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Theorem lshpnel 29100
Description: A hyperplane's generating vector does not belong to the hyperplane. (Contributed by NM, 3-Jul-2014.)
Hypotheses
Ref Expression
lshpnel.v  |-  V  =  ( Base `  W
)
lshpnel.n  |-  N  =  ( LSpan `  W )
lshpnel.p  |-  .(+)  =  (
LSSum `  W )
lshpnel.h  |-  H  =  (LSHyp `  W )
lshpnel.w  |-  ( ph  ->  W  e.  LMod )
lshpnel.u  |-  ( ph  ->  U  e.  H )
lshpnel.x  |-  ( ph  ->  X  e.  V )
lshpnel.e  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  =  V )
Assertion
Ref Expression
lshpnel  |-  ( ph  ->  -.  X  e.  U
)

Proof of Theorem lshpnel
StepHypRef Expression
1 lshpnel.v . . 3  |-  V  =  ( Base `  W
)
2 lshpnel.h . . 3  |-  H  =  (LSHyp `  W )
3 lshpnel.w . . 3  |-  ( ph  ->  W  e.  LMod )
4 lshpnel.u . . 3  |-  ( ph  ->  U  e.  H )
51, 2, 3, 4lshpne 29099 . 2  |-  ( ph  ->  U  =/=  V )
63adantr 452 . . . . . . . 8  |-  ( (
ph  /\  X  e.  U )  ->  W  e.  LMod )
7 eqid 2389 . . . . . . . . 9  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
87lsssssubg 15963 . . . . . . . 8  |-  ( W  e.  LMod  ->  ( LSubSp `  W )  C_  (SubGrp `  W ) )
96, 8syl 16 . . . . . . 7  |-  ( (
ph  /\  X  e.  U )  ->  ( LSubSp `
 W )  C_  (SubGrp `  W ) )
107, 2, 3, 4lshplss 29098 . . . . . . . 8  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
1110adantr 452 . . . . . . 7  |-  ( (
ph  /\  X  e.  U )  ->  U  e.  ( LSubSp `  W )
)
129, 11sseldd 3294 . . . . . 6  |-  ( (
ph  /\  X  e.  U )  ->  U  e.  (SubGrp `  W )
)
13 lshpnel.x . . . . . . . . 9  |-  ( ph  ->  X  e.  V )
1413adantr 452 . . . . . . . 8  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  V )
15 lshpnel.n . . . . . . . . 9  |-  N  =  ( LSpan `  W )
161, 7, 15lspsncl 15982 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
176, 14, 16syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  X  e.  U )  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
189, 17sseldd 3294 . . . . . 6  |-  ( (
ph  /\  X  e.  U )  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
19 simpr 448 . . . . . . 7  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  U )
207, 15, 6, 11, 19lspsnel5a 16001 . . . . . 6  |-  ( (
ph  /\  X  e.  U )  ->  ( N `  { X } )  C_  U
)
21 lshpnel.p . . . . . . 7  |-  .(+)  =  (
LSSum `  W )
2221lsmss2 15229 . . . . . 6  |-  ( ( U  e.  (SubGrp `  W )  /\  ( N `  { X } )  e.  (SubGrp `  W )  /\  ( N `  { X } )  C_  U
)  ->  ( U  .(+) 
( N `  { X } ) )  =  U )
2312, 18, 20, 22syl3anc 1184 . . . . 5  |-  ( (
ph  /\  X  e.  U )  ->  ( U  .(+)  ( N `  { X } ) )  =  U )
24 lshpnel.e . . . . . 6  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  =  V )
2524adantr 452 . . . . 5  |-  ( (
ph  /\  X  e.  U )  ->  ( U  .(+)  ( N `  { X } ) )  =  V )
2623, 25eqtr3d 2423 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  U  =  V )
2726ex 424 . . 3  |-  ( ph  ->  ( X  e.  U  ->  U  =  V ) )
2827necon3ad 2588 . 2  |-  ( ph  ->  ( U  =/=  V  ->  -.  X  e.  U
) )
295, 28mpd 15 1  |-  ( ph  ->  -.  X  e.  U
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2552    C_ wss 3265   {csn 3759   ` cfv 5396  (class class class)co 6022   Basecbs 13398  SubGrpcsubg 14867   LSSumclsm 15197   LModclmod 15879   LSubSpclss 15937   LSpanclspn 15976  LSHypclsh 29092
This theorem is referenced by:  lshpnelb  29101  lshpne0  29103  lshpdisj  29104
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-0g 13656  df-mnd 14619  df-submnd 14668  df-grp 14741  df-minusg 14742  df-sbg 14743  df-subg 14870  df-lsm 15199  df-mgp 15578  df-rng 15592  df-ur 15594  df-lmod 15881  df-lss 15938  df-lsp 15977  df-lshyp 29094
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