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Theorem lsmcv2 29666
Description: Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 23784 analog.) (Contributed by NM, 10-Jan-2015.)
Hypotheses
Ref Expression
lsmcv2.v  |-  V  =  ( Base `  W
)
lsmcv2.s  |-  S  =  ( LSubSp `  W )
lsmcv2.n  |-  N  =  ( LSpan `  W )
lsmcv2.p  |-  .(+)  =  (
LSSum `  W )
lsmcv2.c  |-  C  =  (  <oLL  `  W )
lsmcv2.w  |-  ( ph  ->  W  e.  LVec )
lsmcv2.u  |-  ( ph  ->  U  e.  S )
lsmcv2.x  |-  ( ph  ->  X  e.  V )
lsmcv2.l  |-  ( ph  ->  -.  ( N `  { X } )  C_  U )
Assertion
Ref Expression
lsmcv2  |-  ( ph  ->  U C ( U 
.(+)  ( N `  { X } ) ) )

Proof of Theorem lsmcv2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lsmcv2.l . . 3  |-  ( ph  ->  -.  ( N `  { X } )  C_  U )
2 lsmcv2.p . . . 4  |-  .(+)  =  (
LSSum `  W )
3 lsmcv2.w . . . . . . 7  |-  ( ph  ->  W  e.  LVec )
4 lveclmod 16166 . . . . . . 7  |-  ( W  e.  LVec  ->  W  e. 
LMod )
53, 4syl 16 . . . . . 6  |-  ( ph  ->  W  e.  LMod )
6 lsmcv2.s . . . . . . 7  |-  S  =  ( LSubSp `  W )
76lsssssubg 16022 . . . . . 6  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
85, 7syl 16 . . . . 5  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
9 lsmcv2.u . . . . 5  |-  ( ph  ->  U  e.  S )
108, 9sseldd 3341 . . . 4  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
11 lsmcv2.x . . . . . 6  |-  ( ph  ->  X  e.  V )
12 lsmcv2.v . . . . . . 7  |-  V  =  ( Base `  W
)
13 lsmcv2.n . . . . . . 7  |-  N  =  ( LSpan `  W )
1412, 6, 13lspsncl 16041 . . . . . 6  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  S
)
155, 11, 14syl2anc 643 . . . . 5  |-  ( ph  ->  ( N `  { X } )  e.  S
)
168, 15sseldd 3341 . . . 4  |-  ( ph  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
172, 10, 16lssnle 15294 . . 3  |-  ( ph  ->  ( -.  ( N `
 { X }
)  C_  U  <->  U  C.  ( U  .(+)  ( N `
 { X }
) ) ) )
181, 17mpbid 202 . 2  |-  ( ph  ->  U  C.  ( U 
.(+)  ( N `  { X } ) ) )
19 3simpa 954 . . . . 5  |-  ( (
ph  /\  x  e.  S  /\  ( U  C.  x  /\  x  C_  ( U  .(+)  ( N `  { X } ) ) ) )  ->  ( ph  /\  x  e.  S
) )
20 simp3l 985 . . . . 5  |-  ( (
ph  /\  x  e.  S  /\  ( U  C.  x  /\  x  C_  ( U  .(+)  ( N `  { X } ) ) ) )  ->  U  C.  x )
21 simp3r 986 . . . . 5  |-  ( (
ph  /\  x  e.  S  /\  ( U  C.  x  /\  x  C_  ( U  .(+)  ( N `  { X } ) ) ) )  ->  x  C_  ( U  .(+)  ( N `
 { X }
) ) )
223adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  W  e.  LVec )
239adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  U  e.  S )
24 simpr 448 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
2511adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  X  e.  V )
2612, 6, 13, 2, 22, 23, 24, 25lsmcv 16201 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  U  C.  x  /\  x  C_  ( U  .(+)  ( N `
 { X }
) ) )  ->  x  =  ( U  .(+) 
( N `  { X } ) ) )
2719, 20, 21, 26syl3anc 1184 . . . 4  |-  ( (
ph  /\  x  e.  S  /\  ( U  C.  x  /\  x  C_  ( U  .(+)  ( N `  { X } ) ) ) )  ->  x  =  ( U  .(+)  ( N `  { X } ) ) )
28273exp 1152 . . 3  |-  ( ph  ->  ( x  e.  S  ->  ( ( U  C.  x  /\  x  C_  ( U  .(+)  ( N `  { X } ) ) )  ->  x  =  ( U  .(+)  ( N `
 { X }
) ) ) ) )
2928ralrimiv 2780 . 2  |-  ( ph  ->  A. x  e.  S  ( ( U  C.  x  /\  x  C_  ( U  .(+)  ( N `  { X } ) ) )  ->  x  =  ( U  .(+)  ( N `
 { X }
) ) ) )
30 lsmcv2.c . . 3  |-  C  =  (  <oLL  `  W )
316, 2lsmcl 16143 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  ( N `  { X } )  e.  S
)  ->  ( U  .(+) 
( N `  { X } ) )  e.  S )
325, 9, 15, 31syl3anc 1184 . . 3  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  e.  S
)
336, 30, 3, 9, 32lcvbr2 29659 . 2  |-  ( ph  ->  ( U C ( U  .(+)  ( N `  { X } ) )  <->  ( U  C.  ( U  .(+)  ( N `
 { X }
) )  /\  A. x  e.  S  (
( U  C.  x  /\  x  C_  ( U 
.(+)  ( N `  { X } ) ) )  ->  x  =  ( U  .(+)  ( N `
 { X }
) ) ) ) ) )
3418, 29, 33mpbir2and 889 1  |-  ( ph  ->  U C ( U 
.(+)  ( N `  { X } ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312    C. wpss 3313   {csn 3806   class class class wbr 4204   ` cfv 5445  (class class class)co 6072   Basecbs 13457  SubGrpcsubg 14926   LSSumclsm 15256   LModclmod 15938   LSubSpclss 15996   LSpanclspn 16035   LVecclvec 16162    <oLL clcv 29655
This theorem is referenced by:  lcv1  29678
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-tpos 6470  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-3 10048  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-0g 13715  df-mnd 14678  df-submnd 14727  df-grp 14800  df-minusg 14801  df-sbg 14802  df-subg 14929  df-cntz 15104  df-lsm 15258  df-cmn 15402  df-abl 15403  df-mgp 15637  df-rng 15651  df-ur 15653  df-oppr 15716  df-dvdsr 15734  df-unit 15735  df-invr 15765  df-drng 15825  df-lmod 15940  df-lss 15997  df-lsp 16036  df-lvec 16163  df-lcv 29656
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