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Theorem lshpnel2N 32628
Description: Condition that determines a hyperplane. (Contributed by NM, 3-Oct-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lshpnel2.v  |-  V  =  ( Base `  W
)
lshpnel2.s  |-  S  =  ( LSubSp `  W )
lshpnel2.n  |-  N  =  ( LSpan `  W )
lshpnel2.p  |-  .(+)  =  (
LSSum `  W )
lshpnel2.h  |-  H  =  (LSHyp `  W )
lshpnel2.w  |-  ( ph  ->  W  e.  LVec )
lshpnel2.u  |-  ( ph  ->  U  e.  S )
lshpnel2.t  |-  ( ph  ->  U  =/=  V )
lshpnel2.x  |-  ( ph  ->  X  e.  V )
lshpnel2.e  |-  ( ph  ->  -.  X  e.  U
)
Assertion
Ref Expression
lshpnel2N  |-  ( ph  ->  ( U  e.  H  <->  ( U  .(+)  ( N `  { X } ) )  =  V ) )

Proof of Theorem lshpnel2N
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 lshpnel2.e . . . 4  |-  ( ph  ->  -.  X  e.  U
)
21adantr 465 . . 3  |-  ( (
ph  /\  U  e.  H )  ->  -.  X  e.  U )
3 lshpnel2.v . . . 4  |-  V  =  ( Base `  W
)
4 lshpnel2.n . . . 4  |-  N  =  ( LSpan `  W )
5 lshpnel2.p . . . 4  |-  .(+)  =  (
LSSum `  W )
6 lshpnel2.h . . . 4  |-  H  =  (LSHyp `  W )
7 lshpnel2.w . . . . 5  |-  ( ph  ->  W  e.  LVec )
87adantr 465 . . . 4  |-  ( (
ph  /\  U  e.  H )  ->  W  e.  LVec )
9 simpr 461 . . . 4  |-  ( (
ph  /\  U  e.  H )  ->  U  e.  H )
10 lshpnel2.x . . . . 5  |-  ( ph  ->  X  e.  V )
1110adantr 465 . . . 4  |-  ( (
ph  /\  U  e.  H )  ->  X  e.  V )
123, 4, 5, 6, 8, 9, 11lshpnelb 32627 . . 3  |-  ( (
ph  /\  U  e.  H )  ->  ( -.  X  e.  U  <->  ( U  .(+)  ( N `  { X } ) )  =  V ) )
132, 12mpbid 210 . 2  |-  ( (
ph  /\  U  e.  H )  ->  ( U  .(+)  ( N `  { X } ) )  =  V )
14 lshpnel2.u . . . 4  |-  ( ph  ->  U  e.  S )
1514adantr 465 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  U  e.  S )
16 lshpnel2.t . . . 4  |-  ( ph  ->  U  =/=  V )
1716adantr 465 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  U  =/=  V )
1810adantr 465 . . . 4  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  X  e.  V )
19 lveclmod 17186 . . . . . . . . . . 11  |-  ( W  e.  LVec  ->  W  e. 
LMod )
207, 19syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  e.  LMod )
21 lshpnel2.s . . . . . . . . . . 11  |-  S  =  ( LSubSp `  W )
2221, 4lspid 17062 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( N `  U )  =  U )
2320, 14, 22syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( N `  U
)  =  U )
2423uneq1d 3508 . . . . . . . 8  |-  ( ph  ->  ( ( N `  U )  u.  ( N `  { X } ) )  =  ( U  u.  ( N `  { X } ) ) )
2524fveq2d 5694 . . . . . . 7  |-  ( ph  ->  ( N `  (
( N `  U
)  u.  ( N `
 { X }
) ) )  =  ( N `  ( U  u.  ( N `  { X } ) ) ) )
263, 21lssss 17017 . . . . . . . . 9  |-  ( U  e.  S  ->  U  C_  V )
2714, 26syl 16 . . . . . . . 8  |-  ( ph  ->  U  C_  V )
2810snssd 4017 . . . . . . . 8  |-  ( ph  ->  { X }  C_  V )
293, 4lspun 17067 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  { X }  C_  V )  -> 
( N `  ( U  u.  { X } ) )  =  ( N `  (
( N `  U
)  u.  ( N `
 { X }
) ) ) )
3020, 27, 28, 29syl3anc 1218 . . . . . . 7  |-  ( ph  ->  ( N `  ( U  u.  { X } ) )  =  ( N `  (
( N `  U
)  u.  ( N `
 { X }
) ) ) )
313, 21, 4lspsncl 17057 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  S
)
3220, 10, 31syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( N `  { X } )  e.  S
)
3321, 4, 5lsmsp 17166 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  ( N `  { X } )  e.  S
)  ->  ( U  .(+) 
( N `  { X } ) )  =  ( N `  ( U  u.  ( N `  { X } ) ) ) )
3420, 14, 32, 33syl3anc 1218 . . . . . . 7  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  =  ( N `  ( U  u.  ( N `  { X } ) ) ) )
3525, 30, 343eqtr4rd 2485 . . . . . 6  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  =  ( N `  ( U  u.  { X }
) ) )
3635eqeq1d 2450 . . . . 5  |-  ( ph  ->  ( ( U  .(+)  ( N `  { X } ) )  =  V  <->  ( N `  ( U  u.  { X } ) )  =  V ) )
3736biimpa 484 . . . 4  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  ( N `  ( U  u.  { X } ) )  =  V )
38 sneq 3886 . . . . . . . 8  |-  ( v  =  X  ->  { v }  =  { X } )
3938uneq2d 3509 . . . . . . 7  |-  ( v  =  X  ->  ( U  u.  { v } )  =  ( U  u.  { X } ) )
4039fveq2d 5694 . . . . . 6  |-  ( v  =  X  ->  ( N `  ( U  u.  { v } ) )  =  ( N `
 ( U  u.  { X } ) ) )
4140eqeq1d 2450 . . . . 5  |-  ( v  =  X  ->  (
( N `  ( U  u.  { v } ) )  =  V  <->  ( N `  ( U  u.  { X } ) )  =  V ) )
4241rspcev 3072 . . . 4  |-  ( ( X  e.  V  /\  ( N `  ( U  u.  { X }
) )  =  V )  ->  E. v  e.  V  ( N `  ( U  u.  {
v } ) )  =  V )
4318, 37, 42syl2anc 661 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  E. v  e.  V  ( N `  ( U  u.  {
v } ) )  =  V )
447adantr 465 . . . 4  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  W  e.  LVec )
453, 4, 21, 6islshp 32622 . . . 4  |-  ( W  e.  LVec  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/= 
V  /\  E. v  e.  V  ( N `  ( U  u.  {
v } ) )  =  V ) ) )
4644, 45syl 16 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/= 
V  /\  E. v  e.  V  ( N `  ( U  u.  {
v } ) )  =  V ) ) )
4715, 17, 43, 46mpbir3and 1171 . 2  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  U  e.  H )
4813, 47impbida 828 1  |-  ( ph  ->  ( U  e.  H  <->  ( 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 2605   E.wrex 2715    u. cun 3325    C_ wss 3327   {csn 3876   ` cfv 5417  (class class class)co 6090   Basecbs 14173   LSSumclsm 16132   LModclmod 16947   LSubSpclss 17012   LSpanclspn 17051   LVecclvec 17182  LSHypclsh 32618
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 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6831  df-rdg 6865  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-3 10380  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-0g 14379  df-mnd 15414  df-submnd 15464  df-grp 15544  df-minusg 15545  df-sbg 15546  df-subg 15677  df-cntz 15834  df-lsm 16134  df-cmn 16278  df-abl 16279  df-mgp 16591  df-ur 16603  df-rng 16646  df-oppr 16714  df-dvdsr 16732  df-unit 16733  df-invr 16763  df-drng 16833  df-lmod 16949  df-lss 17013  df-lsp 17052  df-lvec 17183  df-lshyp 32620
This theorem is referenced by: (None)
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