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Theorem lshpnelb 29782
Description: The subspace sum of a hyperplane and the span of an element equals the vector space iff the element is not in the hyperplane. (Contributed by NM, 2-Oct-2014.)
Hypotheses
Ref Expression
lshpnelb.v  |-  V  =  ( Base `  W
)
lshpnelb.n  |-  N  =  ( LSpan `  W )
lshpnelb.p  |-  .(+)  =  (
LSSum `  W )
lshpnelb.h  |-  H  =  (LSHyp `  W )
lshpnelb.w  |-  ( ph  ->  W  e.  LVec )
lshpnelb.u  |-  ( ph  ->  U  e.  H )
lshpnelb.x  |-  ( ph  ->  X  e.  V )
Assertion
Ref Expression
lshpnelb  |-  ( ph  ->  ( -.  X  e.  U  <->  ( U  .(+)  ( N `  { X } ) )  =  V ) )

Proof of Theorem lshpnelb
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 lshpnelb.u . . . . . 6  |-  ( ph  ->  U  e.  H )
2 lshpnelb.v . . . . . . 7  |-  V  =  ( Base `  W
)
3 lshpnelb.n . . . . . . 7  |-  N  =  ( LSpan `  W )
4 eqid 2436 . . . . . . 7  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
5 lshpnelb.p . . . . . . 7  |-  .(+)  =  (
LSSum `  W )
6 lshpnelb.h . . . . . . 7  |-  H  =  (LSHyp `  W )
7 lshpnelb.w . . . . . . . 8  |-  ( ph  ->  W  e.  LVec )
8 lveclmod 16178 . . . . . . . 8  |-  ( W  e.  LVec  ->  W  e. 
LMod )
97, 8syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  LMod )
102, 3, 4, 5, 6, 9islshpsm 29778 . . . . . 6  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  ( LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+) 
( N `  {
v } ) )  =  V ) ) )
111, 10mpbid 202 . . . . 5  |-  ( ph  ->  ( U  e.  (
LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+)  ( N `  { v } ) )  =  V ) )
1211simp3d 971 . . . 4  |-  ( ph  ->  E. v  e.  V  ( U  .(+)  ( N `
 { v } ) )  =  V )
1312adantr 452 . . 3  |-  ( (
ph  /\  -.  X  e.  U )  ->  E. v  e.  V  ( U  .(+) 
( N `  {
v } ) )  =  V )
14 simp1l 981 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ph )
15 simp2 958 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  v  e.  V )
164lsssssubg 16034 . . . . . . . . . . . 12  |-  ( W  e.  LMod  ->  ( LSubSp `  W )  C_  (SubGrp `  W ) )
179, 16syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( LSubSp `  W )  C_  (SubGrp `  W )
)
184, 6, 9, 1lshplss 29779 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
1917, 18sseldd 3349 . . . . . . . . . 10  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
20 lshpnelb.x . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  V )
212, 4, 3lspsncl 16053 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
229, 20, 21syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
2317, 22sseldd 3349 . . . . . . . . . 10  |-  ( ph  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
245lsmub1 15290 . . . . . . . . . 10  |-  ( ( U  e.  (SubGrp `  W )  /\  ( N `  { X } )  e.  (SubGrp `  W ) )  ->  U  C_  ( U  .(+)  ( N `  { X } ) ) )
2519, 23, 24syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  U  C_  ( U  .(+) 
( N `  { X } ) ) )
2625adantr 452 . . . . . . . 8  |-  ( (
ph  /\  -.  X  e.  U )  ->  U  C_  ( U  .(+)  ( N `
 { X }
) ) )
275lsmub2 15291 . . . . . . . . . . . 12  |-  ( ( U  e.  (SubGrp `  W )  /\  ( N `  { X } )  e.  (SubGrp `  W ) )  -> 
( N `  { X } )  C_  ( U  .(+)  ( N `  { X } ) ) )
2819, 23, 27syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  C_  ( U  .(+)  ( N `  { X } ) ) )
292, 3lspsnid 16069 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
309, 20, 29syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  ( N `
 { X }
) )
3128, 30sseldd 3349 . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( U 
.(+)  ( N `  { X } ) ) )
32 nelne1 2693 . . . . . . . . . 10  |-  ( ( X  e.  ( U 
.(+)  ( N `  { X } ) )  /\  -.  X  e.  U )  ->  ( U  .(+)  ( N `  { X } ) )  =/=  U )
3331, 32sylan 458 . . . . . . . . 9  |-  ( (
ph  /\  -.  X  e.  U )  ->  ( U  .(+)  ( N `  { X } ) )  =/=  U )
3433necomd 2687 . . . . . . . 8  |-  ( (
ph  /\  -.  X  e.  U )  ->  U  =/=  ( U  .(+)  ( N `
 { X }
) ) )
35 df-pss 3336 . . . . . . . 8  |-  ( U 
C.  ( U  .(+)  ( N `  { X } ) )  <->  ( U  C_  ( U  .(+)  ( N `
 { X }
) )  /\  U  =/=  ( U  .(+)  ( N `
 { X }
) ) ) )
3626, 34, 35sylanbrc 646 . . . . . . 7  |-  ( (
ph  /\  -.  X  e.  U )  ->  U  C.  ( U  .(+)  ( N `
 { X }
) ) )
37363ad2ant1 978 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  U  C.  ( U  .(+)  ( N `
 { X }
) ) )
384, 5lsmcl 16155 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  U  e.  ( LSubSp `  W )  /\  ( N `  { X } )  e.  (
LSubSp `  W ) )  ->  ( U  .(+)  ( N `  { X } ) )  e.  ( LSubSp `  W )
)
399, 18, 22, 38syl3anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  e.  (
LSubSp `  W ) )
402, 4lssss 16013 . . . . . . . . . . 11  |-  ( ( U  .(+)  ( N `  { X } ) )  e.  ( LSubSp `  W )  ->  ( U  .(+)  ( N `  { X } ) ) 
C_  V )
4139, 40syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  C_  V
)
4241adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( U  .(+) 
( N `  {
v } ) )  =  V )  -> 
( U  .(+)  ( N `
 { X }
) )  C_  V
)
43 simpr 448 . . . . . . . . 9  |-  ( (
ph  /\  ( U  .(+) 
( N `  {
v } ) )  =  V )  -> 
( U  .(+)  ( N `
 { v } ) )  =  V )
4442, 43sseqtr4d 3385 . . . . . . . 8  |-  ( (
ph  /\  ( U  .(+) 
( N `  {
v } ) )  =  V )  -> 
( U  .(+)  ( N `
 { X }
) )  C_  ( U  .(+)  ( N `  { v } ) ) )
4544adantlr 696 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  C_  ( U  .(+)  ( N `
 { v } ) ) )
46453adant2 976 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  C_  ( U  .(+)  ( N `
 { v } ) ) )
477adantr 452 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  W  e.  LVec )
4818adantr 452 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  U  e.  ( LSubSp `  W )
)
4939adantr 452 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  ( U  .(+)  ( N `  { X } ) )  e.  ( LSubSp `  W
) )
50 simpr 448 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  v  e.  V )
512, 4, 3, 5, 47, 48, 49, 50lsmcv 16213 . . . . . 6  |-  ( ( ( ph  /\  v  e.  V )  /\  U  C.  ( U  .(+)  ( N `
 { X }
) )  /\  ( U  .(+)  ( N `  { X } ) ) 
C_  ( U  .(+)  ( N `  { v } ) ) )  ->  ( U  .(+)  ( N `  { X } ) )  =  ( U  .(+)  ( N `
 { v } ) ) )
5214, 15, 37, 46, 51syl211anc 1190 . . . . 5  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  =  ( U  .(+)  ( N `
 { v } ) ) )
53 simp3 959 . . . . 5  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  {
v } ) )  =  V )
5452, 53eqtrd 2468 . . . 4  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  =  V )
5554rexlimdv3a 2832 . . 3  |-  ( (
ph  /\  -.  X  e.  U )  ->  ( E. v  e.  V  ( U  .(+)  ( N `
 { v } ) )  =  V  ->  ( U  .(+)  ( N `  { X } ) )  =  V ) )
5613, 55mpd 15 . 2  |-  ( (
ph  /\  -.  X  e.  U )  ->  ( U  .(+)  ( N `  { X } ) )  =  V )
579adantr 452 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  W  e.  LMod )
581adantr 452 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  U  e.  H )
5920adantr 452 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  X  e.  V )
60 simpr 448 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  ( U  .(+)  ( N `  { X } ) )  =  V )
612, 3, 5, 6, 57, 58, 59, 60lshpnel 29781 . 2  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  -.  X  e.  U )
6256, 61impbida 806 1  |-  ( ph  ->  ( -.  X  e.  U  <->  ( U  .(+)  ( N `  { X } ) )  =  V ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706    C_ wss 3320    C. wpss 3321   {csn 3814   ` cfv 5454  (class class class)co 6081   Basecbs 13469  SubGrpcsubg 14938   LSSumclsm 15268   LModclmod 15950   LSubSpclss 16008   LSpanclspn 16047   LVecclvec 16174  LSHypclsh 29773
This theorem is referenced by:  lshpnel2N  29783  l1cvpat  29852  dochexmidat  32257
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-cntz 15116  df-lsm 15270  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-drng 15837  df-lmod 15952  df-lss 16009  df-lsp 16048  df-lvec 16175  df-lshyp 29775
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