Theorem lcoel0 42217

Description: The zero vector is always a linear combination. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)

Assertion

Ref	Expression
lcoel0	|- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( 0g ` M ) e. ( M LinCo V ) )

Proof of Theorem lcoel0

Dummy variables s v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6201	. . . 4 |- ( 0g ` M ) e. _V
2	1	snid 4208	. . 3 |- ( 0g ` M ) e. { ( 0g ` M ) }
3		oveq2 6658	. . . 4 |- ( V = (/) -> ( M LinCo V ) = ( M LinCo (/) ) )
4		lmodgrp 18870	. . . . . 6 |- ( M e. LMod -> M e. Grp )
5		grpmnd 17429	. . . . . 6 |- ( M e. Grp -> M e. Mnd )
6		lco0 42216	. . . . . 6 |- ( M e. Mnd -> ( M LinCo (/) ) = { ( 0g ` M ) } )
7	4, 5, 6	3syl 18	. . . . 5 |- ( M e. LMod -> ( M LinCo (/) ) = { ( 0g ` M ) } )
8	7	adantr 481	. . . 4 |- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( M LinCo (/) ) = { ( 0g ` M ) } )
9	3, 8	sylan9eq 2676	. . 3 |- ( ( V = (/) /\ ( M e. LMod /\ V e. ~P ( Base ` M ) ) ) -> ( M LinCo V ) = { ( 0g ` M ) } )
10	2, 9	syl5eleqr 2708	. 2 |- ( ( V = (/) /\ ( M e. LMod /\ V e. ~P ( Base ` M ) ) ) -> ( 0g ` M ) e. ( M LinCo V ) )
11		eqid 2622	. . . . . 6 |- ( Base ` M ) = ( Base ` M )
12		eqid 2622	. . . . . 6 |- ( 0g ` M ) = ( 0g ` M )
13	11, 12	lmod0vcl 18892	. . . . 5 |- ( M e. LMod -> ( 0g ` M ) e. ( Base ` M ) )
14	13	adantr 481	. . . 4 |- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( 0g ` M ) e. ( Base ` M ) )
15	14	adantl 482	. . 3 |- ( ( -. V = (/) /\ ( M e. LMod /\ V e. ~P ( Base ` M ) ) ) -> ( 0g ` M ) e. ( Base ` M ) )
16		eqid 2622	. . . . . 6 |- (Scalar ` M ) = (Scalar ` M )
17		eqid 2622	. . . . . 6 |- ( 0g ` (Scalar ` M ) ) = ( 0g ` (Scalar ` M ) )
18		eqidd 2623	. . . . . . 7 |- ( v = w -> ( 0g ` (Scalar ` M ) ) = ( 0g ` (Scalar ` M ) ) )
19	18	cbvmptv 4750	. . . . . 6 |- ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) = ( w e. V |-> ( 0g ` (Scalar ` M ) ) )
20		eqid 2622	. . . . . 6 |- ( Base ` (Scalar ` M ) ) = ( Base ` (Scalar ` M ) )
21	11, 16, 17, 12, 19, 20	lcoc0 42211	. . . . 5 |- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) finSupp ( 0g ` (Scalar ` M ) ) /\ ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) ( linC ` M ) V ) = ( 0g ` M ) ) )
22	21	adantl 482	. . . 4 |- ( ( -. V = (/) /\ ( M e. LMod /\ V e. ~P ( Base ` M ) ) ) -> ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) finSupp ( 0g ` (Scalar ` M ) ) /\ ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) ( linC ` M ) V ) = ( 0g ` M ) ) )
23		simpl 473	. . . . . . . 8 |- ( ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ ( -. V = (/) /\ ( M e. LMod /\ V e. ~P ( Base ` M ) ) ) ) -> ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) e. ( ( Base ` (Scalar ` M ) ) ^m V ) )
24		breq1 4656	. . . . . . . . . 10 |- ( s = ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) -> ( s finSupp ( 0g ` (Scalar ` M ) ) <-> ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) finSupp ( 0g ` (Scalar ` M ) ) ) )
25		oveq1 6657	. . . . . . . . . . . 12 |- ( s = ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) -> ( s ( linC ` M ) V ) = ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) ( linC ` M ) V ) )
26	25	eqeq2d 2632	. . . . . . . . . . 11 |- ( s = ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) -> ( ( 0g ` M ) = ( s ( linC ` M ) V ) <-> ( 0g ` M ) = ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) ( linC ` M ) V ) ) )
27		eqcom 2629	. . . . . . . . . . 11 |- ( ( 0g ` M ) = ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) ( linC ` M ) V ) <-> ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) ( linC ` M ) V ) = ( 0g ` M ) )
28	26, 27	syl6bb 276	. . . . . . . . . 10 |- ( s = ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) -> ( ( 0g ` M ) = ( s ( linC ` M ) V ) <-> ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) ( linC ` M ) V ) = ( 0g ` M ) ) )
29	24, 28	anbi12d 747	. . . . . . . . 9 |- ( s = ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) -> ( ( s finSupp ( 0g ` (Scalar ` M ) ) /\ ( 0g ` M ) = ( s ( linC ` M ) V ) ) <-> ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) finSupp ( 0g ` (Scalar ` M ) ) /\ ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) ( linC ` M ) V ) = ( 0g ` M ) ) ) )
30	29	adantl 482	. . . . . . . 8 |- ( ( ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ ( -. V = (/) /\ ( M e. LMod /\ V e. ~P ( Base ` M ) ) ) ) /\ s = ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) ) -> ( ( s finSupp ( 0g ` (Scalar ` M ) ) /\ ( 0g ` M ) = ( s ( linC ` M ) V ) ) <-> ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) finSupp ( 0g ` (Scalar ` M ) ) /\ ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) ( linC ` M ) V ) = ( 0g ` M ) ) ) )
31	23, 30	rspcedv 3313	. . . . . . 7 |- ( ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ ( -. V = (/) /\ ( M e. LMod /\ V e. ~P ( Base ` M ) ) ) ) -> ( ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) finSupp ( 0g ` (Scalar ` M ) ) /\ ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) ( linC ` M ) V ) = ( 0g ` M ) ) -> E. s e. ( ( Base ` (Scalar ` M ) ) ^m V ) ( s finSupp ( 0g ` (Scalar ` M ) ) /\ ( 0g ` M ) = ( s ( linC ` M ) V ) ) ) )
32	31	ex 450	. . . . . 6 |- ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) e. ( ( Base ` (Scalar ` M ) ) ^m V ) -> ( ( -. V = (/) /\ ( M e. LMod /\ V e. ~P ( Base ` M ) ) ) -> ( ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) finSupp ( 0g ` (Scalar ` M ) ) /\ ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) ( linC ` M ) V ) = ( 0g ` M ) ) -> E. s e. ( ( Base ` (Scalar ` M ) ) ^m V ) ( s finSupp ( 0g ` (Scalar ` M ) ) /\ ( 0g ` M ) = ( s ( linC ` M ) V ) ) ) ) )
33	32	com23 86	. . . . 5 |- ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) e. ( ( Base ` (Scalar ` M ) ) ^m V ) -> ( ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) finSupp ( 0g ` (Scalar ` M ) ) /\ ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) ( linC ` M ) V ) = ( 0g ` M ) ) -> ( ( -. V = (/) /\ ( M e. LMod /\ V e. ~P ( Base ` M ) ) ) -> E. s e. ( ( Base ` (Scalar ` M ) ) ^m V ) ( s finSupp ( 0g ` (Scalar ` M ) ) /\ ( 0g ` M ) = ( s ( linC ` M ) V ) ) ) ) )
34	33	3impib 1262	. . . 4 |- ( ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) finSupp ( 0g ` (Scalar ` M ) ) /\ ( ( v e. V |-> ( 0g ` (Scalar ` M ) ) ) ( linC ` M ) V ) = ( 0g ` M ) ) -> ( ( -. V = (/) /\ ( M e. LMod /\ V e. ~P ( Base ` M ) ) ) -> E. s e. ( ( Base ` (Scalar ` M ) ) ^m V ) ( s finSupp ( 0g ` (Scalar ` M ) ) /\ ( 0g ` M ) = ( s ( linC ` M ) V ) ) ) )
35	22, 34	mpcom 38	. . 3 |- ( ( -. V = (/) /\ ( M e. LMod /\ V e. ~P ( Base ` M ) ) ) -> E. s e. ( ( Base ` (Scalar ` M ) ) ^m V ) ( s finSupp ( 0g ` (Scalar ` M ) ) /\ ( 0g ` M ) = ( s ( linC ` M ) V ) ) )
36	11, 16, 20	lcoval 42201	. . . 4 |- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( ( 0g ` M ) e. ( M LinCo V ) <-> ( ( 0g ` M ) e. ( Base ` M ) /\ E. s e. ( ( Base ` (Scalar ` M ) ) ^m V ) ( s finSupp ( 0g ` (Scalar ` M ) ) /\ ( 0g ` M ) = ( s ( linC ` M ) V ) ) ) ) )
37	36	adantl 482	. . 3 |- ( ( -. V = (/) /\ ( M e. LMod /\ V e. ~P ( Base ` M ) ) ) -> ( ( 0g ` M ) e. ( M LinCo V ) <-> ( ( 0g ` M ) e. ( Base ` M ) /\ E. s e. ( ( Base ` (Scalar ` M ) ) ^m V ) ( s finSupp ( 0g ` (Scalar ` M ) ) /\ ( 0g ` M ) = ( s ( linC ` M ) V ) ) ) ) )
38	15, 35, 37	mpbir2and 957	. 2 |- ( ( -. V = (/) /\ ( M e. LMod /\ V e. ~P ( Base ` M ) ) ) -> ( 0g ` M ) e. ( M LinCo V ) )
39	10, 38	pm2.61ian 831	1 |- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( 0g ` M ) e. ( M LinCo V ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   finSupp cfsupp 8275   Basecbs 15857  Scalarcsca 15944   0gc0g 16100   Mndcmnd 17294   Grpcgrp 17422   LModclmod 18863   linC clinc 42193   LinCo clinco 42194

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

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-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-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-map 7859  df-en 7956  df-fin 7959  df-fsupp 8276  df-seq 12802  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ring 18549  df-lmod 18865  df-linc 42195  df-lco 42196

This theorem is referenced by:  lincolss  42223