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Theorem lspsnel6 15767
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 15711 . . . 4  |-  ( ( U  e.  S  /\  X  e.  U )  ->  X  e.  V )
51, 4sylan 457 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  V )
6 lspsnel5.w . . . . 5  |-  ( ph  ->  W  e.  LMod )
76adantr 451 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  W  e.  LMod )
81adantr 451 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  U  e.  S )
9 simpr 447 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  U )
10 lspsnel5.n . . . . 5  |-  N  =  ( LSpan `  W )
113, 10lspsnss 15763 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  X  e.  U )  ->  ( N `  { X } )  C_  U
)
127, 8, 9, 11syl3anc 1182 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  ( N `  { X } )  C_  U
)
135, 12jca 518 . 2  |-  ( (
ph  /\  X  e.  U )  ->  ( X  e.  V  /\  ( N `  { X } )  C_  U
) )
142, 10lspsnid 15766 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
156, 14sylan 457 . . . 4  |-  ( (
ph  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
16 ssel 3187 . . . 4  |-  ( ( N `  { X } )  C_  U  ->  ( X  e.  ( N `  { X } )  ->  X  e.  U ) )
1715, 16syl5com 26 . . 3  |-  ( (
ph  /\  X  e.  V )  ->  (
( N `  { X } )  C_  U  ->  X  e.  U ) )
1817impr 602 . 2  |-  ( (
ph  /\  ( X  e.  V  /\  ( N `  { X } )  C_  U
) )  ->  X  e.  U )
1913, 18impbida 805 1  |-  ( ph  ->  ( X  e.  U  <->  ( X  e.  V  /\  ( N `  { X } )  C_  U
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   {csn 3653   ` cfv 5271   Basecbs 13164   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744
This theorem is referenced by:  lspsnel5  15768  lsmelval2  15854  dihjat1lem  32240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320  df-0g 13420  df-mnd 14383  df-grp 14505  df-lmod 15645  df-lss 15706  df-lsp 15745
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