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Theorem lsmsatcv 29822
Description: Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 22247 analog.) Explicit atom version of lsmcv 15910. (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 2296 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2296 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 lsmsatcv.a . . . . 5  |-  A  =  (LSAtoms `  W )
63, 4, 5islsati 29806 . . . 4  |-  ( ( W  e.  LVec  /\  Q  e.  A )  ->  E. v  e.  ( Base `  W
) Q  =  ( ( LSpan `  W ) `  { v } ) )
71, 2, 6syl2anc 642 . . 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 451 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  W  e.  LVec )
11 lsmsatcv.t . . . . . . . . 9  |-  ( ph  ->  T  e.  S )
1211adantr 451 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  T  e.  S )
13 lsmsatcv.u . . . . . . . . 9  |-  ( ph  ->  U  e.  S )
1413adantr 451 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  U  e.  S )
15 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  v  e.  ( Base `  W )
)
163, 8, 4, 9, 10, 12, 14, 15lsmcv 15910 . . . . . . 7  |-  ( ( ( ph  /\  v  e.  ( Base `  W
) )  /\  T  C.  U  /\  U  C_  ( T  .(+)  ( (
LSpan `  W ) `  { v } ) ) )  ->  U  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) )
17163expib 1154 . . . . . 6  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  ( ( T  C.  U  /\  U  C_  ( T  .(+)  ( (
LSpan `  W ) `  { v } ) ) )  ->  U  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) ) )
18173adant3 975 . . . . 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 5882 . . . . . . . . 9  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( T  .(+)  Q )  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) )
2019sseq2d 3219 . . . . . . . 8  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( U  C_  ( T  .(+)  Q )  <->  U  C_  ( T  .(+)  ( ( LSpan `  W ) `  {
v } ) ) ) )
2120anbi2d 684 . . . . . . 7  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( ( T  C.  U  /\  U  C_  ( T  .(+)  Q ) )  <-> 
( T  C.  U  /\  U  C_  ( T 
.(+)  ( ( LSpan `  W ) `  {
v } ) ) ) ) )
2219eqeq2d 2307 . . . . . . 7  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( U  =  ( T  .(+)  Q )  <->  U  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) ) )
2321, 22imbi12d 311 . . . . . 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 978 . . . . 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 223 . . . 4  |-  ( (
ph  /\  v  e.  ( Base `  W )  /\  Q  =  (
( LSpan `  W ) `  { v } ) )  ->  ( ( T  C.  U  /\  U  C_  ( T  .(+)  Q ) )  ->  U  =  ( T  .(+)  Q ) ) )
2625rexlimdv3a 2682 . . 3  |-  ( ph  ->  ( E. v  e.  ( Base `  W
) Q  =  ( ( LSpan `  W ) `  { v } )  ->  ( ( T 
C.  U  /\  U  C_  ( T  .(+)  Q ) )  ->  U  =  ( T  .(+)  Q ) ) ) )
277, 26mpd 14 . 2  |-  ( ph  ->  ( ( T  C.  U  /\  U  C_  ( T  .(+)  Q ) )  ->  U  =  ( T  .(+)  Q )
) )
28273impib 1149 1  |-  ( (
ph  /\  T  C.  U  /\  U  C_  ( T  .(+)  Q ) )  ->  U  =  ( T  .(+)  Q )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165    C. wpss 3166   {csn 3653   ` cfv 5271  (class class class)co 5874   Basecbs 13164   LSSumclsm 14961   LSubSpclss 15705   LSpanclspn 15744   LVecclvec 15871  LSAtomsclsa 29786
This theorem is referenced by:  dochsat  32195
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-tpos 6250  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-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872  df-lsatoms 29788
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