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Theorem lshpnelb 34182
Description: The subspace sum of a hyperplane and the span of an element equals the vector space iff the element is not in the hyperplane. (Contributed by NM, 2-Oct-2014.)
Hypotheses
Ref Expression
lshpnelb.v  |-  V  =  ( Base `  W
)
lshpnelb.n  |-  N  =  ( LSpan `  W )
lshpnelb.p  |-  .(+)  =  (
LSSum `  W )
lshpnelb.h  |-  H  =  (LSHyp `  W )
lshpnelb.w  |-  ( ph  ->  W  e.  LVec )
lshpnelb.u  |-  ( ph  ->  U  e.  H )
lshpnelb.x  |-  ( ph  ->  X  e.  V )
Assertion
Ref Expression
lshpnelb  |-  ( ph  ->  ( -.  X  e.  U  <->  ( U  .(+)  ( N `  { X } ) )  =  V ) )

Proof of Theorem lshpnelb
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 lshpnelb.u . . . . . 6  |-  ( ph  ->  U  e.  H )
2 lshpnelb.v . . . . . . 7  |-  V  =  ( Base `  W
)
3 lshpnelb.n . . . . . . 7  |-  N  =  ( LSpan `  W )
4 eqid 2467 . . . . . . 7  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
5 lshpnelb.p . . . . . . 7  |-  .(+)  =  (
LSSum `  W )
6 lshpnelb.h . . . . . . 7  |-  H  =  (LSHyp `  W )
7 lshpnelb.w . . . . . . . 8  |-  ( ph  ->  W  e.  LVec )
8 lveclmod 17623 . . . . . . . 8  |-  ( W  e.  LVec  ->  W  e. 
LMod )
97, 8syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  LMod )
102, 3, 4, 5, 6, 9islshpsm 34178 . . . . . 6  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  ( LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+) 
( N `  {
v } ) )  =  V ) ) )
111, 10mpbid 210 . . . . 5  |-  ( ph  ->  ( U  e.  (
LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+)  ( N `  { v } ) )  =  V ) )
1211simp3d 1010 . . . 4  |-  ( ph  ->  E. v  e.  V  ( U  .(+)  ( N `
 { v } ) )  =  V )
1312adantr 465 . . 3  |-  ( (
ph  /\  -.  X  e.  U )  ->  E. v  e.  V  ( U  .(+) 
( N `  {
v } ) )  =  V )
14 simp1l 1020 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ph )
15 simp2 997 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  v  e.  V )
164lsssssubg 17475 . . . . . . . . . . . 12  |-  ( W  e.  LMod  ->  ( LSubSp `  W )  C_  (SubGrp `  W ) )
179, 16syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( LSubSp `  W )  C_  (SubGrp `  W )
)
184, 6, 9, 1lshplss 34179 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
1917, 18sseldd 3510 . . . . . . . . . 10  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
20 lshpnelb.x . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  V )
212, 4, 3lspsncl 17494 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
229, 20, 21syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
2317, 22sseldd 3510 . . . . . . . . . 10  |-  ( ph  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
245lsmub1 16549 . . . . . . . . . 10  |-  ( ( U  e.  (SubGrp `  W )  /\  ( N `  { X } )  e.  (SubGrp `  W ) )  ->  U  C_  ( U  .(+)  ( N `  { X } ) ) )
2519, 23, 24syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  U  C_  ( U  .(+) 
( N `  { X } ) ) )
2625adantr 465 . . . . . . . 8  |-  ( (
ph  /\  -.  X  e.  U )  ->  U  C_  ( U  .(+)  ( N `
 { X }
) ) )
275lsmub2 16550 . . . . . . . . . . . 12  |-  ( ( U  e.  (SubGrp `  W )  /\  ( N `  { X } )  e.  (SubGrp `  W ) )  -> 
( N `  { X } )  C_  ( U  .(+)  ( N `  { X } ) ) )
2819, 23, 27syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  C_  ( U  .(+)  ( N `  { X } ) ) )
292, 3lspsnid 17510 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
309, 20, 29syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  ( N `
 { X }
) )
3128, 30sseldd 3510 . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( U 
.(+)  ( N `  { X } ) ) )
32 nelne1 2796 . . . . . . . . . 10  |-  ( ( X  e.  ( U 
.(+)  ( N `  { X } ) )  /\  -.  X  e.  U )  ->  ( U  .(+)  ( N `  { X } ) )  =/=  U )
3331, 32sylan 471 . . . . . . . . 9  |-  ( (
ph  /\  -.  X  e.  U )  ->  ( U  .(+)  ( N `  { X } ) )  =/=  U )
3433necomd 2738 . . . . . . . 8  |-  ( (
ph  /\  -.  X  e.  U )  ->  U  =/=  ( U  .(+)  ( N `
 { X }
) ) )
35 df-pss 3497 . . . . . . . 8  |-  ( U 
C.  ( U  .(+)  ( N `  { X } ) )  <->  ( U  C_  ( U  .(+)  ( N `
 { X }
) )  /\  U  =/=  ( U  .(+)  ( N `
 { X }
) ) ) )
3626, 34, 35sylanbrc 664 . . . . . . 7  |-  ( (
ph  /\  -.  X  e.  U )  ->  U  C.  ( U  .(+)  ( N `
 { X }
) ) )
37363ad2ant1 1017 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  U  C.  ( U  .(+)  ( N `  { X } ) ) )
384, 5lsmcl 17600 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  U  e.  ( LSubSp `  W )  /\  ( N `  { X } )  e.  (
LSubSp `  W ) )  ->  ( U  .(+)  ( N `  { X } ) )  e.  ( LSubSp `  W )
)
399, 18, 22, 38syl3anc 1228 . . . . . . . . . . 11  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  e.  (
LSubSp `  W ) )
402, 4lssss 17454 . . . . . . . . . . 11  |-  ( ( U  .(+)  ( N `  { X } ) )  e.  ( LSubSp `  W )  ->  ( U  .(+)  ( N `  { X } ) ) 
C_  V )
4139, 40syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  C_  V
)
4241adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( U  .(+) 
( N `  {
v } ) )  =  V )  -> 
( U  .(+)  ( N `
 { X }
) )  C_  V
)
43 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  ( U  .(+) 
( N `  {
v } ) )  =  V )  -> 
( U  .(+)  ( N `
 { v } ) )  =  V )
4442, 43sseqtr4d 3546 . . . . . . . 8  |-  ( (
ph  /\  ( U  .(+) 
( N `  {
v } ) )  =  V )  -> 
( U  .(+)  ( N `
 { X }
) )  C_  ( U  .(+)  ( N `  { v } ) ) )
4544adantlr 714 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  C_  ( U  .(+)  ( N `
 { v } ) ) )
46453adant2 1015 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  C_  ( U  .(+)  ( N `
 { v } ) ) )
477adantr 465 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  W  e.  LVec )
4818adantr 465 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  U  e.  ( LSubSp `  W )
)
4939adantr 465 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  ( U  .(+)  ( N `  { X } ) )  e.  ( LSubSp `  W
) )
50 simpr 461 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  v  e.  V )
512, 4, 3, 5, 47, 48, 49, 50lsmcv 17658 . . . . . 6  |-  ( ( ( ph  /\  v  e.  V )  /\  U  C.  ( U  .(+)  ( N `
 { X }
) )  /\  ( U  .(+)  ( N `  { X } ) ) 
C_  ( U  .(+)  ( N `  { v } ) ) )  ->  ( U  .(+)  ( N `  { X } ) )  =  ( U  .(+)  ( N `
 { v } ) ) )
5214, 15, 37, 46, 51syl211anc 1234 . . . . 5  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  =  ( U  .(+)  ( N `
 { v } ) ) )
53 simp3 998 . . . . 5  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  {
v } ) )  =  V )
5452, 53eqtrd 2508 . . . 4  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  =  V )
5554rexlimdv3a 2961 . . 3  |-  ( (
ph  /\  -.  X  e.  U )  ->  ( E. v  e.  V  ( U  .(+)  ( N `
 { v } ) )  =  V  ->  ( U  .(+)  ( N `  { X } ) )  =  V ) )
5613, 55mpd 15 . 2  |-  ( (
ph  /\  -.  X  e.  U )  ->  ( U  .(+)  ( N `  { X } ) )  =  V )
579adantr 465 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  W  e.  LMod )
581adantr 465 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  U  e.  H )
5920adantr 465 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  X  e.  V )
60 simpr 461 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  ( U  .(+)  ( N `  { X } ) )  =  V )
612, 3, 5, 6, 57, 58, 59, 60lshpnel 34181 . 2  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  -.  X  e.  U )
6256, 61impbida 830 1  |-  ( ph  ->  ( -.  X  e.  U  <->  ( U  .(+)  ( N `  { X } ) )  =  V ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818    C_ wss 3481    C. wpss 3482   {csn 4033   ` cfv 5594  (class class class)co 6295   Basecbs 14507  SubGrpcsubg 16067   LSSumclsm 16527   LModclmod 17383   LSubSpclss 17449   LSpanclspn 17488   LVecclvec 17619  LSHypclsh 34173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-cntz 16227  df-lsm 16529  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-drng 17269  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lvec 17620  df-lshyp 34175
This theorem is referenced by:  lshpnel2N  34183  l1cvpat  34252  dochexmidat  36657
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