Theorem islindeps 42242

Description: The property of being a linearly dependent subset. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)

Hypotheses

Ref	Expression
islindeps.b	|- B = ( Base ` M )
islindeps.z	|- Z = ( 0g ` M )
islindeps.r	|- R = (Scalar ` M )
islindeps.e	|- E = ( Base ` R )
islindeps.0	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
islindeps	|- ( ( M e. W /\ S e. ~P B ) -> ( S linDepS M <-> E. f e. ( E ^m S ) ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z /\ E. x e. S ( f ` x ) =/= .0. ) ) )

Distinct variable groups:   f, E   f, M, x   S, f, x

Allowed substitution hints:   B( x, f)   R( x, f)   E( x)   W( x, f)   .0. ( x, f)   Z( x, f)

Proof of Theorem islindeps

Step	Hyp	Ref	Expression
1		lindepsnlininds 42241	. . 3 |- ( ( S e. ~P B /\ M e. W ) -> ( S linDepS M <-> -. S linIndS M ) )
2	1	ancoms 469	. 2 |- ( ( M e. W /\ S e. ~P B ) -> ( S linDepS M <-> -. S linIndS M ) )
3		islindeps.b	. . . . . 6 |- B = ( Base ` M )
4		islindeps.z	. . . . . 6 |- Z = ( 0g ` M )
5		islindeps.r	. . . . . 6 |- R = (Scalar ` M )
6		islindeps.e	. . . . . 6 |- E = ( Base ` R )
7		islindeps.0	. . . . . 6 |- .0. = ( 0g ` R )
8	3, 4, 5, 6, 7	islininds 42235	. . . . 5 |- ( ( S e. ~P B /\ M e. W ) -> ( S linIndS M <-> ( S e. ~P B /\ A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) ) ) )
9	8	ancoms 469	. . . 4 |- ( ( M e. W /\ S e. ~P B ) -> ( S linIndS M <-> ( S e. ~P B /\ A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) ) ) )
10		ibar 525	. . . . . 6 |- ( S e. ~P B -> ( A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) <-> ( S e. ~P B /\ A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) ) ) )
11	10	bicomd 213	. . . . 5 |- ( S e. ~P B -> ( ( S e. ~P B /\ A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) ) <-> A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) ) )
12	11	adantl 482	. . . 4 |- ( ( M e. W /\ S e. ~P B ) -> ( ( S e. ~P B /\ A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) ) <-> A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) ) )
13	9, 12	bitrd 268	. . 3 |- ( ( M e. W /\ S e. ~P B ) -> ( S linIndS M <-> A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) ) )
14	13	notbid 308	. 2 |- ( ( M e. W /\ S e. ~P B ) -> ( -. S linIndS M <-> -. A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) ) )
15		rexnal 2995	. . . 4 |- ( E. f e. ( E ^m S ) -. ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) <-> -. A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) )
16		df-ne 2795	. . . . . . . . 9 |- ( ( f ` x ) =/= .0. <-> -. ( f ` x ) = .0. )
17	16	rexbii 3041	. . . . . . . 8 |- ( E. x e. S ( f ` x ) =/= .0. <-> E. x e. S -. ( f ` x ) = .0. )
18		rexnal 2995	. . . . . . . 8 |- ( E. x e. S -. ( f ` x ) = .0. <-> -. A. x e. S ( f ` x ) = .0. )
19	17, 18	bitr2i 265	. . . . . . 7 |- ( -. A. x e. S ( f ` x ) = .0. <-> E. x e. S ( f ` x ) =/= .0. )
20	19	anbi2i 730	. . . . . 6 |- ( ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) /\ -. A. x e. S ( f ` x ) = .0. ) <-> ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) /\ E. x e. S ( f ` x ) =/= .0. ) )
21		pm4.61 442	. . . . . 6 |- ( -. ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) <-> ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) /\ -. A. x e. S ( f ` x ) = .0. ) )
22		df-3an 1039	. . . . . 6 |- ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z /\ E. x e. S ( f ` x ) =/= .0. ) <-> ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) /\ E. x e. S ( f ` x ) =/= .0. ) )
23	20, 21, 22	3bitr4i 292	. . . . 5 |- ( -. ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) <-> ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z /\ E. x e. S ( f ` x ) =/= .0. ) )
24	23	rexbii 3041	. . . 4 |- ( E. f e. ( E ^m S ) -. ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) <-> E. f e. ( E ^m S ) ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z /\ E. x e. S ( f ` x ) =/= .0. ) )
25	15, 24	bitr3i 266	. . 3 |- ( -. A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) <-> E. f e. ( E ^m S ) ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z /\ E. x e. S ( f ` x ) =/= .0. ) )
26	25	a1i 11	. 2 |- ( ( M e. W /\ S e. ~P B ) -> ( -. A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) <-> E. f e. ( E ^m S ) ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z /\ E. x e. S ( f ` x ) =/= .0. ) ) )
27	2, 14, 26	3bitrd 294	1 |- ( ( M e. W /\ S e. ~P B ) -> ( S linDepS M <-> E. f e. ( E ^m S ) ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z /\ E. x e. S ( f ` x ) =/= .0. ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   ~Pcpw 4158   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   finSupp cfsupp 8275   Basecbs 15857  Scalarcsca 15944   0gc0g 16100   linC clinc 42193   linIndS clininds 42229   linDepS clindeps 42230

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  ax-pr 4906

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-mo 2475  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-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-opab 4713  df-iota 5851  df-fv 5896  df-ov 6653  df-lininds 42231  df-lindeps 42233

This theorem is referenced by:  el0ldep  42255  ldepspr  42262  islindeps2  42272  isldepslvec2  42274  zlmodzxzldep  42293