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Theorem lspsnel6 13560
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.w . . . . 5  |-  ( ph  ->  W  e.  LMod )
21adantr 276 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  W  e.  LMod )
3 lspsnel5.a . . . . 5  |-  ( ph  ->  U  e.  S )
43adantr 276 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  U  e.  S )
5 simpr 110 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  U )
6 lspsnel5.v . . . . 5  |-  V  =  ( Base `  W
)
7 lspsnel5.s . . . . 5  |-  S  =  ( LSubSp `  W )
86, 7lsselg 13513 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  X  e.  U )  ->  X  e.  V )
92, 4, 5, 8syl3anc 1248 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  V )
10 lspsnel5.n . . . . 5  |-  N  =  ( LSpan `  W )
117, 10lspsnss 13556 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  X  e.  U )  ->  ( N `  { X } )  C_  U
)
122, 4, 5, 11syl3anc 1248 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  ( N `  { X } )  C_  U
)
139, 12jca 306 . 2  |-  ( (
ph  /\  X  e.  U )  ->  ( X  e.  V  /\  ( N `  { X } )  C_  U
) )
146, 10lspsnid 13559 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
151, 14sylan 283 . . . 4  |-  ( (
ph  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
16 ssel 3161 . . . 4  |-  ( ( N `  { X } )  C_  U  ->  ( X  e.  ( N `  { X } )  ->  X  e.  U ) )
1715, 16syl5com 29 . . 3  |-  ( (
ph  /\  X  e.  V )  ->  (
( N `  { X } )  C_  U  ->  X  e.  U ) )
1817impr 379 . 2  |-  ( (
ph  /\  ( X  e.  V  /\  ( N `  { X } )  C_  U
) )  ->  X  e.  U )
1913, 18impbida 596 1  |-  ( ph  ->  ( X  e.  U  <->  ( X  e.  V  /\  ( N `  { X } )  C_  U
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1363    e. wcel 2158    C_ wss 3141   {csn 3604   ` cfv 5228   Basecbs 12475   LModclmod 13439   LSubSpclss 13504   LSpanclspn 13538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-5 8994  df-6 8995  df-ndx 12478  df-slot 12479  df-base 12481  df-plusg 12563  df-mulr 12564  df-sca 12566  df-vsca 12567  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12837  df-grp 12899  df-lmod 13441  df-lssm 13505  df-lsp 13539
This theorem is referenced by:  lspsnel5  13561
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