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Theorem lshpnel2N 34811
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 34810 . . 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 17878 . . . . . . . . . . 11  |-  ( W  e.  LVec  ->  W  e. 
LMod )
207, 19syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  e.  LMod )
21 lshpnel2.s . . . . . . . . . . 11  |-  S  =  ( LSubSp `  W )
2221, 4lspid 17754 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( N `  U )  =  U )
2320, 14, 22syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( N `  U
)  =  U )
2423uneq1d 3653 . . . . . . . 8  |-  ( ph  ->  ( ( N `  U )  u.  ( N `  { X } ) )  =  ( U  u.  ( N `  { X } ) ) )
2524fveq2d 5876 . . . . . . 7  |-  ( ph  ->  ( N `  (
( N `  U
)  u.  ( N `
 { X }
) ) )  =  ( N `  ( U  u.  ( N `  { X } ) ) ) )
263, 21lssss 17709 . . . . . . . . 9  |-  ( U  e.  S  ->  U  C_  V )
2714, 26syl 16 . . . . . . . 8  |-  ( ph  ->  U  C_  V )
2810snssd 4177 . . . . . . . 8  |-  ( ph  ->  { X }  C_  V )
293, 4lspun 17759 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  { X }  C_  V )  -> 
( N `  ( U  u.  { X } ) )  =  ( N `  (
( N `  U
)  u.  ( N `
 { X }
) ) ) )
3020, 27, 28, 29syl3anc 1228 . . . . . . 7  |-  ( ph  ->  ( N `  ( U  u.  { X } ) )  =  ( N `  (
( N `  U
)  u.  ( N `
 { X }
) ) ) )
313, 21, 4lspsncl 17749 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  S
)
3220, 10, 31syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( N `  { X } )  e.  S
)
3321, 4, 5lsmsp 17858 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  ( N `  { X } )  e.  S
)  ->  ( U  .(+) 
( N `  { X } ) )  =  ( N `  ( U  u.  ( N `  { X } ) ) ) )
3420, 14, 32, 33syl3anc 1228 . . . . . . 7  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  =  ( N `  ( U  u.  ( N `  { X } ) ) ) )
3525, 30, 343eqtr4rd 2509 . . . . . 6  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  =  ( N `  ( U  u.  { X }
) ) )
3635eqeq1d 2459 . . . . 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 4042 . . . . . . . 8  |-  ( v  =  X  ->  { v }  =  { X } )
3938uneq2d 3654 . . . . . . 7  |-  ( v  =  X  ->  ( U  u.  { v } )  =  ( U  u.  { X } ) )
4039fveq2d 5876 . . . . . 6  |-  ( v  =  X  ->  ( N `  ( U  u.  { v } ) )  =  ( N `
 ( U  u.  { X } ) ) )
4140eqeq1d 2459 . . . . 5  |-  ( v  =  X  ->  (
( N `  ( U  u.  { v } ) )  =  V  <->  ( N `  ( U  u.  { X } ) )  =  V ) )
4241rspcev 3210 . . . 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 34805 . . . 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 1179 . 2  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  U  e.  H )
4813, 47impbida 832 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808    u. cun 3469    C_ wss 3471   {csn 4032   ` cfv 5594  (class class class)co 6296   Basecbs 14643   LSSumclsm 16780   LModclmod 17638   LSubSpclss 17704   LSpanclspn 17743   LVecclvec 17874  LSHypclsh 34801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-0g 14858  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-grp 16183  df-minusg 16184  df-sbg 16185  df-subg 16324  df-cntz 16481  df-lsm 16782  df-cmn 16926  df-abl 16927  df-mgp 17268  df-ur 17280  df-ring 17326  df-oppr 17398  df-dvdsr 17416  df-unit 17417  df-invr 17447  df-drng 17524  df-lmod 17640  df-lss 17705  df-lsp 17744  df-lvec 17875  df-lshyp 34803
This theorem is referenced by: (None)
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