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Theorem lshpnel2N 29101
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 29100 . . 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 16106 . . . . . . . . . . 11  |-  ( W  e.  LVec  ->  W  e. 
LMod )
207, 19syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  e.  LMod )
21 lshpnel2.s . . . . . . . . . . 11  |-  S  =  ( LSubSp `  W )
2221, 4lspid 15986 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( N `  U )  =  U )
2320, 14, 22syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( N `  U
)  =  U )
2423uneq1d 3444 . . . . . . . 8  |-  ( ph  ->  ( ( N `  U )  u.  ( N `  { X } ) )  =  ( U  u.  ( N `  { X } ) ) )
2524fveq2d 5673 . . . . . . 7  |-  ( ph  ->  ( N `  (
( N `  U
)  u.  ( N `
 { X }
) ) )  =  ( N `  ( U  u.  ( N `  { X } ) ) ) )
263, 21lssss 15941 . . . . . . . . 9  |-  ( U  e.  S  ->  U  C_  V )
2714, 26syl 16 . . . . . . . 8  |-  ( ph  ->  U  C_  V )
2810snssd 3887 . . . . . . . 8  |-  ( ph  ->  { X }  C_  V )
293, 4lspun 15991 . . . . . . . 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 15981 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  S
)
3220, 10, 31syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( N `  { X } )  e.  S
)
3321, 4, 5lsmsp 16086 . . . . . . . 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 2431 . . . . . 6  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  =  ( N `  ( U  u.  { X }
) ) )
3635eqeq1d 2396 . . . . 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 3769 . . . . . . . 8  |-  ( v  =  X  ->  { v }  =  { X } )
3938uneq2d 3445 . . . . . . 7  |-  ( v  =  X  ->  ( U  u.  { v } )  =  ( U  u.  { X } ) )
4039fveq2d 5673 . . . . . 6  |-  ( v  =  X  ->  ( N `  ( U  u.  { v } ) )  =  ( N `
 ( U  u.  { X } ) ) )
4140eqeq1d 2396 . . . . 5  |-  ( v  =  X  ->  (
( N `  ( U  u.  { v } ) )  =  V  <->  ( N `  ( U  u.  { X } ) )  =  V ) )
4241rspcev 2996 . . . 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 29095 . . . 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 1649    e. wcel 1717    =/= wne 2551   E.wrex 2651    u. cun 3262    C_ wss 3264   {csn 3758   ` cfv 5395  (class class class)co 6021   Basecbs 13397   LSSumclsm 15196   LModclmod 15878   LSubSpclss 15936   LSpanclspn 15975   LVecclvec 16102  LSHypclsh 29091
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-tpos 6416  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-3 9992  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-0g 13655  df-mnd 14618  df-submnd 14667  df-grp 14740  df-minusg 14741  df-sbg 14742  df-subg 14869  df-cntz 15044  df-lsm 15198  df-cmn 15342  df-abl 15343  df-mgp 15577  df-rng 15591  df-ur 15593  df-oppr 15656  df-dvdsr 15674  df-unit 15675  df-invr 15705  df-drng 15765  df-lmod 15880  df-lss 15937  df-lsp 15976  df-lvec 16103  df-lshyp 29093
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