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Theorem lbsextlem1 16157
Description: Lemma for lbsext 16162. The set  S is the set of all linearly independent sets containing 
C; we show here that it is nonempty. (Contributed by Mario Carneiro, 25-Jun-2014.)
Hypotheses
Ref Expression
lbsext.v  |-  V  =  ( Base `  W
)
lbsext.j  |-  J  =  (LBasis `  W )
lbsext.n  |-  N  =  ( LSpan `  W )
lbsext.w  |-  ( ph  ->  W  e.  LVec )
lbsext.c  |-  ( ph  ->  C  C_  V )
lbsext.x  |-  ( ph  ->  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x } ) ) )
lbsext.s  |-  S  =  { z  e.  ~P V  |  ( C  C_  z  /\  A. x  e.  z  -.  x  e.  ( N `  (
z  \  { x } ) ) ) }
Assertion
Ref Expression
lbsextlem1  |-  ( ph  ->  S  =/=  (/) )
Distinct variable groups:    x, J    ph, x    x, S    x, z, C    x, N, z   
x, V, z    x, W
Allowed substitution hints:    ph( z)    S( z)    J( z)    W( z)

Proof of Theorem lbsextlem1
StepHypRef Expression
1 lbsext.c . . . 4  |-  ( ph  ->  C  C_  V )
2 lbsext.v . . . . . 6  |-  V  =  ( Base `  W
)
3 fvex 5682 . . . . . 6  |-  ( Base `  W )  e.  _V
42, 3eqeltri 2457 . . . . 5  |-  V  e. 
_V
54elpw2 4305 . . . 4  |-  ( C  e.  ~P V  <->  C  C_  V
)
61, 5sylibr 204 . . 3  |-  ( ph  ->  C  e.  ~P V
)
7 lbsext.x . . . 4  |-  ( ph  ->  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x } ) ) )
8 ssid 3310 . . . 4  |-  C  C_  C
97, 8jctil 524 . . 3  |-  ( ph  ->  ( C  C_  C  /\  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x } ) ) ) )
10 sseq2 3313 . . . . 5  |-  ( z  =  C  ->  ( C  C_  z  <->  C  C_  C
) )
11 difeq1 3401 . . . . . . . . 9  |-  ( z  =  C  ->  (
z  \  { x } )  =  ( C  \  { x } ) )
1211fveq2d 5672 . . . . . . . 8  |-  ( z  =  C  ->  ( N `  ( z  \  { x } ) )  =  ( N `
 ( C  \  { x } ) ) )
1312eleq2d 2454 . . . . . . 7  |-  ( z  =  C  ->  (
x  e.  ( N `
 ( z  \  { x } ) )  <->  x  e.  ( N `  ( C  \  { x } ) ) ) )
1413notbid 286 . . . . . 6  |-  ( z  =  C  ->  ( -.  x  e.  ( N `  ( z  \  { x } ) )  <->  -.  x  e.  ( N `  ( C 
\  { x }
) ) ) )
1514raleqbi1dv 2855 . . . . 5  |-  ( z  =  C  ->  ( A. x  e.  z  -.  x  e.  ( N `  ( z  \  { x } ) )  <->  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x } ) ) ) )
1610, 15anbi12d 692 . . . 4  |-  ( z  =  C  ->  (
( C  C_  z  /\  A. x  e.  z  -.  x  e.  ( N `  ( z 
\  { x }
) ) )  <->  ( C  C_  C  /\  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x }
) ) ) ) )
17 lbsext.s . . . 4  |-  S  =  { z  e.  ~P V  |  ( C  C_  z  /\  A. x  e.  z  -.  x  e.  ( N `  (
z  \  { x } ) ) ) }
1816, 17elrab2 3037 . . 3  |-  ( C  e.  S  <->  ( C  e.  ~P V  /\  ( C  C_  C  /\  A. x  e.  C  -.  x  e.  ( N `  ( C  \  {
x } ) ) ) ) )
196, 9, 18sylanbrc 646 . 2  |-  ( ph  ->  C  e.  S )
20 ne0i 3577 . 2  |-  ( C  e.  S  ->  S  =/=  (/) )
2119, 20syl 16 1  |-  ( ph  ->  S  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   {crab 2653   _Vcvv 2899    \ cdif 3260    C_ wss 3263   (/)c0 3571   ~Pcpw 3742   {csn 3757   ` cfv 5394   Basecbs 13396   LSpanclspn 15974  LBasisclbs 16073   LVecclvec 16101
This theorem is referenced by:  lbsextlem4  16160
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402
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