Theorem linc0scn0 42212

Description: If a set contains the zero element of a module, there is a linear combination being 0 where not all scalars are 0. (Contributed by AV, 13-Apr-2019.)

Hypotheses

Ref	Expression
linc0scn0.b	|- B = ( Base ` M )
linc0scn0.s	|- S = (Scalar ` M )
linc0scn0.0	|- .0. = ( 0g ` S )
linc0scn0.1	|- .1. = ( 1r ` S )
linc0scn0.z	|- Z = ( 0g ` M )
linc0scn0.f	|- F = ( x e. V |-> if ( x = Z , .1. , .0. ) )

Assertion

Ref	Expression
linc0scn0	|- ( ( M e. LMod /\ V e. ~P B ) -> ( F ( linC ` M ) V ) = Z )

Distinct variable groups:   x, B   x, M   x, V   x, Z   x, .0.   x, .1.

Allowed substitution hints:   S( x)   F( x)

Proof of Theorem linc0scn0

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( M e. LMod /\ V e. ~P B ) -> M e. LMod )
2		linc0scn0.s	. . . . . . . . 9 |- S = (Scalar ` M )
3	2	lmodring 18871	. . . . . . . 8 |- ( M e. LMod -> S e. Ring )
4	2	eqcomi 2631	. . . . . . . . . . 11 |- (Scalar ` M ) = S
5	4	fveq2i 6194	. . . . . . . . . 10 |- ( Base ` (Scalar ` M ) ) = ( Base ` S )
6		linc0scn0.1	. . . . . . . . . 10 |- .1. = ( 1r ` S )
7	5, 6	ringidcl 18568	. . . . . . . . 9 |- ( S e. Ring -> .1. e. ( Base ` (Scalar ` M ) ) )
8		linc0scn0.0	. . . . . . . . . 10 |- .0. = ( 0g ` S )
9	5, 8	ring0cl 18569	. . . . . . . . 9 |- ( S e. Ring -> .0. e. ( Base ` (Scalar ` M ) ) )
10	7, 9	jca 554	. . . . . . . 8 |- ( S e. Ring -> ( .1. e. ( Base ` (Scalar ` M ) ) /\ .0. e. ( Base ` (Scalar ` M ) ) ) )
11	3, 10	syl 17	. . . . . . 7 |- ( M e. LMod -> ( .1. e. ( Base ` (Scalar ` M ) ) /\ .0. e. ( Base ` (Scalar ` M ) ) ) )
12	11	ad2antrr 762	. . . . . 6 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ x e. V ) -> ( .1. e. ( Base ` (Scalar ` M ) ) /\ .0. e. ( Base ` (Scalar ` M ) ) ) )
13		ifcl 4130	. . . . . 6 |- ( ( .1. e. ( Base ` (Scalar ` M ) ) /\ .0. e. ( Base ` (Scalar ` M ) ) ) -> if ( x = Z , .1. , .0. ) e. ( Base ` (Scalar ` M ) ) )
14	12, 13	syl 17	. . . . 5 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ x e. V ) -> if ( x = Z , .1. , .0. ) e. ( Base ` (Scalar ` M ) ) )
15		linc0scn0.f	. . . . 5 |- F = ( x e. V |-> if ( x = Z , .1. , .0. ) )
16	14, 15	fmptd 6385	. . . 4 |- ( ( M e. LMod /\ V e. ~P B ) -> F : V --> ( Base ` (Scalar ` M ) ) )
17		fvex 6201	. . . . . 6 |- ( Base ` (Scalar ` M ) ) e. _V
18	17	a1i 11	. . . . 5 |- ( M e. LMod -> ( Base ` (Scalar ` M ) ) e. _V )
19		elmapg 7870	. . . . 5 |- ( ( ( Base ` (Scalar ` M ) ) e. _V /\ V e. ~P B ) -> ( F e. ( ( Base ` (Scalar ` M ) ) ^m V ) <-> F : V --> ( Base ` (Scalar ` M ) ) ) )
20	18, 19	sylan 488	. . . 4 |- ( ( M e. LMod /\ V e. ~P B ) -> ( F e. ( ( Base ` (Scalar ` M ) ) ^m V ) <-> F : V --> ( Base ` (Scalar ` M ) ) ) )
21	16, 20	mpbird 247	. . 3 |- ( ( M e. LMod /\ V e. ~P B ) -> F e. ( ( Base ` (Scalar ` M ) ) ^m V ) )
22		linc0scn0.b	. . . . . . 7 |- B = ( Base ` M )
23	22	pweqi 4162	. . . . . 6 |- ~P B = ~P ( Base ` M )
24	23	eleq2i 2693	. . . . 5 |- ( V e. ~P B <-> V e. ~P ( Base ` M ) )
25	24	biimpi 206	. . . 4 |- ( V e. ~P B -> V e. ~P ( Base ` M ) )
26	25	adantl 482	. . 3 |- ( ( M e. LMod /\ V e. ~P B ) -> V e. ~P ( Base ` M ) )
27		lincval 42198	. . 3 |- ( ( M e. LMod /\ F e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ V e. ~P ( Base ` M ) ) -> ( F ( linC ` M ) V ) = ( M gsumg ( v e. V |-> ( ( F ` v ) ( .s ` M ) v ) ) ) )
28	1, 21, 26, 27	syl3anc 1326	. 2 |- ( ( M e. LMod /\ V e. ~P B ) -> ( F ( linC ` M ) V ) = ( M gsumg ( v e. V |-> ( ( F ` v ) ( .s ` M ) v ) ) ) )
29		simpr 477	. . . . . . 7 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) -> v e. V )
30		fvex 6201	. . . . . . . . 9 |- ( 1r ` S ) e. _V
31	6, 30	eqeltri 2697	. . . . . . . 8 |- .1. e. _V
32		fvex 6201	. . . . . . . . 9 |- ( 0g ` S ) e. _V
33	8, 32	eqeltri 2697	. . . . . . . 8 |- .0. e. _V
34	31, 33	ifex 4156	. . . . . . 7 |- if ( v = Z , .1. , .0. ) e. _V
35		eqeq1 2626	. . . . . . . . 9 |- ( x = v -> ( x = Z <-> v = Z ) )
36	35	ifbid 4108	. . . . . . . 8 |- ( x = v -> if ( x = Z , .1. , .0. ) = if ( v = Z , .1. , .0. ) )
37	36, 15	fvmptg 6280	. . . . . . 7 |- ( ( v e. V /\ if ( v = Z , .1. , .0. ) e. _V ) -> ( F ` v ) = if ( v = Z , .1. , .0. ) )
38	29, 34, 37	sylancl 694	. . . . . 6 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) -> ( F ` v ) = if ( v = Z , .1. , .0. ) )
39	38	oveq1d 6665	. . . . 5 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) -> ( ( F ` v ) ( .s ` M ) v ) = ( if ( v = Z , .1. , .0. ) ( .s ` M ) v ) )
40		ovif 6737	. . . . . 6 |- ( if ( v = Z , .1. , .0. ) ( .s ` M ) v ) = if ( v = Z , ( .1. ( .s ` M ) v ) , ( .0. ( .s ` M ) v ) )
41	40	a1i 11	. . . . 5 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) -> ( if ( v = Z , .1. , .0. ) ( .s ` M ) v ) = if ( v = Z , ( .1. ( .s ` M ) v ) , ( .0. ( .s ` M ) v ) ) )
42		oveq2 6658	. . . . . . . 8 |- ( v = Z -> ( .1. ( .s ` M ) v ) = ( .1. ( .s ` M ) Z ) )
43	42	adantl 482	. . . . . . 7 |- ( ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) /\ v = Z ) -> ( .1. ( .s ` M ) v ) = ( .1. ( .s ` M ) Z ) )
44		eqid 2622	. . . . . . . . . . . 12 |- ( Base ` S ) = ( Base ` S )
45	2, 44, 6	lmod1cl 18890	. . . . . . . . . . 11 |- ( M e. LMod -> .1. e. ( Base ` S ) )
46	45	ancli 574	. . . . . . . . . 10 |- ( M e. LMod -> ( M e. LMod /\ .1. e. ( Base ` S ) ) )
47	46	adantr 481	. . . . . . . . 9 |- ( ( M e. LMod /\ V e. ~P B ) -> ( M e. LMod /\ .1. e. ( Base ` S ) ) )
48	47	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) /\ v = Z ) -> ( M e. LMod /\ .1. e. ( Base ` S ) ) )
49		eqid 2622	. . . . . . . . 9 |- ( .s ` M ) = ( .s ` M )
50		linc0scn0.z	. . . . . . . . 9 |- Z = ( 0g ` M )
51	2, 49, 44, 50	lmodvs0 18897	. . . . . . . 8 |- ( ( M e. LMod /\ .1. e. ( Base ` S ) ) -> ( .1. ( .s ` M ) Z ) = Z )
52	48, 51	syl 17	. . . . . . 7 |- ( ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) /\ v = Z ) -> ( .1. ( .s ` M ) Z ) = Z )
53	43, 52	eqtrd 2656	. . . . . 6 |- ( ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) /\ v = Z ) -> ( .1. ( .s ` M ) v ) = Z )
54	1	adantr 481	. . . . . . . 8 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) -> M e. LMod )
55		elelpwi 4171	. . . . . . . . . . 11 |- ( ( v e. V /\ V e. ~P B ) -> v e. B )
56	55	expcom 451	. . . . . . . . . 10 |- ( V e. ~P B -> ( v e. V -> v e. B ) )
57	56	adantl 482	. . . . . . . . 9 |- ( ( M e. LMod /\ V e. ~P B ) -> ( v e. V -> v e. B ) )
58	57	imp 445	. . . . . . . 8 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) -> v e. B )
59	22, 2, 49, 8, 50	lmod0vs 18896	. . . . . . . 8 |- ( ( M e. LMod /\ v e. B ) -> ( .0. ( .s ` M ) v ) = Z )
60	54, 58, 59	syl2anc 693	. . . . . . 7 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) -> ( .0. ( .s ` M ) v ) = Z )
61	60	adantr 481	. . . . . 6 |- ( ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) /\ -. v = Z ) -> ( .0. ( .s ` M ) v ) = Z )
62	53, 61	ifeqda 4121	. . . . 5 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) -> if ( v = Z , ( .1. ( .s ` M ) v ) , ( .0. ( .s ` M ) v ) ) = Z )
63	39, 41, 62	3eqtrd 2660	. . . 4 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) -> ( ( F ` v ) ( .s ` M ) v ) = Z )
64	63	mpteq2dva 4744	. . 3 |- ( ( M e. LMod /\ V e. ~P B ) -> ( v e. V |-> ( ( F ` v ) ( .s ` M ) v ) ) = ( v e. V |-> Z ) )
65	64	oveq2d 6666	. 2 |- ( ( M e. LMod /\ V e. ~P B ) -> ( M gsumg ( v e. V |-> ( ( F ` v ) ( .s ` M ) v ) ) ) = ( M gsumg ( v e. V |-> Z ) ) )
66		lmodgrp 18870	. . . 4 |- ( M e. LMod -> M e. Grp )
67		grpmnd 17429	. . . 4 |- ( M e. Grp -> M e. Mnd )
68	66, 67	syl 17	. . 3 |- ( M e. LMod -> M e. Mnd )
69	50	gsumz 17374	. . 3 |- ( ( M e. Mnd /\ V e. ~P B ) -> ( M gsumg ( v e. V |-> Z ) ) = Z )
70	68, 69	sylan 488	. 2 |- ( ( M e. LMod /\ V e. ~P B ) -> ( M gsumg ( v e. V |-> Z ) ) = Z )
71	28, 65, 70	3eqtrd 2660	1 |- ( ( M e. LMod /\ V e. ~P B ) -> ( F ( linC ` M ) V ) = Z )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   ~Pcpw 4158   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294   Grpcgrp 17422   1rcur 18501   Ringcrg 18547   LModclmod 18863   linC clinc 42193

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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  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-seq 12802  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-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-linc 42195

This theorem is referenced by:  el0ldep  42255