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Theorem lshpnelb 32725
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 2443 . . . . . . 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 17209 . . . . . . . 8  |-  ( W  e.  LVec  ->  W  e. 
LMod )
97, 8syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  LMod )
102, 3, 4, 5, 6, 9islshpsm 32721 . . . . . 6  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  ( LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+) 
( N `  {
v } ) )  =  V ) ) )
111, 10mpbid 210 . . . . 5  |-  ( ph  ->  ( U  e.  (
LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+)  ( N `  { v } ) )  =  V ) )
1211simp3d 1002 . . . 4  |-  ( ph  ->  E. v  e.  V  ( U  .(+)  ( N `
 { v } ) )  =  V )
1312adantr 465 . . 3  |-  ( (
ph  /\  -.  X  e.  U )  ->  E. v  e.  V  ( U  .(+) 
( N `  {
v } ) )  =  V )
14 simp1l 1012 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ph )
15 simp2 989 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  v  e.  V )
164lsssssubg 17061 . . . . . . . . . . . 12  |-  ( W  e.  LMod  ->  ( LSubSp `  W )  C_  (SubGrp `  W ) )
179, 16syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( LSubSp `  W )  C_  (SubGrp `  W )
)
184, 6, 9, 1lshplss 32722 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
1917, 18sseldd 3378 . . . . . . . . . 10  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
20 lshpnelb.x . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  V )
212, 4, 3lspsncl 17080 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
229, 20, 21syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
2317, 22sseldd 3378 . . . . . . . . . 10  |-  ( ph  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
245lsmub1 16176 . . . . . . . . . 10  |-  ( ( U  e.  (SubGrp `  W )  /\  ( N `  { X } )  e.  (SubGrp `  W ) )  ->  U  C_  ( U  .(+)  ( N `  { X } ) ) )
2519, 23, 24syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  U  C_  ( U  .(+) 
( N `  { X } ) ) )
2625adantr 465 . . . . . . . 8  |-  ( (
ph  /\  -.  X  e.  U )  ->  U  C_  ( U  .(+)  ( N `
 { X }
) ) )
275lsmub2 16177 . . . . . . . . . . . 12  |-  ( ( U  e.  (SubGrp `  W )  /\  ( N `  { X } )  e.  (SubGrp `  W ) )  -> 
( N `  { X } )  C_  ( U  .(+)  ( N `  { X } ) ) )
2819, 23, 27syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  C_  ( U  .(+)  ( N `  { X } ) ) )
292, 3lspsnid 17096 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
309, 20, 29syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  ( N `
 { X }
) )
3128, 30sseldd 3378 . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( U 
.(+)  ( N `  { X } ) ) )
32 nelne1 2722 . . . . . . . . . 10  |-  ( ( X  e.  ( U 
.(+)  ( N `  { X } ) )  /\  -.  X  e.  U )  ->  ( U  .(+)  ( N `  { X } ) )  =/=  U )
3331, 32sylan 471 . . . . . . . . 9  |-  ( (
ph  /\  -.  X  e.  U )  ->  ( U  .(+)  ( N `  { X } ) )  =/=  U )
3433necomd 2640 . . . . . . . 8  |-  ( (
ph  /\  -.  X  e.  U )  ->  U  =/=  ( U  .(+)  ( N `
 { X }
) ) )
35 df-pss 3365 . . . . . . . 8  |-  ( U 
C.  ( U  .(+)  ( N `  { X } ) )  <->  ( U  C_  ( U  .(+)  ( N `
 { X }
) )  /\  U  =/=  ( U  .(+)  ( N `
 { X }
) ) ) )
3626, 34, 35sylanbrc 664 . . . . . . 7  |-  ( (
ph  /\  -.  X  e.  U )  ->  U  C.  ( U  .(+)  ( N `
 { X }
) ) )
37363ad2ant1 1009 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  U  C.  ( U  .(+)  ( N `  { X } ) ) )
384, 5lsmcl 17186 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  U  e.  ( LSubSp `  W )  /\  ( N `  { X } )  e.  (
LSubSp `  W ) )  ->  ( U  .(+)  ( N `  { X } ) )  e.  ( LSubSp `  W )
)
399, 18, 22, 38syl3anc 1218 . . . . . . . . . . 11  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  e.  (
LSubSp `  W ) )
402, 4lssss 17040 . . . . . . . . . . 11  |-  ( ( U  .(+)  ( N `  { X } ) )  e.  ( LSubSp `  W )  ->  ( U  .(+)  ( N `  { X } ) ) 
C_  V )
4139, 40syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  C_  V
)
4241adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( U  .(+) 
( N `  {
v } ) )  =  V )  -> 
( U  .(+)  ( N `
 { X }
) )  C_  V
)
43 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  ( U  .(+) 
( N `  {
v } ) )  =  V )  -> 
( U  .(+)  ( N `
 { v } ) )  =  V )
4442, 43sseqtr4d 3414 . . . . . . . 8  |-  ( (
ph  /\  ( U  .(+) 
( N `  {
v } ) )  =  V )  -> 
( U  .(+)  ( N `
 { X }
) )  C_  ( U  .(+)  ( N `  { v } ) ) )
4544adantlr 714 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  C_  ( U  .(+)  ( N `
 { v } ) ) )
46453adant2 1007 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  C_  ( U  .(+)  ( N `
 { v } ) ) )
477adantr 465 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  W  e.  LVec )
4818adantr 465 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  U  e.  ( LSubSp `  W )
)
4939adantr 465 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  ( U  .(+)  ( N `  { X } ) )  e.  ( LSubSp `  W
) )
50 simpr 461 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  v  e.  V )
512, 4, 3, 5, 47, 48, 49, 50lsmcv 17244 . . . . . 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 1224 . . . . 5  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  =  ( U  .(+)  ( N `
 { v } ) ) )
53 simp3 990 . . . . 5  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  {
v } ) )  =  V )
5452, 53eqtrd 2475 . . . 4  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  =  V )
5554rexlimdv3a 2864 . . 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 465 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  W  e.  LMod )
581adantr 465 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  U  e.  H )
5920adantr 465 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  X  e.  V )
60 simpr 461 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  ( U  .(+)  ( N `  { X } ) )  =  V )
612, 3, 5, 6, 57, 58, 59, 60lshpnel 32724 . 2  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  -.  X  e.  U )
6256, 61impbida 828 1  |-  ( ph  ->  ( -.  X  e.  U  <->  ( U  .(+)  ( N `  { X } ) )  =  V ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620   E.wrex 2737    C_ wss 3349    C. wpss 3350   {csn 3898   ` cfv 5439  (class class class)co 6112   Basecbs 14195  SubGrpcsubg 15696   LSSumclsm 16154   LModclmod 16970   LSubSpclss 17035   LSpanclspn 17074   LVecclvec 17205  LSHypclsh 32716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-tpos 6766  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-0g 14401  df-mnd 15436  df-submnd 15486  df-grp 15566  df-minusg 15567  df-sbg 15568  df-subg 15699  df-cntz 15856  df-lsm 16156  df-cmn 16300  df-abl 16301  df-mgp 16614  df-ur 16626  df-rng 16669  df-oppr 16737  df-dvdsr 16755  df-unit 16756  df-invr 16786  df-drng 16856  df-lmod 16972  df-lss 17036  df-lsp 17075  df-lvec 17206  df-lshyp 32718
This theorem is referenced by:  lshpnel2N  32726  l1cvpat  32795  dochexmidat  35200
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