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Theorem lspsnel6 17509
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 17453 . . . 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 17505 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  X  e.  U )  ->  ( N `  { X } )  C_  U
)
127, 8, 9, 11syl3anc 1228 . . 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 17508 . . . . 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 3503 . . . 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 830 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 1379    e. wcel 1767    C_ wss 3481   {csn 4033   ` cfv 5594   Basecbs 14506   LModclmod 17381   LSubSpclss 17447   LSpanclspn 17486
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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-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-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  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-0g 14713  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-grp 15928  df-lmod 17383  df-lss 17448  df-lsp 17487
This theorem is referenced by:  lspsnel5  17510  lsmelval2  17600  dihjat1lem  36631
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