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Theorem lshpnel2N 29710
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 452 . . 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 452 . . . 4  |-  ( (
ph  /\  U  e.  H )  ->  W  e.  LVec )
9 simpr 448 . . . 4  |-  ( (
ph  /\  U  e.  H )  ->  U  e.  H )
10 lshpnel2.x . . . . 5  |-  ( ph  ->  X  e.  V )
1110adantr 452 . . . 4  |-  ( (
ph  /\  U  e.  H )  ->  X  e.  V )
123, 4, 5, 6, 8, 9, 11lshpnelb 29709 . . 3  |-  ( (
ph  /\  U  e.  H )  ->  ( -.  X  e.  U  <->  ( U  .(+)  ( N `  { X } ) )  =  V ) )
132, 12mpbid 202 . 2  |-  ( (
ph  /\  U  e.  H )  ->  ( U  .(+)  ( N `  { X } ) )  =  V )
14 lshpnel2.u . . . 4  |-  ( ph  ->  U  e.  S )
1514adantr 452 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  U  e.  S )
16 lshpnel2.t . . . 4  |-  ( ph  ->  U  =/=  V )
1716adantr 452 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  U  =/=  V )
1810adantr 452 . . . 4  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  X  e.  V )
19 lveclmod 16170 . . . . . . . . . . 11  |-  ( W  e.  LVec  ->  W  e. 
LMod )
207, 19syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  e.  LMod )
21 lshpnel2.s . . . . . . . . . . 11  |-  S  =  ( LSubSp `  W )
2221, 4lspid 16050 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( N `  U )  =  U )
2320, 14, 22syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( N `  U
)  =  U )
2423uneq1d 3492 . . . . . . . 8  |-  ( ph  ->  ( ( N `  U )  u.  ( N `  { X } ) )  =  ( U  u.  ( N `  { X } ) ) )
2524fveq2d 5724 . . . . . . 7  |-  ( ph  ->  ( N `  (
( N `  U
)  u.  ( N `
 { X }
) ) )  =  ( N `  ( U  u.  ( N `  { X } ) ) ) )
263, 21lssss 16005 . . . . . . . . 9  |-  ( U  e.  S  ->  U  C_  V )
2714, 26syl 16 . . . . . . . 8  |-  ( ph  ->  U  C_  V )
2810snssd 3935 . . . . . . . 8  |-  ( ph  ->  { X }  C_  V )
293, 4lspun 16055 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  { X }  C_  V )  -> 
( N `  ( U  u.  { X } ) )  =  ( N `  (
( N `  U
)  u.  ( N `
 { X }
) ) ) )
3020, 27, 28, 29syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( N `  ( U  u.  { X } ) )  =  ( N `  (
( N `  U
)  u.  ( N `
 { X }
) ) ) )
313, 21, 4lspsncl 16045 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  S
)
3220, 10, 31syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( N `  { X } )  e.  S
)
3321, 4, 5lsmsp 16150 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  ( N `  { X } )  e.  S
)  ->  ( U  .(+) 
( N `  { X } ) )  =  ( N `  ( U  u.  ( N `  { X } ) ) ) )
3420, 14, 32, 33syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  =  ( N `  ( U  u.  ( N `  { X } ) ) ) )
3525, 30, 343eqtr4rd 2478 . . . . . 6  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  =  ( N `  ( U  u.  { X }
) ) )
3635eqeq1d 2443 . . . . 5  |-  ( ph  ->  ( ( U  .(+)  ( N `  { X } ) )  =  V  <->  ( N `  ( U  u.  { X } ) )  =  V ) )
3736biimpa 471 . . . 4  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  ( N `  ( U  u.  { X } ) )  =  V )
38 sneq 3817 . . . . . . . 8  |-  ( v  =  X  ->  { v }  =  { X } )
3938uneq2d 3493 . . . . . . 7  |-  ( v  =  X  ->  ( U  u.  { v } )  =  ( U  u.  { X } ) )
4039fveq2d 5724 . . . . . 6  |-  ( v  =  X  ->  ( N `  ( U  u.  { v } ) )  =  ( N `
 ( U  u.  { X } ) ) )
4140eqeq1d 2443 . . . . 5  |-  ( v  =  X  ->  (
( N `  ( U  u.  { v } ) )  =  V  <->  ( N `  ( U  u.  { X } ) )  =  V ) )
4241rspcev 3044 . . . 4  |-  ( ( X  e.  V  /\  ( N `  ( U  u.  { X }
) )  =  V )  ->  E. v  e.  V  ( N `  ( U  u.  {
v } ) )  =  V )
4318, 37, 42syl2anc 643 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  E. v  e.  V  ( N `  ( U  u.  {
v } ) )  =  V )
447adantr 452 . . . 4  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  W  e.  LVec )
453, 4, 21, 6islshp 29704 . . . 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 1137 . 2  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  U  e.  H )
4813, 47impbida 806 1  |-  ( ph  ->  ( U  e.  H  <->  ( 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 2598   E.wrex 2698    u. cun 3310    C_ wss 3312   {csn 3806   ` cfv 5446  (class class class)co 6073   Basecbs 13461   LSSumclsm 15260   LModclmod 15942   LSubSpclss 16000   LSpanclspn 16039   LVecclvec 16166  LSHypclsh 29700
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-0g 13719  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-cntz 15108  df-lsm 15262  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167  df-lshyp 29702
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