Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lsmsatcv Unicode version

Theorem lsmsatcv 27889
Description: Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 22079 analog.) Explicit atom version of lsmcv 15729. (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 2253 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2253 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 lsmsatcv.a . . . . 5  |-  A  =  (LSAtoms `  W )
63, 4, 5islsati 27873 . . . 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 15729 . . . . . . 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 5718 . . . . . . . . 9  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( T  .(+)  Q )  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) )
2019sseq2d 3127 . . . . . . . 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 2264 . . . . . . 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 2631 . . 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 2510    C_ wss 3078    C. wpss 3079   {csn 3544   ` cfv 4592  (class class class)co 5710   Basecbs 13022   LSSumclsm 14780   LSubSpclss 15524   LSpanclspn 15563   LVecclvec 15690  LSAtomsclsa 27853
This theorem is referenced by:  dochsat  30262
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-tpos 6086  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-2 9684  df-3 9685  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-0g 13278  df-mnd 14202  df-submnd 14251  df-grp 14324  df-minusg 14325  df-sbg 14326  df-subg 14453  df-lsm 14782  df-cmn 14926  df-abl 14927  df-mgp 15161  df-ring 15175  df-ur 15177  df-oppr 15240  df-dvdsr 15258  df-unit 15259  df-invr 15289  df-drng 15349  df-lmod 15464  df-lss 15525  df-lsp 15564  df-lvec 15691  df-lsatoms 27855
  Copyright terms: Public domain W3C validator