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Theorem lshpnelb 29174
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 2283 . . . . . . 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 15859 . . . . . . . 8  |-  ( W  e.  LVec  ->  W  e. 
LMod )
97, 8syl 15 . . . . . . 7  |-  ( ph  ->  W  e.  LMod )
102, 3, 4, 5, 6, 9islshpsm 29170 . . . . . 6  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  ( LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+) 
( N `  {
v } ) )  =  V ) ) )
111, 10mpbid 201 . . . . 5  |-  ( ph  ->  ( U  e.  (
LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+)  ( N `  { v } ) )  =  V ) )
1211simp3d 969 . . . 4  |-  ( ph  ->  E. v  e.  V  ( U  .(+)  ( N `
 { v } ) )  =  V )
1312adantr 451 . . 3  |-  ( (
ph  /\  -.  X  e.  U )  ->  E. v  e.  V  ( U  .(+) 
( N `  {
v } ) )  =  V )
14 simp1l 979 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ph )
15 simp2 956 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  v  e.  V )
164lsssssubg 15715 . . . . . . . . . . . 12  |-  ( W  e.  LMod  ->  ( LSubSp `  W )  C_  (SubGrp `  W ) )
179, 16syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( LSubSp `  W )  C_  (SubGrp `  W )
)
18 eqid 2283 . . . . . . . . . . . 12  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
1918, 6, 9, 1lshplss 29171 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
2017, 19sseldd 3181 . . . . . . . . . 10  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
21 lshpnelb.x . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  V )
222, 4, 3lspsncl 15734 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
239, 21, 22syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
2417, 23sseldd 3181 . . . . . . . . . 10  |-  ( ph  ->  ( N `  { X } )  e.  (SubGrp `  W ) )
255lsmub1 14967 . . . . . . . . . 10  |-  ( ( U  e.  (SubGrp `  W )  /\  ( N `  { X } )  e.  (SubGrp `  W ) )  ->  U  C_  ( U  .(+)  ( N `  { X } ) ) )
2620, 24, 25syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  U  C_  ( U  .(+) 
( N `  { X } ) ) )
2726adantr 451 . . . . . . . 8  |-  ( (
ph  /\  -.  X  e.  U )  ->  U  C_  ( U  .(+)  ( N `
 { X }
) ) )
285lsmub2 14968 . . . . . . . . . . . 12  |-  ( ( U  e.  (SubGrp `  W )  /\  ( N `  { X } )  e.  (SubGrp `  W ) )  -> 
( N `  { X } )  C_  ( U  .(+)  ( N `  { X } ) ) )
2920, 24, 28syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  C_  ( U  .(+)  ( N `  { X } ) ) )
302, 3lspsnid 15750 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
319, 21, 30syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  ( N `
 { X }
) )
3229, 31sseldd 3181 . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( U 
.(+)  ( N `  { X } ) ) )
33 nelne1 2535 . . . . . . . . . 10  |-  ( ( X  e.  ( U 
.(+)  ( N `  { X } ) )  /\  -.  X  e.  U )  ->  ( U  .(+)  ( N `  { X } ) )  =/=  U )
3432, 33sylan 457 . . . . . . . . 9  |-  ( (
ph  /\  -.  X  e.  U )  ->  ( U  .(+)  ( N `  { X } ) )  =/=  U )
3534necomd 2529 . . . . . . . 8  |-  ( (
ph  /\  -.  X  e.  U )  ->  U  =/=  ( U  .(+)  ( N `
 { X }
) ) )
36 df-pss 3168 . . . . . . . 8  |-  ( U 
C.  ( U  .(+)  ( N `  { X } ) )  <->  ( U  C_  ( U  .(+)  ( N `
 { X }
) )  /\  U  =/=  ( U  .(+)  ( N `
 { X }
) ) ) )
3727, 35, 36sylanbrc 645 . . . . . . 7  |-  ( (
ph  /\  -.  X  e.  U )  ->  U  C.  ( U  .(+)  ( N `
 { X }
) ) )
38373ad2ant1 976 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  U  C.  ( U  .(+)  ( N `
 { X }
) ) )
392, 18, 3lspsncl 15734 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
409, 21, 39syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
414, 5lsmcl 15836 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  U  e.  ( LSubSp `  W )  /\  ( N `  { X } )  e.  (
LSubSp `  W ) )  ->  ( U  .(+)  ( N `  { X } ) )  e.  ( LSubSp `  W )
)
429, 19, 40, 41syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  e.  (
LSubSp `  W ) )
432, 4lssss 15694 . . . . . . . . . . 11  |-  ( ( U  .(+)  ( N `  { X } ) )  e.  ( LSubSp `  W )  ->  ( U  .(+)  ( N `  { X } ) ) 
C_  V )
4442, 43syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  C_  V
)
4544adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( U  .(+) 
( N `  {
v } ) )  =  V )  -> 
( U  .(+)  ( N `
 { X }
) )  C_  V
)
46 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  ( U  .(+) 
( N `  {
v } ) )  =  V )  -> 
( U  .(+)  ( N `
 { v } ) )  =  V )
4745, 46sseqtr4d 3215 . . . . . . . 8  |-  ( (
ph  /\  ( U  .(+) 
( N `  {
v } ) )  =  V )  -> 
( U  .(+)  ( N `
 { X }
) )  C_  ( U  .(+)  ( N `  { v } ) ) )
4847adantlr 695 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  C_  ( U  .(+)  ( N `
 { v } ) ) )
49483adant2 974 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  C_  ( U  .(+)  ( N `
 { v } ) ) )
507adantr 451 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  W  e.  LVec )
514, 6, 9, 1lshplss 29171 . . . . . . . 8  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
5251adantr 451 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  U  e.  ( LSubSp `  W )
)
539, 19, 23, 41syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  e.  (
LSubSp `  W ) )
5453adantr 451 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  ( U  .(+)  ( N `  { X } ) )  e.  ( LSubSp `  W
) )
55 simpr 447 . . . . . . 7  |-  ( (
ph  /\  v  e.  V )  ->  v  e.  V )
562, 4, 3, 5, 50, 52, 54, 55lsmcv 15894 . . . . . 6  |-  ( ( ( ph  /\  v  e.  V )  /\  U  C.  ( U  .(+)  ( N `
 { X }
) )  /\  ( U  .(+)  ( N `  { X } ) ) 
C_  ( U  .(+)  ( N `  { v } ) ) )  ->  ( U  .(+)  ( N `  { X } ) )  =  ( U  .(+)  ( N `
 { v } ) ) )
5714, 15, 38, 49, 56syl211anc 1188 . . . . 5  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  =  ( U  .(+)  ( N `
 { v } ) ) )
58 simp3 957 . . . . 5  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  {
v } ) )  =  V )
5957, 58eqtrd 2315 . . . 4  |-  ( ( ( ph  /\  -.  X  e.  U )  /\  v  e.  V  /\  ( U  .(+)  ( N `
 { v } ) )  =  V )  ->  ( U  .(+) 
( N `  { X } ) )  =  V )
6059rexlimdv3a 2669 . . 3  |-  ( (
ph  /\  -.  X  e.  U )  ->  ( E. v  e.  V  ( U  .(+)  ( N `
 { v } ) )  =  V  ->  ( U  .(+)  ( N `  { X } ) )  =  V ) )
6113, 60mpd 14 . 2  |-  ( (
ph  /\  -.  X  e.  U )  ->  ( U  .(+)  ( N `  { X } ) )  =  V )
629adantr 451 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  W  e.  LMod )
631adantr 451 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  U  e.  H )
6421adantr 451 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  X  e.  V )
65 simpr 447 . . 3  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  ( U  .(+)  ( N `  { X } ) )  =  V )
662, 3, 5, 6, 62, 63, 64, 65lshpnel 29173 . 2  |-  ( (
ph  /\  ( U  .(+) 
( N `  { X } ) )  =  V )  ->  -.  X  e.  U )
6761, 66impbida 805 1  |-  ( ph  ->  ( -.  X  e.  U  <->  ( U  .(+)  ( N `  { X } ) )  =  V ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    C_ wss 3152    C. wpss 3153   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148  SubGrpcsubg 14615   LSSumclsm 14945   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728   LVecclvec 15855  LSHypclsh 29165
This theorem is referenced by:  lshpnel2N  29175  l1cvpat  29244  dochexmidat  31649
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lshyp 29167
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