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Theorem lsmsatcv 34213
Description: Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 26401 analog.) Explicit atom version of lsmcv 17656. (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 2467 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2467 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 lsmsatcv.a . . . . 5  |-  A  =  (LSAtoms `  W )
63, 4, 5islsati 34197 . . . 4  |-  ( ( W  e.  LVec  /\  Q  e.  A )  ->  E. v  e.  ( Base `  W
) Q  =  ( ( LSpan `  W ) `  { v } ) )
71, 2, 6syl2anc 661 . . 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 465 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  W  e.  LVec )
11 lsmsatcv.t . . . . . . . . 9  |-  ( ph  ->  T  e.  S )
1211adantr 465 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  T  e.  S )
13 lsmsatcv.u . . . . . . . . 9  |-  ( ph  ->  U  e.  S )
1413adantr 465 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  U  e.  S )
15 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  v  e.  ( Base `  W )
)
163, 8, 4, 9, 10, 12, 14, 15lsmcv 17656 . . . . . . 7  |-  ( ( ( ph  /\  v  e.  ( Base `  W
) )  /\  T  C.  U  /\  U  C_  ( T  .(+)  ( (
LSpan `  W ) `  { v } ) ) )  ->  U  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) )
17163expib 1199 . . . . . 6  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  ( ( T  C.  U  /\  U  C_  ( T  .(+)  ( (
LSpan `  W ) `  { v } ) ) )  ->  U  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) ) )
18173adant3 1016 . . . . 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 6303 . . . . . . . . 9  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( T  .(+)  Q )  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) )
2019sseq2d 3537 . . . . . . . 8  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( U  C_  ( T  .(+)  Q )  <->  U  C_  ( T  .(+)  ( ( LSpan `  W ) `  {
v } ) ) ) )
2120anbi2d 703 . . . . . . 7  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( ( T  C.  U  /\  U  C_  ( T  .(+)  Q ) )  <-> 
( T  C.  U  /\  U  C_  ( T 
.(+)  ( ( LSpan `  W ) `  {
v } ) ) ) ) )
2219eqeq2d 2481 . . . . . . 7  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( U  =  ( T  .(+)  Q )  <->  U  =  ( T  .(+)  ( ( LSpan `  W ) `  { v } ) ) ) )
2321, 22imbi12d 320 . . . . . 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 1019 . . . . 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 232 . . . 4  |-  ( (
ph  /\  v  e.  ( Base `  W )  /\  Q  =  (
( LSpan `  W ) `  { v } ) )  ->  ( ( T  C.  U  /\  U  C_  ( T  .(+)  Q ) )  ->  U  =  ( T  .(+)  Q ) ) )
2625rexlimdv3a 2961 . . 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 1194 1  |-  ( (
ph  /\  T  C.  U  /\  U  C_  ( T 
.(+)  Q ) )  ->  U  =  ( T  .(+) 
Q ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2818    C_ wss 3481    C. wpss 3482   {csn 4033   ` cfv 5594  (class class class)co 6295   Basecbs 14506   LSSumclsm 16525   LSubSpclss 17447   LSpanclspn 17486   LVecclvec 17617  LSAtomsclsa 34177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-0g 14713  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-submnd 15839  df-grp 15928  df-minusg 15929  df-sbg 15930  df-subg 16069  df-lsm 16527  df-cmn 16671  df-abl 16672  df-mgp 17012  df-ur 17024  df-ring 17070  df-oppr 17142  df-dvdsr 17160  df-unit 17161  df-invr 17191  df-drng 17267  df-lmod 17383  df-lss 17448  df-lsp 17487  df-lvec 17618  df-lsatoms 34179
This theorem is referenced by:  dochsat  36586
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