Theorem linindsi 42236

Description: The implications of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)

Hypotheses

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

Assertion

Ref	Expression
linindsi	|- ( 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. ) ) )

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

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

Proof of Theorem linindsi

Step	Hyp	Ref	Expression
1		linindsv 42234	. . 3 |- ( S linIndS M -> ( S e. _V /\ M e. _V ) )
2		islininds.b	. . . 4 |- B = ( Base ` M )
3		islininds.z	. . . 4 |- Z = ( 0g ` M )
4		islininds.r	. . . 4 |- R = (Scalar ` M )
5		islininds.e	. . . 4 |- E = ( Base ` R )
6		islininds.0	. . . 4 |- .0. = ( 0g ` R )
7	2, 3, 4, 5, 6	islininds 42235	. . 3 |- ( ( S e. _V /\ M e. _V ) -> ( 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. ) ) ) )
8	1, 7	syl 17	. 2 |- ( S linIndS M -> ( 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	ibi 256	1 |- ( 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. ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   ~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

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-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-xp 5120  df-rel 5121  df-iota 5851  df-fv 5896  df-ov 6653  df-lininds 42231

This theorem is referenced by:  linindslinci  42237  linindscl  42240