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Theorem lspsnel6 17074
Description: Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
Hypotheses
Ref Expression
lspsnel5.v  |-  V  =  ( Base `  W
)
lspsnel5.s  |-  S  =  ( LSubSp `  W )
lspsnel5.n  |-  N  =  ( LSpan `  W )
lspsnel5.w  |-  ( ph  ->  W  e.  LMod )
lspsnel5.a  |-  ( ph  ->  U  e.  S )
Assertion
Ref Expression
lspsnel6  |-  ( ph  ->  ( X  e.  U  <->  ( X  e.  V  /\  ( N `  { X } )  C_  U
) ) )

Proof of Theorem lspsnel6
StepHypRef Expression
1 lspsnel5.a . . . 4  |-  ( ph  ->  U  e.  S )
2 lspsnel5.v . . . . 5  |-  V  =  ( Base `  W
)
3 lspsnel5.s . . . . 5  |-  S  =  ( LSubSp `  W )
42, 3lssel 17018 . . . 4  |-  ( ( U  e.  S  /\  X  e.  U )  ->  X  e.  V )
51, 4sylan 471 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  V )
6 lspsnel5.w . . . . 5  |-  ( ph  ->  W  e.  LMod )
76adantr 465 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  W  e.  LMod )
81adantr 465 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  U  e.  S )
9 simpr 461 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  U )
10 lspsnel5.n . . . . 5  |-  N  =  ( LSpan `  W )
113, 10lspsnss 17070 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  X  e.  U )  ->  ( N `  { X } )  C_  U
)
127, 8, 9, 11syl3anc 1218 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  ( N `  { X } )  C_  U
)
135, 12jca 532 . 2  |-  ( (
ph  /\  X  e.  U )  ->  ( X  e.  V  /\  ( N `  { X } )  C_  U
) )
142, 10lspsnid 17073 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
156, 14sylan 471 . . . 4  |-  ( (
ph  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
16 ssel 3349 . . . 4  |-  ( ( N `  { X } )  C_  U  ->  ( X  e.  ( N `  { X } )  ->  X  e.  U ) )
1715, 16syl5com 30 . . 3  |-  ( (
ph  /\  X  e.  V )  ->  (
( N `  { X } )  C_  U  ->  X  e.  U ) )
1817impr 619 . 2  |-  ( (
ph  /\  ( X  e.  V  /\  ( N `  { X } )  C_  U
) )  ->  X  e.  U )
1913, 18impbida 828 1  |-  ( ph  ->  ( X  e.  U  <->  ( X  e.  V  /\  ( N `  { X } )  C_  U
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3327   {csn 3876   ` cfv 5417   Basecbs 14173   LModclmod 16947   LSubSpclss 17012   LSpanclspn 17051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-0g 14379  df-mnd 15414  df-grp 15544  df-lmod 16949  df-lss 17013  df-lsp 17052
This theorem is referenced by:  lspsnel5  17075  lsmelval2  17165  dihjat1lem  35071
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