lsmsatcv.s . . . . . . . 8  |-  S  =  ( LSubSp `  W )
9 lsmsatcv.p . . . . . . . 8  |-  .(+)  =  (
LSSum `  W )
101adantr 452 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  W  e.  LVec )
11 lsmsatcv.t . . . . . . . . 9  |-  ( ph  ->  T  e.  S )
1211adantr 452 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  T  e.  S )
13 lsmsatcv.u . . . . . . . . 9  |-  ( ph  ->  U  e.  S )
1413adantr 452 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  U  e.  S )
15 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  v  e.  ( Base `  W )
)
163, 8, 4, 9, 10, 12, 14, 15lsmcv 16168 . . . . . . 7  |-  ( ( ( ph  /\  v  e.  ( Base `  W
) )  /\  T  C.  U  /\  U  C_  ( T  .(+)  ( (
LSpan `  W ) `  { v } ) ) )  ->  U  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) )
17163expib 1156 . . . . . 6  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  ( ( T  C.  U  /\  U  C_  ( T  .(+)  ( (
LSpan `  W ) `  { v } ) ) )  ->  U  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) ) )
18173adant3 977 . . . . 5  |-  ( (
ph  /\  v  e.  ( Base `  W )  /\  Q  =  (
( LSpan `  W ) `  { v } ) )  ->  ( ( T  C.  U  /\  U  C_  ( T  .(+)  ( (
LSpan `  W ) `  { v } ) ) )  ->  U  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) ) )
19 oveq2 6048 . . . . . . . . 9  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( T  .(+)  Q )  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) )
2019sseq2d 3336 . . . . . . . 8  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( U  C_  ( T  .(+)  Q )  <->  U  C_  ( T  .(+)  ( ( LSpan `  W ) `  {
v } ) ) ) )
2120anbi2d 685 . . . . . . 7  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( ( T  C.  U  /\  U  C_  ( T  .(+)  Q ) )  <-> 
( T  C.  U  /\  U  C_  ( T 
.(+)  ( ( LSpan `  W ) `  {
v } ) ) ) ) )
2219eqeq2d 2415 . . . . . . 7  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( U  =  ( T  .(+)  Q )  <->  U  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) ) )
2321, 22imbi12d 312 . . . . . 6  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( ( ( T 
C.  U  /\  U  C_  ( T  .(+)  Q ) )  ->  U  =  ( T  .(+)  Q ) )  <->  ( ( T 
C.  U  /\  U  C_  ( T  .(+)  ( (
LSpan `  W ) `  { v } ) ) )  ->  U  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) ) ) )
24233ad2ant3 980 . . . . 5  |-  ( (
ph  /\  v  e.  ( Base `  W )  /\  Q  =  (
( LSpan `  W ) `  { v } ) )  ->  ( (
( T  C.  U  /\  U  C_  ( T 
.(+)  Q ) )  ->  U  =  ( T  .(+) 
Q ) )  <->  ( ( T  C.  U  /\  U  C_  ( T  .(+)  ( (
LSpan `  W ) `  { v } ) ) )  ->  U  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) ) ) )
2518, 24mpbird 224 . . . 4  |-  ( (
ph  /\  v  e.  ( Base `  W )  /\  Q  =  (
( LSpan `  W ) `  { v } ) )  ->  ( ( T  C.  U  /\  U  C_  ( T  .(+)  Q ) )  ->  U  =  ( T  .(+)  Q ) ) )
2625rexlimdv3a 2792 . . 3  |-  ( ph  ->  ( E. v  e.  ( Base `  W
) Q  =  ( ( LSpan `  W ) `  { v } )  ->  ( ( T 
C.  U  /\  U  C_  ( T  .(+)  Q ) )  ->  U  =  ( T  .(+)  Q ) ) ) )
277, 26mpd 15 . 2  |-  ( ph  ->  ( ( T  C.  U  /\  U  C_  ( T  .(+)  Q ) )  ->  U  =  ( T  .(+)  Q )
) )
28273impib 1151 1  |-  ( (
ph  /\  T  C.  U  /\  U  C_  ( T  .(+)  Q ) )  ->  U  =  ( T  .(+)  Q )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2667    C_ wss 3280    C. wpss 3281   {csn 3774   ` cfv 5413  (class class class)co 6040   Basecbs 13424   LSSumclsm 15223   LSubSpclss 15963   LSpanclspn 16002   LVecclvec 16129  LSAtomsclsa 29457
This theorem is referenced by:  dochsat  31866
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-tpos 6438  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-3 10015  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-0g 13682  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-lsm 15225  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-drng 15792  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lvec 16130  df-lsatoms 29459
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