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Theorem lspsnel6 16062
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 16006 . . . 4  |-  ( ( U  e.  S  /\  X  e.  U )  ->  X  e.  V )
51, 4sylan 458 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  V )
6 lspsnel5.w . . . . 5  |-  ( ph  ->  W  e.  LMod )
76adantr 452 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  W  e.  LMod )
81adantr 452 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  U  e.  S )
9 simpr 448 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  U )
10 lspsnel5.n . . . . 5  |-  N  =  ( LSpan `  W )
113, 10lspsnss 16058 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  X  e.  U )  ->  ( N `  { X } )  C_  U
)
127, 8, 9, 11syl3anc 1184 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  ( N `  { X } )  C_  U
)
135, 12jca 519 . 2  |-  ( (
ph  /\  X  e.  U )  ->  ( X  e.  V  /\  ( N `  { X } )  C_  U
) )
142, 10lspsnid 16061 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
156, 14sylan 458 . . . 4  |-  ( (
ph  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
16 ssel 3334 . . . 4  |-  ( ( N `  { X } )  C_  U  ->  ( X  e.  ( N `  { X } )  ->  X  e.  U ) )
1715, 16syl5com 28 . . 3  |-  ( (
ph  /\  X  e.  V )  ->  (
( N `  { X } )  C_  U  ->  X  e.  U ) )
1817impr 603 . 2  |-  ( (
ph  /\  ( X  e.  V  /\  ( N `  { X } )  C_  U
) )  ->  X  e.  U )
1913, 18impbida 806 1  |-  ( ph  ->  ( X  e.  U  <->  ( X  e.  V  /\  ( N `  { X } )  C_  U
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312   {csn 3806   ` cfv 5446   Basecbs 13461   LModclmod 15942   LSubSpclss 16000   LSpanclspn 16039
This theorem is referenced by:  lspsnel5  16063  lsmelval2  16149  dihjat1lem  32163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-riota 6541  df-0g 13719  df-mnd 14682  df-grp 14804  df-lmod 15944  df-lss 16001  df-lsp 16040
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