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Theorem lbsextlem1 15927
Description: Lemma for lbsext 15932. 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 5555 . . . . . 6  |-  ( Base `  W )  e.  _V
42, 3eqeltri 2366 . . . . 5  |-  V  e. 
_V
54elpw2 4191 . . . 4  |-  ( C  e.  ~P V  <->  C  C_  V
)
61, 5sylibr 203 . . 3  |-  ( ph  ->  C  e.  ~P V
)
7 lbsext.x . . . 4  |-  ( ph  ->  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x } ) ) )
8 ssid 3210 . . . 4  |-  C  C_  C
97, 8jctil 523 . . 3  |-  ( ph  ->  ( C  C_  C  /\  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x } ) ) ) )
10 sseq2 3213 . . . . 5  |-  ( z  =  C  ->  ( C  C_  z  <->  C  C_  C
) )
11 difeq1 3300 . . . . . . . . 9  |-  ( z  =  C  ->  (
z  \  { x } )  =  ( C  \  { x } ) )
1211fveq2d 5545 . . . . . . . 8  |-  ( z  =  C  ->  ( N `  ( z  \  { x } ) )  =  ( N `
 ( C  \  { x } ) ) )
1312eleq2d 2363 . . . . . . 7  |-  ( z  =  C  ->  (
x  e.  ( N `
 ( z  \  { x } ) )  <->  x  e.  ( N `  ( C  \  { x } ) ) ) )
1413notbid 285 . . . . . 6  |-  ( z  =  C  ->  ( -.  x  e.  ( N `  ( z  \  { x } ) )  <->  -.  x  e.  ( N `  ( C 
\  { x }
) ) ) )
1514raleqbi1dv 2757 . . . . 5  |-  ( z  =  C  ->  ( A. x  e.  z  -.  x  e.  ( N `  ( z  \  { x } ) )  <->  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x } ) ) ) )
1610, 15anbi12d 691 . . . 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 2938 . . 3  |-  ( C  e.  S  <->  ( C  e.  ~P V  /\  ( C  C_  C  /\  A. x  e.  C  -.  x  e.  ( N `  ( C  \  {
x } ) ) ) ) )
196, 9, 18sylanbrc 645 . 2  |-  ( ph  ->  C  e.  S )
20 ne0i 3474 . 2  |-  ( C  e.  S  ->  S  =/=  (/) )
2119, 20syl 15 1  |-  ( ph  ->  S  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560   _Vcvv 2801    \ cdif 3162    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   ` cfv 5271   Basecbs 13164   LSpanclspn 15744  LBasisclbs 15843   LVecclvec 15871
This theorem is referenced by:  lbsextlem4  15930
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165
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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  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-br 4040  df-iota 5235  df-fv 5279
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