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Theorem lbsextlem1 17999
Description: Lemma for lbsext 18004. 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 5858 . . . . . 6  |-  ( Base `  W )  e.  _V
42, 3eqeltri 2538 . . . . 5  |-  V  e. 
_V
54elpw2 4601 . . . 4  |-  ( C  e.  ~P V  <->  C  C_  V
)
61, 5sylibr 212 . . 3  |-  ( ph  ->  C  e.  ~P V
)
7 lbsext.x . . . 4  |-  ( ph  ->  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x } ) ) )
8 ssid 3508 . . . 4  |-  C  C_  C
97, 8jctil 535 . . 3  |-  ( ph  ->  ( C  C_  C  /\  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x } ) ) ) )
10 sseq2 3511 . . . . 5  |-  ( z  =  C  ->  ( C  C_  z  <->  C  C_  C
) )
11 difeq1 3601 . . . . . . . . 9  |-  ( z  =  C  ->  (
z  \  { x } )  =  ( C  \  { x } ) )
1211fveq2d 5852 . . . . . . . 8  |-  ( z  =  C  ->  ( N `  ( z  \  { x } ) )  =  ( N `
 ( C  \  { x } ) ) )
1312eleq2d 2524 . . . . . . 7  |-  ( z  =  C  ->  (
x  e.  ( N `
 ( z  \  { x } ) )  <->  x  e.  ( N `  ( C  \  { x } ) ) ) )
1413notbid 292 . . . . . 6  |-  ( z  =  C  ->  ( -.  x  e.  ( N `  ( z  \  { x } ) )  <->  -.  x  e.  ( N `  ( C 
\  { x }
) ) ) )
1514raleqbi1dv 3059 . . . . 5  |-  ( z  =  C  ->  ( A. x  e.  z  -.  x  e.  ( N `  ( z  \  { x } ) )  <->  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x } ) ) ) )
1610, 15anbi12d 708 . . . 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 3256 . . 3  |-  ( C  e.  S  <->  ( C  e.  ~P V  /\  ( C  C_  C  /\  A. x  e.  C  -.  x  e.  ( N `  ( C  \  {
x } ) ) ) ) )
196, 9, 18sylanbrc 662 . 2  |-  ( ph  ->  C  e.  S )
20 ne0i 3789 . 2  |-  ( C  e.  S  ->  S  =/=  (/) )
2119, 20syl 16 1  |-  ( ph  ->  S  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   {crab 2808   _Vcvv 3106    \ cdif 3458    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016   ` cfv 5570   Basecbs 14716   LSpanclspn 17812  LBasisclbs 17915   LVecclvec 17943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578
This theorem is referenced by:  lbsextlem4  18002
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