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Theorem lcv1 29231
Description: Covering property of a subspace plus an atom. (chcv1 22935 analog.) (Contributed by NM, 10-Jan-2015.)
Hypotheses
Ref Expression
lcv1.s  |-  S  =  ( LSubSp `  W )
lcv1.p  |-  .(+)  =  (
LSSum `  W )
lcv1.a  |-  A  =  (LSAtoms `  W )
lcv1.c  |-  C  =  (  <oLL  `  W )
lcv1.w  |-  ( ph  ->  W  e.  LVec )
lcv1.u  |-  ( ph  ->  U  e.  S )
lcv1.q  |-  ( ph  ->  Q  e.  A )
Assertion
Ref Expression
lcv1  |-  ( ph  ->  ( -.  Q  C_  U 
<->  U C ( U 
.(+)  Q ) ) )

Proof of Theorem lcv1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lcv1.q . . . . 5  |-  ( ph  ->  Q  e.  A )
2 lcv1.w . . . . . 6  |-  ( ph  ->  W  e.  LVec )
3 eqid 2283 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2283 . . . . . . 7  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 eqid 2283 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
6 lcv1.a . . . . . . 7  |-  A  =  (LSAtoms `  W )
73, 4, 5, 6islsat 29181 . . . . . 6  |-  ( W  e.  LVec  ->  ( Q  e.  A  <->  E. x  e.  ( ( Base `  W
)  \  { ( 0g `  W ) } ) Q  =  ( ( LSpan `  W ) `  { x } ) ) )
82, 7syl 15 . . . . 5  |-  ( ph  ->  ( Q  e.  A  <->  E. x  e.  ( (
Base `  W )  \  { ( 0g `  W ) } ) Q  =  ( (
LSpan `  W ) `  { x } ) ) )
91, 8mpbid 201 . . . 4  |-  ( ph  ->  E. x  e.  ( ( Base `  W
)  \  { ( 0g `  W ) } ) Q  =  ( ( LSpan `  W ) `  { x } ) )
109adantr 451 . . 3  |-  ( (
ph  /\  -.  Q  C_  U )  ->  E. x  e.  ( ( Base `  W
)  \  { ( 0g `  W ) } ) Q  =  ( ( LSpan `  W ) `  { x } ) )
11 lcv1.s . . . . . 6  |-  S  =  ( LSubSp `  W )
12 lcv1.p . . . . . 6  |-  .(+)  =  (
LSSum `  W )
13 lcv1.c . . . . . 6  |-  C  =  (  <oLL  `  W )
142adantr 451 . . . . . . 7  |-  ( (
ph  /\  -.  Q  C_  U )  ->  W  e.  LVec )
15143ad2ant1 976 . . . . . 6  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  W  e.  LVec )
16 lcv1.u . . . . . . . 8  |-  ( ph  ->  U  e.  S )
1716adantr 451 . . . . . . 7  |-  ( (
ph  /\  -.  Q  C_  U )  ->  U  e.  S )
18173ad2ant1 976 . . . . . 6  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  U  e.  S )
19 eldifi 3298 . . . . . . 7  |-  ( x  e.  ( ( Base `  W )  \  {
( 0g `  W
) } )  ->  x  e.  ( Base `  W ) )
20193ad2ant2 977 . . . . . 6  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  x  e.  ( Base `  W )
)
21 simp1r 980 . . . . . . 7  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  -.  Q  C_  U )
22 simp3 957 . . . . . . . 8  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  Q  =  ( ( LSpan `  W
) `  { x } ) )
2322sseq1d 3205 . . . . . . 7  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  ( Q  C_  U  <->  ( ( LSpan `  W ) `  {
x } )  C_  U ) )
2421, 23mtbid 291 . . . . . 6  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  -.  (
( LSpan `  W ) `  { x } ) 
C_  U )
253, 11, 4, 12, 13, 15, 18, 20, 24lsmcv2 29219 . . . . 5  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  U C
( U  .(+)  ( (
LSpan `  W ) `  { x } ) ) )
2622oveq2d 5874 . . . . 5  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  ( U  .(+) 
Q )  =  ( U  .(+)  ( ( LSpan `  W ) `  { x } ) ) )
2725, 26breqtrrd 4049 . . . 4  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  U C
( U  .(+)  Q ) )
2827rexlimdv3a 2669 . . 3  |-  ( (
ph  /\  -.  Q  C_  U )  ->  ( E. x  e.  (
( Base `  W )  \  { ( 0g `  W ) } ) Q  =  ( (
LSpan `  W ) `  { x } )  ->  U C ( U  .(+)  Q )
) )
2910, 28mpd 14 . 2  |-  ( (
ph  /\  -.  Q  C_  U )  ->  U C ( U  .(+)  Q ) )
302adantr 451 . . . 4  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  W  e.  LVec )
3116adantr 451 . . . 4  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  U  e.  S )
32 lveclmod 15859 . . . . . . 7  |-  ( W  e.  LVec  ->  W  e. 
LMod )
332, 32syl 15 . . . . . 6  |-  ( ph  ->  W  e.  LMod )
3411, 6, 33, 1lsatlssel 29187 . . . . . 6  |-  ( ph  ->  Q  e.  S )
3511, 12lsmcl 15836 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  Q  e.  S )  ->  ( U  .(+)  Q )  e.  S )
3633, 16, 34, 35syl3anc 1182 . . . . 5  |-  ( ph  ->  ( U  .(+)  Q )  e.  S )
3736adantr 451 . . . 4  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  ( U  .(+) 
Q )  e.  S
)
38 simpr 447 . . . 4  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  U C
( U  .(+)  Q ) )
3911, 13, 30, 31, 37, 38lcvpss 29214 . . 3  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  U  C.  ( U  .(+)  Q ) )
4011lsssssubg 15715 . . . . . . 7  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
4133, 40syl 15 . . . . . 6  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
4241, 16sseldd 3181 . . . . 5  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
4341, 34sseldd 3181 . . . . 5  |-  ( ph  ->  Q  e.  (SubGrp `  W ) )
4412, 42, 43lssnle 14983 . . . 4  |-  ( ph  ->  ( -.  Q  C_  U 
<->  U  C.  ( U 
.(+)  Q ) ) )
4544adantr 451 . . 3  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  ( -.  Q  C_  U  <->  U  C.  ( U  .(+)  Q ) ) )
4639, 45mpbird 223 . 2  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  -.  Q  C_  U )
4729, 46impbida 805 1  |-  ( ph  ->  ( -.  Q  C_  U 
<->  U C ( U 
.(+)  Q ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    \ cdif 3149    C_ wss 3152    C. wpss 3153   {csn 3640   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   0gc0g 13400  SubGrpcsubg 14615   LSSumclsm 14945   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728   LVecclvec 15855  LSAtomsclsa 29164    <oLL clcv 29208
This theorem is referenced by:  lcv2  29232  lsatnle  29234  lsatcvat3  29242
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-lsatoms 29166  df-lcv 29209
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