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Theorem lsmsatcv 28450
Description: Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 22192 analog.) Explicit atom version of lsmcv 15857. (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
StepHypRef Expression
1 lsmsatcv.w . . . 4  |-  ( ph  ->  W  e.  LVec )
2 lsmsatcv.x . . . 4  |-  ( ph  ->  Q  e.  A )
3 eqid 2258 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2258 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 lsmsatcv.a . . . . 5  |-  A  =  (LSAtoms `  W )
63, 4, 5islsati 28434 . . . 4  |-  ( ( W  e.  LVec  /\  Q  e.  A )  ->  E. v  e.  ( Base `  W
) Q  =  ( ( LSpan `  W ) `  { v } ) )
71, 2, 6syl2anc 645 . . 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 453 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  W  e.  LVec )
11 lsmsatcv.t . . . . . . . . 9  |-  ( ph  ->  T  e.  S )
1211adantr 453 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  T  e.  S )
13 lsmsatcv.u . . . . . . . . 9  |-  ( ph  ->  U  e.  S )
1413adantr 453 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  U  e.  S )
15 simpr 449 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  v  e.  ( Base `  W )
)
163, 8, 4, 9, 10, 12, 14, 15lsmcv 15857 . . . . . . 7  |-  ( ( ( ph  /\  v  e.  ( Base `  W
) )  /\  T  C.  U  /\  U  C_  ( T  .(+)  ( (
LSpan `  W ) `  { v } ) ) )  ->  U  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) )
17163expib 1159 . . . . . 6  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  ( ( T  C.  U  /\  U  C_  ( T  .(+)  ( (
LSpan `  W ) `  { v } ) ) )  ->  U  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) ) )
18173adant3 980 . . . . 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 5800 . . . . . . . . 9  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( T  .(+)  Q )  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) )
2019sseq2d 3181 . . . . . . . 8  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( U  C_  ( T  .(+)  Q )  <->  U  C_  ( T  .(+)  ( ( LSpan `  W ) `  {
v } ) ) ) )
2120anbi2d 687 . . . . . . 7  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( ( T  C.  U  /\  U  C_  ( T  .(+)  Q ) )  <-> 
( T  C.  U  /\  U  C_  ( T 
.(+)  ( ( LSpan `  W ) `  {
v } ) ) ) ) )
2219eqeq2d 2269 . . . . . . 7  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( U  =  ( T  .(+)  Q )  <->  U  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) ) )
2321, 22imbi12d 313 . . . . . 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 983 . . . . 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 225 . . . 4  |-  ( (
ph  /\  v  e.  ( Base `  W )  /\  Q  =  (
( LSpan `  W ) `  { v } ) )  ->  ( ( T  C.  U  /\  U  C_  ( T  .(+)  Q ) )  ->  U  =  ( T  .(+)  Q ) ) )
2625rexlimdv3a 2644 . . 3  |-  ( ph  ->  ( E. v  e.  ( Base `  W
) Q  =  ( ( LSpan `  W ) `  { v } )  ->  ( ( T 
C.  U  /\  U  C_  ( T  .(+)  Q ) )  ->  U  =  ( T  .(+)  Q ) ) ) )
277, 26mpd 16 . 2  |-  ( ph  ->  ( ( T  C.  U  /\  U  C_  ( T  .(+)  Q ) )  ->  U  =  ( T  .(+)  Q )
) )
28273impib 1154 1  |-  ( (
ph  /\  T  C.  U  /\  U  C_  ( T  .(+)  Q ) )  ->  U  =  ( T  .(+)  Q )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   E.wrex 2519    C_ wss 3127    C. wpss 3128   {csn 3614   ` cfv 4673  (class class class)co 5792   Basecbs 13111   LSSumclsm 14908   LSubSpclss 15652   LSpanclspn 15691   LVecclvec 15818  LSAtomsclsa 28414
This theorem is referenced by:  dochsat  30823
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-tpos 6168  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-2 9772  df-3 9773  df-ndx 13114  df-slot 13115  df-base 13116  df-sets 13117  df-ress 13118  df-plusg 13184  df-mulr 13185  df-0g 13367  df-mnd 14330  df-submnd 14379  df-grp 14452  df-minusg 14453  df-sbg 14454  df-subg 14581  df-lsm 14910  df-cmn 15054  df-abl 15055  df-mgp 15289  df-ring 15303  df-ur 15305  df-oppr 15368  df-dvdsr 15386  df-unit 15387  df-invr 15417  df-drng 15477  df-lmod 15592  df-lss 15653  df-lsp 15692  df-lvec 15819  df-lsatoms 28416
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