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Theorem lshpnelb 29796
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 2296 . . . . . . 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 15875 . . . . . . . 8  |-  ( W  e.  LVec  ->  W  e. 
LMod )
97, 8syl 15 . . . . . . 7  |-  ( ph  ->  W  e.  LMod )
102, 3, 4, 5, 6, 9islshpsm 29792 . . . . . 6  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  ( LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+) 
( N `  {
v } ) )  =  V ) ) )
111, 10mpbid 201 . . . . 5  |-  ( ph  ->  ( U  e.  (
LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+)  ( N `  { v } ) )  =  V ) )
1211simp3d 969 . . . 4  |-  ( ph  ->  E. v  e.  V  ( U  .(+)  ( N `
 { v } ) )  =  V )
1312adantr 451 . . 3  |-  ( (
ph  /\  -.  X  e.  U )  ->  E. v  e.  V  ( U  .(+) 
( N `  {
v } ) )  =  V )
14 simp1l 979 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ph )
15 simp2 956 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  v  e.  V )
164lsssssubg 15731 . . . . . . . . . . . 12  |-  ( W  e.  LMod  ->  ( LSubSp `  W )  C_  (SubGrp `  W ) )
179, 16syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( LSubSp `  W )  C_  (SubGrp `  W )
)
18 eqid 2296 . . . . . . . . . . . 12  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
1918, 6, 9, 1lshplss 29793 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
2017, 19sseldd 3194 . . . . . . . . . 10  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
21 lshpnelb.x . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  V )
222, 4, 3lspsncl 15750 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
239, 21, 22syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
2417, 23sseldd 3194 . . . . . . . . . 10  |-  ( ph  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
255lsmub1 14983 . . . . . . . . . 10  |-  ( ( U  e.  (SubGrp `  W )  /\  ( N `  { X } )  e.  (SubGrp `  W ) )  ->  U  C_  ( U  .(+)  ( N `  { X } ) ) )
2620, 24, 25syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  U  C_  ( U  .(+) 
( N `  { X } ) ) )
2726adantr 451 . . . . . . . 8  |-  ( (
ph  /\  -.  X  e.  U )  ->  U  C_  ( U  .(+)  ( N `
 { X }
) ) )
285lsmub2 14984 . . . . . . . . . . . 12  |-  ( ( U  e.  (SubGrp `  W )  /\  ( N `  { X } )  e.  (SubGrp `  W ) )  -> 
( N `  { X } )  C_  ( U  .(+)  ( N `  { X } ) ) )
2920, 24, 28syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  C_  ( U  .(+)  ( N `  { X } ) ) )
302, 3lspsnid 15766 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
319, 21, 30syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  ( N `
 { X }
) )
3229, 31sseldd 3194 . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( U 
.(+)  ( N `  { X } ) ) )
33 nelne1 2548 . . . . . . . . . 10  |-  ( ( X  e.  ( U 
.(+)  ( N `  { X } ) )  /\  -.  X  e.  U )  ->  ( U  .(+)  ( N `  { X } ) )  =/=  U )
3432, 33sylan 457 . . . . . . . . 9  |-  ( (
ph  /\  -.  X  e.  U )  ->  ( U  .(+)  ( N `  { X } ) )  =/=  U )
3534necomd 2542 . . . . . . . 8  |-  ( (
ph  /\  -.  X  e.  U )  ->  U  =/=  ( U  .(+)  ( N `
 { X }
) ) )
36 df-pss 3181 . . . . . . . 8  |-  ( U 
C.  ( U  .(+)  ( N `  { X } ) )  <->  ( U  C_  ( U  .(+)  ( N `
 { X }
) )  /\  U  =/=  ( U  .(+)  ( N `
 { X }
) ) ) )
3727, 35, 36sylanbrc 645 . . . . . . 7  |-  ( (
ph  /\  -.  X  e.  U )  ->  U  C.  ( U  .(+)  ( N `
 { X }
) ) )
38373ad2ant1 976 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  U  C.  ( U  .(+)  ( N `
 { X }
) ) )
392, 18, 3lspsncl 15750 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
409, 21, 39syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
414, 5lsmcl 15852 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  U  e.  ( LSubSp `  W )  /\  ( N `  { X } )  e.  (
LSubSp `  W ) )  ->  ( U  .(+)  ( N `  { X } ) )  e.  ( LSubSp `  W )
)
429, 19, 40, 41syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  e.  (
LSubSp `  W ) )
432, 4lssss 15710 . . . . . . . . . . 11  |-  ( ( U  .(+)  ( N `  { X } ) )  e.  ( LSubSp `  W )  ->  ( U  .(+)  ( N `  { X } ) ) 
C_  V )
4442, 43syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  C_  V
)
4544adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( U  .(+) 
( N `  {
v } ) )  =  V )  -> 
( U  .(+)  ( N `
 { X }
) )  C_  V
)
46 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  ( U  .(+) 
( N `  {
v } ) )  =  V )  -> 
( U  .(+)  ( N `
 { v } ) )  =  V )
4745, 46sseqtr4d 3228 . . . . . . . 8  |-  ( (
ph  /\  ( U  .(+) 
( N `  {
v } ) )  =  V )  -> 
( U  .(+)  ( N `
 { X }
) )  C_  ( U  .(+)  ( N `  { v } ) ) )
4847adantlr 695 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  C_  ( U  .(+)  ( N `
 { v } ) ) )
49483adant2 974 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  C_  ( U  .(+)  ( N `
 { v } ) ) )
507adantr 451 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  W  e.  LVec )
514, 6, 9, 1lshplss 29793 . . . . . . . 8  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
5251adantr 451 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  U  e.  ( LSubSp `  W )
)
539, 19, 23, 41syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  e.  (
LSubSp `  W ) )
5453adantr 451 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  ( U  .(+)  ( N `  { X } ) )  e.  ( LSubSp `  W
) )
55 simpr 447 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  v  e.  V )
562, 4, 3, 5, 50, 52, 54, 55lsmcv 15910 . . . . . 6  |-  ( ( ( ph  /\  v  e.  V )  /\  U  C.  ( U  .(+)  ( N `
 { X }
) )  /\  ( U  .(+)  ( N `  { X } ) ) 
C_  ( U  .(+)  ( N `  { v } ) ) )  ->  ( U  .(+)  ( N `  { X } ) )  =  ( U  .(+)  ( N `
 { v } ) ) )
5714, 15, 38, 49, 56syl211anc 1188 . . . . 5  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  =  ( U  .(+)  ( N `
 { v } ) ) )
58 simp3 957 . . . . 5  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  {
v } ) )  =  V )
5957, 58eqtrd 2328 . . . 4  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  =  V )
6059rexlimdv3a 2682 . . 3  |-  ( (
ph  /\  -.  X  e.  U )  ->  ( E. v  e.  V  ( U  .(+)  ( N `
 { v } ) )  =  V  ->  ( U  .(+)  ( N `  { X } ) )  =  V ) )
6113, 60mpd 14 . 2  |-  ( (
ph  /\  -.  X  e.  U )  ->  ( U  .(+)  ( N `  { X } ) )  =  V )
629adantr 451 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  W  e.  LMod )
631adantr 451 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  U  e.  H )
6421adantr 451 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  X  e.  V )
65 simpr 447 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  ( U  .(+)  ( N `  { X } ) )  =  V )
662, 3, 5, 6, 62, 63, 64, 65lshpnel 29795 . 2  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  -.  X  e.  U )
6761, 66impbida 805 1  |-  ( ph  ->  ( -.  X  e.  U  <->  ( U  .(+)  ( N `  { X } ) )  =  V ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    C_ wss 3165    C. wpss 3166   {csn 3653   ` cfv 5271  (class class class)co 5874   Basecbs 13164  SubGrpcsubg 14631   LSSumclsm 14961   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744   LVecclvec 15871  LSHypclsh 29787
This theorem is referenced by:  lshpnel2N  29797  l1cvpat  29866  dochexmidat  32271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872  df-lshyp 29789
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