Theorem lbsextlem1 19158

Description: Lemma for lbsext 19163. 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

Step	Hyp	Ref	Expression
1		lbsext.c	. . . 4 |- ( ph -> C C_ V )
2		lbsext.v	. . . . . 6 |- V = ( Base ` W )
3		fvex 6201	. . . . . 6 |- ( Base ` W ) e. _V
4	2, 3	eqeltri 2697	. . . . 5 |- V e. _V
5	4	elpw2 4828	. . . 4 |- ( C e. ~P V <-> C C_ V )
6	1, 5	sylibr 224	. . 3 |- ( ph -> C e. ~P V )
7		lbsext.x	. . . 4 |- ( ph -> A. x e. C -. x e. ( N ` ( C \ { x } ) ) )
8		ssid 3624	. . . 4 |- C C_ C
9	7, 8	jctil 560	. . 3 |- ( ph -> ( C C_ C /\ A. x e. C -. x e. ( N ` ( C \ { x } ) ) ) )
10		sseq2 3627	. . . . 5 |- ( z = C -> ( C C_ z <-> C C_ C ) )
11		difeq1 3721	. . . . . . . . 9 |- ( z = C -> ( z \ { x } ) = ( C \ { x } ) )
12	11	fveq2d 6195	. . . . . . . 8 |- ( z = C -> ( N ` ( z \ { x } ) ) = ( N ` ( C \ { x } ) ) )
13	12	eleq2d 2687	. . . . . . 7 |- ( z = C -> ( x e. ( N ` ( z \ { x } ) ) <-> x e. ( N ` ( C \ { x } ) ) ) )
14	13	notbid 308	. . . . . 6 |- ( z = C -> ( -. x e. ( N ` ( z \ { x } ) ) <-> -. x e. ( N ` ( C \ { x } ) ) ) )
15	14	raleqbi1dv 3146	. . . . 5 |- ( z = C -> ( A. x e. z -. x e. ( N ` ( z \ { x } ) ) <-> A. x e. C -. x e. ( N ` ( C \ { x } ) ) ) )
16	10, 15	anbi12d 747	. . . 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 } ) ) ) }
18	16, 17	elrab2 3366	. . 3 |- ( C e. S <-> ( C e. ~P V /\ ( C C_ C /\ A. x e. C -. x e. ( N ` ( C \ { x } ) ) ) ) )
19	6, 9, 18	sylanbrc 698	. 2 |- ( ph -> C e. S )
20		ne0i 3921	. 2 |- ( C e. S -> S =/= (/) )
21	19, 20	syl 17	1 |- ( ph -> S =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   ` cfv 5888   Basecbs 15857   LSpanclspn 18971  LBasisclbs 19074   LVecclvec 19102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  lbsextlem4  19161