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Theorem lbsextlem1 15911
Description: Lemma for lbsext 15916. 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 5539 . . . . . 6  |-  ( Base `  W )  e.  _V
42, 3eqeltri 2353 . . . . 5  |-  V  e. 
_V
54elpw2 4175 . . . 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 3197 . . . 4  |-  C  C_  C
97, 8jctil 523 . . 3  |-  ( ph  ->  ( C  C_  C  /\  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x } ) ) ) )
10 sseq2 3200 . . . . 5  |-  ( z  =  C  ->  ( C  C_  z  <->  C  C_  C
) )
11 difeq1 3287 . . . . . . . . 9  |-  ( z  =  C  ->  (
z  \  { x } )  =  ( C  \  { x } ) )
1211fveq2d 5529 . . . . . . . 8  |-  ( z  =  C  ->  ( N `  ( z  \  { x } ) )  =  ( N `
 ( C  \  { x } ) ) )
1312eleq2d 2350 . . . . . . 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 2744 . . . . 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 2925 . . 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 3461 . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   ` cfv 5255   Basecbs 13148   LSpanclspn 15728  LBasisclbs 15827   LVecclvec 15855
This theorem is referenced by:  lbsextlem4  15914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263
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