Theorem linds0 42254

Description: The empty set is always a linearly independet subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)

Assertion

Ref	Expression
linds0	|- ( M e. V -> (/) linIndS M )

Proof of Theorem linds0

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

Step	Hyp	Ref	Expression
1		ral0 4076	. . . . . 6 |- A. x e. (/) ( (/) ` x ) = ( 0g ` (Scalar ` M ) )
2	1	2a1i 12	. . . . 5 |- ( M e. V -> ( ( (/) finSupp ( 0g ` (Scalar ` M ) ) /\ ( (/) ( linC ` M ) (/) ) = ( 0g ` M ) ) -> A. x e. (/) ( (/) ` x ) = ( 0g ` (Scalar ` M ) ) ) )
3		0ex 4790	. . . . . 6 |- (/) e. _V
4		breq1 4656	. . . . . . . . 9 |- ( f = (/) -> ( f finSupp ( 0g ` (Scalar ` M ) ) <-> (/) finSupp ( 0g ` (Scalar ` M ) ) ) )
5		oveq1 6657	. . . . . . . . . 10 |- ( f = (/) -> ( f ( linC ` M ) (/) ) = ( (/) ( linC ` M ) (/) ) )
6	5	eqeq1d 2624	. . . . . . . . 9 |- ( f = (/) -> ( ( f ( linC ` M ) (/) ) = ( 0g ` M ) <-> ( (/) ( linC ` M ) (/) ) = ( 0g ` M ) ) )
7	4, 6	anbi12d 747	. . . . . . . 8 |- ( f = (/) -> ( ( f finSupp ( 0g ` (Scalar ` M ) ) /\ ( f ( linC ` M ) (/) ) = ( 0g ` M ) ) <-> ( (/) finSupp ( 0g ` (Scalar ` M ) ) /\ ( (/) ( linC ` M ) (/) ) = ( 0g ` M ) ) ) )
8		fveq1 6190	. . . . . . . . . 10 |- ( f = (/) -> ( f ` x ) = ( (/) ` x ) )
9	8	eqeq1d 2624	. . . . . . . . 9 |- ( f = (/) -> ( ( f ` x ) = ( 0g ` (Scalar ` M ) ) <-> ( (/) ` x ) = ( 0g ` (Scalar ` M ) ) ) )
10	9	ralbidv 2986	. . . . . . . 8 |- ( f = (/) -> ( A. x e. (/) ( f ` x ) = ( 0g ` (Scalar ` M ) ) <-> A. x e. (/) ( (/) ` x ) = ( 0g ` (Scalar ` M ) ) ) )
11	7, 10	imbi12d 334	. . . . . . 7 |- ( f = (/) -> ( ( ( f finSupp ( 0g ` (Scalar ` M ) ) /\ ( f ( linC ` M ) (/) ) = ( 0g ` M ) ) -> A. x e. (/) ( f ` x ) = ( 0g ` (Scalar ` M ) ) ) <-> ( ( (/) finSupp ( 0g ` (Scalar ` M ) ) /\ ( (/) ( linC ` M ) (/) ) = ( 0g ` M ) ) -> A. x e. (/) ( (/) ` x ) = ( 0g ` (Scalar ` M ) ) ) ) )
12	11	ralsng 4218	. . . . . 6 |- ( (/) e. _V -> ( A. f e. { (/) } ( ( f finSupp ( 0g ` (Scalar ` M ) ) /\ ( f ( linC ` M ) (/) ) = ( 0g ` M ) ) -> A. x e. (/) ( f ` x ) = ( 0g ` (Scalar ` M ) ) ) <-> ( ( (/) finSupp ( 0g ` (Scalar ` M ) ) /\ ( (/) ( linC ` M ) (/) ) = ( 0g ` M ) ) -> A. x e. (/) ( (/) ` x ) = ( 0g ` (Scalar ` M ) ) ) ) )
13	3, 12	mp1i 13	. . . . 5 |- ( M e. V -> ( A. f e. { (/) } ( ( f finSupp ( 0g ` (Scalar ` M ) ) /\ ( f ( linC ` M ) (/) ) = ( 0g ` M ) ) -> A. x e. (/) ( f ` x ) = ( 0g ` (Scalar ` M ) ) ) <-> ( ( (/) finSupp ( 0g ` (Scalar ` M ) ) /\ ( (/) ( linC ` M ) (/) ) = ( 0g ` M ) ) -> A. x e. (/) ( (/) ` x ) = ( 0g ` (Scalar ` M ) ) ) ) )
14	2, 13	mpbird 247	. . . 4 |- ( M e. V -> A. f e. { (/) } ( ( f finSupp ( 0g ` (Scalar ` M ) ) /\ ( f ( linC ` M ) (/) ) = ( 0g ` M ) ) -> A. x e. (/) ( f ` x ) = ( 0g ` (Scalar ` M ) ) ) )
15		fvex 6201	. . . . . . 7 |- ( Base ` (Scalar ` M ) ) e. _V
16		map0e 7895	. . . . . . 7 |- ( ( Base ` (Scalar ` M ) ) e. _V -> ( ( Base ` (Scalar ` M ) ) ^m (/) ) = 1o )
17	15, 16	mp1i 13	. . . . . 6 |- ( M e. V -> ( ( Base ` (Scalar ` M ) ) ^m (/) ) = 1o )
18		df1o2 7572	. . . . . 6 |- 1o = { (/) }
19	17, 18	syl6eq 2672	. . . . 5 |- ( M e. V -> ( ( Base ` (Scalar ` M ) ) ^m (/) ) = { (/) } )
20	19	raleqdv 3144	. . . 4 |- ( M e. V -> ( A. f e. ( ( Base ` (Scalar ` M ) ) ^m (/) ) ( ( f finSupp ( 0g ` (Scalar ` M ) ) /\ ( f ( linC ` M ) (/) ) = ( 0g ` M ) ) -> A. x e. (/) ( f ` x ) = ( 0g ` (Scalar ` M ) ) ) <-> A. f e. { (/) } ( ( f finSupp ( 0g ` (Scalar ` M ) ) /\ ( f ( linC ` M ) (/) ) = ( 0g ` M ) ) -> A. x e. (/) ( f ` x ) = ( 0g ` (Scalar ` M ) ) ) ) )
21	14, 20	mpbird 247	. . 3 |- ( M e. V -> A. f e. ( ( Base ` (Scalar ` M ) ) ^m (/) ) ( ( f finSupp ( 0g ` (Scalar ` M ) ) /\ ( f ( linC ` M ) (/) ) = ( 0g ` M ) ) -> A. x e. (/) ( f ` x ) = ( 0g ` (Scalar ` M ) ) ) )
22		0elpw 4834	. . 3 |- (/) e. ~P ( Base ` M )
23	21, 22	jctil 560	. 2 |- ( M e. V -> ( (/) e. ~P ( Base ` M ) /\ A. f e. ( ( Base ` (Scalar ` M ) ) ^m (/) ) ( ( f finSupp ( 0g ` (Scalar ` M ) ) /\ ( f ( linC ` M ) (/) ) = ( 0g ` M ) ) -> A. x e. (/) ( f ` x ) = ( 0g ` (Scalar ` M ) ) ) ) )
24		eqid 2622	. . . 4 |- ( Base ` M ) = ( Base ` M )
25		eqid 2622	. . . 4 |- ( 0g ` M ) = ( 0g ` M )
26		eqid 2622	. . . 4 |- (Scalar ` M ) = (Scalar ` M )
27		eqid 2622	. . . 4 |- ( Base ` (Scalar ` M ) ) = ( Base ` (Scalar ` M ) )
28		eqid 2622	. . . 4 |- ( 0g ` (Scalar ` M ) ) = ( 0g ` (Scalar ` M ) )
29	24, 25, 26, 27, 28	islininds 42235	. . 3 |- ( ( (/) e. _V /\ M e. V ) -> ( (/) linIndS M <-> ( (/) e. ~P ( Base ` M ) /\ A. f e. ( ( Base ` (Scalar ` M ) ) ^m (/) ) ( ( f finSupp ( 0g ` (Scalar ` M ) ) /\ ( f ( linC ` M ) (/) ) = ( 0g ` M ) ) -> A. x e. (/) ( f ` x ) = ( 0g ` (Scalar ` M ) ) ) ) ) )
30	3, 29	mpan 706	. 2 |- ( M e. V -> ( (/) linIndS M <-> ( (/) e. ~P ( Base ` M ) /\ A. f e. ( ( Base ` (Scalar ` M ) ) ^m (/) ) ( ( f finSupp ( 0g ` (Scalar ` M ) ) /\ ( f ( linC ` M ) (/) ) = ( 0g ` M ) ) -> A. x e. (/) ( f ` x ) = ( 0g ` (Scalar ` M ) ) ) ) ) )
31	23, 30	mpbird 247	1 |- ( M e. V -> (/) linIndS M )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   1oc1o 7553   ^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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-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-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-map 7859  df-lininds 42231

This theorem is referenced by: (None)