Theorem linindscl 42240

Description: A linearly independent set is a subset of (the base set of) a module. (Contributed by AV, 13-Apr-2019.)

Assertion

Ref	Expression
linindscl	|- ( S linIndS M -> S e. ~P ( Base ` M ) )

Proof of Theorem linindscl

Dummy variables x f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( Base ` M ) = ( Base ` M )
2		eqid 2622	. . 3 |- ( 0g ` M ) = ( 0g ` M )
3		eqid 2622	. . 3 |- (Scalar ` M ) = (Scalar ` M )
4		eqid 2622	. . 3 |- ( Base ` (Scalar ` M ) ) = ( Base ` (Scalar ` M ) )
5		eqid 2622	. . 3 |- ( 0g ` (Scalar ` M ) ) = ( 0g ` (Scalar ` M ) )
6	1, 2, 3, 4, 5	linindsi 42236	. 2 |- ( S linIndS M -> ( S e. ~P ( Base ` M ) /\ A. f e. ( ( Base ` (Scalar ` M ) ) ^m S ) ( ( f finSupp ( 0g ` (Scalar ` M ) ) /\ ( f ( linC ` M ) S ) = ( 0g ` M ) ) -> A. x e. S ( f ` x ) = ( 0g ` (Scalar ` M ) ) ) ) )
7	6	simpld 475	1 |- ( S linIndS M -> S e. ~P ( Base ` M ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   ~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: (None)