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Theorem lshpne0 29152
Description: The member of the span in the hyperplane definition does not belong to the hyperplane. (Contributed by NM, 14-Jul-2014.)
Hypotheses
Ref Expression
lshpne0.v  |-  V  =  ( Base `  W
)
lshpne0.n  |-  N  =  ( LSpan `  W )
lshpne0.p  |-  .(+)  =  (
LSSum `  W )
lshpne0.h  |-  H  =  (LSHyp `  W )
lshpne0.o  |-  .0.  =  ( 0g `  W )
lshpne0.w  |-  ( ph  ->  W  e.  LMod )
lshpne0.u  |-  ( ph  ->  U  e.  H )
lshpne0.x  |-  ( ph  ->  X  e.  V )
lshpne0.e  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  =  V )
Assertion
Ref Expression
lshpne0  |-  ( ph  ->  X  =/=  .0.  )

Proof of Theorem lshpne0
StepHypRef Expression
1 lshpne0.v . . 3  |-  V  =  ( Base `  W
)
2 lshpne0.n . . 3  |-  N  =  ( LSpan `  W )
3 lshpne0.p . . 3  |-  .(+)  =  (
LSSum `  W )
4 lshpne0.h . . 3  |-  H  =  (LSHyp `  W )
5 lshpne0.w . . 3  |-  ( ph  ->  W  e.  LMod )
6 lshpne0.u . . 3  |-  ( ph  ->  U  e.  H )
7 lshpne0.x . . 3  |-  ( ph  ->  X  e.  V )
8 lshpne0.e . . 3  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  =  V )
91, 2, 3, 4, 5, 6, 7, 8lshpnel 29149 . 2  |-  ( ph  ->  -.  X  e.  U
)
10 lshpne0.o . . . 4  |-  .0.  =  ( 0g `  W )
11 eqid 2380 . . . 4  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
125adantr 452 . . . 4  |-  ( (
ph  /\  -.  X  e.  U )  ->  W  e.  LMod )
1311, 4, 5, 6lshplss 29147 . . . . 5  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
1413adantr 452 . . . 4  |-  ( (
ph  /\  -.  X  e.  U )  ->  U  e.  ( LSubSp `  W )
)
157adantr 452 . . . 4  |-  ( (
ph  /\  -.  X  e.  U )  ->  X  e.  V )
16 simpr 448 . . . 4  |-  ( (
ph  /\  -.  X  e.  U )  ->  -.  X  e.  U )
171, 10, 11, 12, 14, 15, 16lssneln0 15948 . . 3  |-  ( (
ph  /\  -.  X  e.  U )  ->  X  e.  ( V  \  {  .0.  } ) )
18 eldifsni 3864 . . 3  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  =/=  .0.  )
1917, 18syl 16 . 2  |-  ( (
ph  /\  -.  X  e.  U )  ->  X  =/=  .0.  )
209, 19mpdan 650 1  |-  ( ph  ->  X  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2543    \ cdif 3253   {csn 3750   ` cfv 5387  (class class class)co 6013   Basecbs 13389   0gc0g 13643   LSSumclsm 15188   LModclmod 15870   LSubSpclss 15928   LSpanclspn 15967  LSHypclsh 29141
This theorem is referenced by:  lshpsmreu  29275  lshpkrlem5  29280
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-0g 13647  df-mnd 14610  df-submnd 14659  df-grp 14732  df-minusg 14733  df-sbg 14734  df-subg 14861  df-lsm 15190  df-mgp 15569  df-rng 15583  df-ur 15585  df-lmod 15872  df-lss 15929  df-lsp 15968  df-lshyp 29143
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