Theorem el0ldep 42255

Description: A set containing the zero element of a module is always linearly dependent, if the underlying ring has at least two elements. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)

Assertion

Ref	Expression
el0ldep	|- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> S linDepS M )

Proof of Theorem el0ldep

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( Base ` M ) = ( Base ` M )
2		eqid 2622	. . . . 5 |- (Scalar ` M ) = (Scalar ` M )
3		eqid 2622	. . . . 5 |- ( 0g ` (Scalar ` M ) ) = ( 0g ` (Scalar ` M ) )
4		eqid 2622	. . . . 5 |- ( 1r ` (Scalar ` M ) ) = ( 1r ` (Scalar ` M ) )
5		eqeq1 2626	. . . . . . 7 |- ( s = y -> ( s = ( 0g ` M ) <-> y = ( 0g ` M ) ) )
6	5	ifbid 4108	. . . . . 6 |- ( s = y -> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) = if ( y = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) )
7	6	cbvmptv 4750	. . . . 5 |- ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) = ( y e. S |-> if ( y = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) )
8	1, 2, 3, 4, 7	mptcfsupp 42161	. . . 4 |- ( ( M e. LMod /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) finSupp ( 0g ` (Scalar ` M ) ) )
9	8	3adant1r 1319	. . 3 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) finSupp ( 0g ` (Scalar ` M ) ) )
10		simp1l 1085	. . . 4 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> M e. LMod )
11		simp2 1062	. . . 4 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> S e. ~P ( Base ` M ) )
12		eqid 2622	. . . . 5 |- ( 0g ` M ) = ( 0g ` M )
13		eqid 2622	. . . . 5 |- ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) = ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) )
14	1, 2, 3, 4, 12, 13	linc0scn0 42212	. . . 4 |- ( ( M e. LMod /\ S e. ~P ( Base ` M ) ) -> ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ( linC ` M ) S ) = ( 0g ` M ) )
15	10, 11, 14	syl2anc 693	. . 3 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ( linC ` M ) S ) = ( 0g ` M ) )
16		simp3 1063	. . . 4 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> ( 0g ` M ) e. S )
17		fveq2 6191	. . . . . 6 |- ( x = ( 0g ` M ) -> ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ` x ) = ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ` ( 0g ` M ) ) )
18	17	neeq1d 2853	. . . . 5 |- ( x = ( 0g ` M ) -> ( ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ` x ) =/= ( 0g ` (Scalar ` M ) ) <-> ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ` ( 0g ` M ) ) =/= ( 0g ` (Scalar ` M ) ) ) )
19	18	adantl 482	. . . 4 |- ( ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) /\ x = ( 0g ` M ) ) -> ( ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ` x ) =/= ( 0g ` (Scalar ` M ) ) <-> ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ` ( 0g ` M ) ) =/= ( 0g ` (Scalar ` M ) ) ) )
20		fvexd 6203	. . . . . 6 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> ( 1r ` (Scalar ` M ) ) e. _V )
21		iftrue 4092	. . . . . . 7 |- ( s = ( 0g ` M ) -> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) = ( 1r ` (Scalar ` M ) ) )
22	21, 13	fvmptg 6280	. . . . . 6 |- ( ( ( 0g ` M ) e. S /\ ( 1r ` (Scalar ` M ) ) e. _V ) -> ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ` ( 0g ` M ) ) = ( 1r ` (Scalar ` M ) ) )
23	16, 20, 22	syl2anc 693	. . . . 5 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ` ( 0g ` M ) ) = ( 1r ` (Scalar ` M ) ) )
24	2	lmodring 18871	. . . . . . . 8 |- ( M e. LMod -> (Scalar ` M ) e. Ring )
25	24	anim1i 592	. . . . . . 7 |- ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) -> ( (Scalar ` M ) e. Ring /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) )
26	25	3ad2ant1 1082	. . . . . 6 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> ( (Scalar ` M ) e. Ring /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) )
27		eqid 2622	. . . . . . 7 |- ( Base ` (Scalar ` M ) ) = ( Base ` (Scalar ` M ) )
28	27, 4, 3	ring1ne0 18591	. . . . . 6 |- ( ( (Scalar ` M ) e. Ring /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) -> ( 1r ` (Scalar ` M ) ) =/= ( 0g ` (Scalar ` M ) ) )
29	26, 28	syl 17	. . . . 5 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> ( 1r ` (Scalar ` M ) ) =/= ( 0g ` (Scalar ` M ) ) )
30	23, 29	eqnetrd 2861	. . . 4 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ` ( 0g ` M ) ) =/= ( 0g ` (Scalar ` M ) ) )
31	16, 19, 30	rspcedvd 3317	. . 3 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> E. x e. S ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ` x ) =/= ( 0g ` (Scalar ` M ) ) )
32	2, 27, 4	lmod1cl 18890	. . . . . . . . . 10 |- ( M e. LMod -> ( 1r ` (Scalar ` M ) ) e. ( Base ` (Scalar ` M ) ) )
33	2, 27, 3	lmod0cl 18889	. . . . . . . . . 10 |- ( M e. LMod -> ( 0g ` (Scalar ` M ) ) e. ( Base ` (Scalar ` M ) ) )
34	32, 33	ifcld 4131	. . . . . . . . 9 |- ( M e. LMod -> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) e. ( Base ` (Scalar ` M ) ) )
35	34	adantr 481	. . . . . . . 8 |- ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) -> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) e. ( Base ` (Scalar ` M ) ) )
36	35	3ad2ant1 1082	. . . . . . 7 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) e. ( Base ` (Scalar ` M ) ) )
37	36	adantr 481	. . . . . 6 |- ( ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) /\ s e. S ) -> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) e. ( Base ` (Scalar ` M ) ) )
38	37, 13	fmptd 6385	. . . . 5 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) : S --> ( Base ` (Scalar ` M ) ) )
39		fvexd 6203	. . . . . 6 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> ( Base ` (Scalar ` M ) ) e. _V )
40	39, 11	elmapd 7871	. . . . 5 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) e. ( ( Base ` (Scalar ` M ) ) ^m S ) <-> ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) : S --> ( Base ` (Scalar ` M ) ) ) )
41	38, 40	mpbird 247	. . . 4 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) e. ( ( Base ` (Scalar ` M ) ) ^m S ) )
42		breq1 4656	. . . . . 6 |- ( f = ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) -> ( f finSupp ( 0g ` (Scalar ` M ) ) <-> ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) finSupp ( 0g ` (Scalar ` M ) ) ) )
43		oveq1 6657	. . . . . . 7 |- ( f = ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) -> ( f ( linC ` M ) S ) = ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ( linC ` M ) S ) )
44	43	eqeq1d 2624	. . . . . 6 |- ( f = ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) -> ( ( f ( linC ` M ) S ) = ( 0g ` M ) <-> ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ( linC ` M ) S ) = ( 0g ` M ) ) )
45		fveq1 6190	. . . . . . . 8 |- ( f = ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) -> ( f ` x ) = ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ` x ) )
46	45	neeq1d 2853	. . . . . . 7 |- ( f = ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) -> ( ( f ` x ) =/= ( 0g ` (Scalar ` M ) ) <-> ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ` x ) =/= ( 0g ` (Scalar ` M ) ) ) )
47	46	rexbidv 3052	. . . . . 6 |- ( f = ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) -> ( E. x e. S ( f ` x ) =/= ( 0g ` (Scalar ` M ) ) <-> E. x e. S ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ` x ) =/= ( 0g ` (Scalar ` M ) ) ) )
48	42, 44, 47	3anbi123d 1399	. . . . 5 |- ( f = ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) -> ( ( f finSupp ( 0g ` (Scalar ` M ) ) /\ ( f ( linC ` M ) S ) = ( 0g ` M ) /\ E. x e. S ( f ` x ) =/= ( 0g ` (Scalar ` M ) ) ) <-> ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) finSupp ( 0g ` (Scalar ` M ) ) /\ ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ( linC ` M ) S ) = ( 0g ` M ) /\ E. x e. S ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ` x ) =/= ( 0g ` (Scalar ` M ) ) ) ) )
49	48	adantl 482	. . . 4 |- ( ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) /\ f = ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ) -> ( ( f finSupp ( 0g ` (Scalar ` M ) ) /\ ( f ( linC ` M ) S ) = ( 0g ` M ) /\ E. x e. S ( f ` x ) =/= ( 0g ` (Scalar ` M ) ) ) <-> ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) finSupp ( 0g ` (Scalar ` M ) ) /\ ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ( linC ` M ) S ) = ( 0g ` M ) /\ E. x e. S ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ` x ) =/= ( 0g ` (Scalar ` M ) ) ) ) )
50	41, 49	rspcedv 3313	. . 3 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> ( ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) finSupp ( 0g ` (Scalar ` M ) ) /\ ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ( linC ` M ) S ) = ( 0g ` M ) /\ E. x e. S ( ( s e. S |-> if ( s = ( 0g ` M ) , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ` x ) =/= ( 0g ` (Scalar ` M ) ) ) -> E. f e. ( ( Base ` (Scalar ` M ) ) ^m S ) ( f finSupp ( 0g ` (Scalar ` M ) ) /\ ( f ( linC ` M ) S ) = ( 0g ` M ) /\ E. x e. S ( f ` x ) =/= ( 0g ` (Scalar ` M ) ) ) ) )
51	9, 15, 31, 50	mp3and 1427	. 2 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> E. f e. ( ( Base ` (Scalar ` M ) ) ^m S ) ( f finSupp ( 0g ` (Scalar ` M ) ) /\ ( f ( linC ` M ) S ) = ( 0g ` M ) /\ E. x e. S ( f ` x ) =/= ( 0g ` (Scalar ` M ) ) ) )
52	1, 12, 2, 27, 3	islindeps 42242	. . 3 |- ( ( M e. LMod /\ S e. ~P ( Base ` M ) ) -> ( S linDepS M <-> E. f e. ( ( Base ` (Scalar ` M ) ) ^m S ) ( f finSupp ( 0g ` (Scalar ` M ) ) /\ ( f ( linC ` M ) S ) = ( 0g ` M ) /\ E. x e. S ( f ` x ) =/= ( 0g ` (Scalar ` M ) ) ) ) )
53	10, 11, 52	syl2anc 693	. 2 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> ( S linDepS M <-> E. f e. ( ( Base ` (Scalar ` M ) ) ^m S ) ( f finSupp ( 0g ` (Scalar ` M ) ) /\ ( f ( linC ` M ) S ) = ( 0g ` M ) /\ E. x e. S ( f ` x ) =/= ( 0g ` (Scalar ` M ) ) ) ) )
54	51, 53	mpbird 247	1 |- ( ( ( M e. LMod /\ 1 < ( # ` ( Base ` (Scalar ` M ) ) ) ) /\ S e. ~P ( Base ` M ) /\ ( 0g ` M ) e. S ) -> S linDepS M )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   ifcif 4086   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   finSupp cfsupp 8275   1c1 9937   < clt 10074   #chash 13117   Basecbs 15857  Scalarcsca 15944   0gc0g 16100   1rcur 18501   Ringcrg 18547   LModclmod 18863   linC clinc 42193   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-hash 13118  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-linc 42195  df-lininds 42231  df-lindeps 42233

This theorem is referenced by:  el0ldepsnzr  42256