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Theorem lsmcv2 29219
Description: Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 22873 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 15859 . . . . . . 7  |-  ( W  e.  LVec  ->  W  e. 
LMod )
53, 4syl 15 . . . . . 6  |-  ( ph  ->  W  e.  LMod )
6 lsmcv2.s . . . . . . 7  |-  S  =  ( LSubSp `  W )
76lsssssubg 15715 . . . . . 6  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
85, 7syl 15 . . . . 5  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
9 lsmcv2.u . . . . 5  |-  ( ph  ->  U  e.  S )
108, 9sseldd 3181 . . . 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 15734 . . . . . 6  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  S
)
155, 11, 14syl2anc 642 . . . . 5  |-  ( ph  ->  ( N `  { X } )  e.  S
)
168, 15sseldd 3181 . . . 4  |-  ( ph  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
172, 10, 16lssnle 14983 . . 3  |-  ( ph  ->  ( -.  ( N `
 { X }
)  C_  U  <->  U  C.  ( U  .(+)  ( N `
 { X }
) ) ) )
181, 17mpbid 201 . 2  |-  ( ph  ->  U  C.  ( U 
.(+)  ( N `  { X } ) ) )
19 3simpa 952 . . . . 5  |-  ( (
ph  /\  x  e.  S  /\  ( U  C.  x  /\  x  C_  ( U  .(+)  ( N `  { X } ) ) ) )  ->  ( ph  /\  x  e.  S
) )
20 simp3l 983 . . . . 5  |-  ( (
ph  /\  x  e.  S  /\  ( U  C.  x  /\  x  C_  ( U  .(+)  ( N `  { X } ) ) ) )  ->  U  C.  x )
21 simp3r 984 . . . . 5  |-  ( (
ph  /\  x  e.  S  /\  ( U  C.  x  /\  x  C_  ( U  .(+)  ( N `  { X } ) ) ) )  ->  x  C_  ( U  .(+)  ( N `
 { X }
) ) )
223adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  W  e.  LVec )
239adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  U  e.  S )
24 simpr 447 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
2511adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  X  e.  V )
2612, 6, 13, 2, 22, 23, 24, 25lsmcv 15894 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  U  C.  x  /\  x  C_  ( U  .(+)  ( N `
 { X }
) ) )  ->  x  =  ( U  .(+) 
( N `  { X } ) ) )
2719, 20, 21, 26syl3anc 1182 . . . 4  |-  ( (
ph  /\  x  e.  S  /\  ( U  C.  x  /\  x  C_  ( U  .(+)  ( N `  { X } ) ) ) )  ->  x  =  ( U  .(+)  ( N `  { X } ) ) )
28273exp 1150 . . 3  |-  ( ph  ->  ( x  e.  S  ->  ( ( U  C.  x  /\  x  C_  ( U  .(+)  ( N `  { X } ) ) )  ->  x  =  ( U  .(+)  ( N `
 { X }
) ) ) ) )
2928ralrimiv 2625 . 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 15836 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  ( N `  { X } )  e.  S
)  ->  ( U  .(+) 
( N `  { X } ) )  e.  S )
325, 9, 15, 31syl3anc 1182 . . 3  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  e.  S
)
336, 30, 3, 9, 32lcvbr2 29212 . 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 888 1  |-  ( ph  ->  U C ( U 
.(+)  ( N `  { X } ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152    C. wpss 3153   {csn 3640   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148  SubGrpcsubg 14615   LSSumclsm 14945   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728   LVecclvec 15855    <oLL clcv 29208
This theorem is referenced by:  lcv1  29231
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lcv 29209
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