Theorem lcoc0 42211

Description: Properties of a linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 28-Jul-2019.)

Hypotheses

Ref	Expression
lincvalsc0.b	|- B = ( Base ` M )
lincvalsc0.s	|- S = (Scalar ` M )
lincvalsc0.0	|- .0. = ( 0g ` S )
lincvalsc0.z	|- Z = ( 0g ` M )
lincvalsc0.f	|- F = ( x e. V |-> .0. )
lcoc0.r	|- R = ( Base ` S )

Assertion

Ref	Expression
lcoc0	|- ( ( M e. LMod /\ V e. ~P B ) -> ( F e. ( R ^m V ) /\ F finSupp .0. /\ ( F ( linC ` M ) V ) = Z ) )

Distinct variable groups:   x, B   x, M   x, V   x, .0.   x, F   x, R

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

Proof of Theorem lcoc0

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lincvalsc0.s	. . . . . 6 |- S = (Scalar ` M )
2		lcoc0.r	. . . . . 6 |- R = ( Base ` S )
3		lincvalsc0.0	. . . . . 6 |- .0. = ( 0g ` S )
4	1, 2, 3	lmod0cl 18889	. . . . 5 |- ( M e. LMod -> .0. e. R )
5	4	ad2antrr 762	. . . 4 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ x e. V ) -> .0. e. R )
6		lincvalsc0.f	. . . 4 |- F = ( x e. V |-> .0. )
7	5, 6	fmptd 6385	. . 3 |- ( ( M e. LMod /\ V e. ~P B ) -> F : V --> R )
8		fvex 6201	. . . . . 6 |- ( Base ` S ) e. _V
9	2, 8	eqeltri 2697	. . . . 5 |- R e. _V
10	9	a1i 11	. . . 4 |- ( M e. LMod -> R e. _V )
11		elmapg 7870	. . . 4 |- ( ( R e. _V /\ V e. ~P B ) -> ( F e. ( R ^m V ) <-> F : V --> R ) )
12	10, 11	sylan 488	. . 3 |- ( ( M e. LMod /\ V e. ~P B ) -> ( F e. ( R ^m V ) <-> F : V --> R ) )
13	7, 12	mpbird 247	. 2 |- ( ( M e. LMod /\ V e. ~P B ) -> F e. ( R ^m V ) )
14		eqidd 2623	. . . . . . 7 |- ( x = v -> .0. = .0. )
15	14	cbvmptv 4750	. . . . . 6 |- ( x e. V |-> .0. ) = ( v e. V |-> .0. )
16	6, 15	eqtri 2644	. . . . 5 |- F = ( v e. V |-> .0. )
17		simpr 477	. . . . 5 |- ( ( M e. LMod /\ V e. ~P B ) -> V e. ~P B )
18		fvex 6201	. . . . . . 7 |- ( 0g ` S ) e. _V
19	3, 18	eqeltri 2697	. . . . . 6 |- .0. e. _V
20	19	a1i 11	. . . . 5 |- ( ( M e. LMod /\ V e. ~P B ) -> .0. e. _V )
21	19	a1i 11	. . . . 5 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) -> .0. e. _V )
22	16, 17, 20, 21	mptsuppd 7318	. . . 4 |- ( ( M e. LMod /\ V e. ~P B ) -> ( F supp .0. ) = { v e. V | .0. =/= .0. } )
23		neirr 2803	. . . . . . . 8 |- -. .0. =/= .0.
24	23	a1i 11	. . . . . . 7 |- ( ( M e. LMod /\ V e. ~P B ) -> -. .0. =/= .0. )
25	24	ralrimivw 2967	. . . . . 6 |- ( ( M e. LMod /\ V e. ~P B ) -> A. v e. V -. .0. =/= .0. )
26		rabeq0 3957	. . . . . 6 |- ( { v e. V | .0. =/= .0. } = (/) <-> A. v e. V -. .0. =/= .0. )
27	25, 26	sylibr 224	. . . . 5 |- ( ( M e. LMod /\ V e. ~P B ) -> { v e. V | .0. =/= .0. } = (/) )
28		0fin 8188	. . . . . 6 |- (/) e. Fin
29	28	a1i 11	. . . . 5 |- ( ( M e. LMod /\ V e. ~P B ) -> (/) e. Fin )
30	27, 29	eqeltrd 2701	. . . 4 |- ( ( M e. LMod /\ V e. ~P B ) -> { v e. V | .0. =/= .0. } e. Fin )
31	22, 30	eqeltrd 2701	. . 3 |- ( ( M e. LMod /\ V e. ~P B ) -> ( F supp .0. ) e. Fin )
32	6	funmpt2 5927	. . . . 5 |- Fun F
33	32	a1i 11	. . . 4 |- ( ( M e. LMod /\ V e. ~P B ) -> Fun F )
34		funisfsupp 8280	. . . 4 |- ( ( Fun F /\ F e. ( R ^m V ) /\ .0. e. _V ) -> ( F finSupp .0. <-> ( F supp .0. ) e. Fin ) )
35	33, 13, 20, 34	syl3anc 1326	. . 3 |- ( ( M e. LMod /\ V e. ~P B ) -> ( F finSupp .0. <-> ( F supp .0. ) e. Fin ) )
36	31, 35	mpbird 247	. 2 |- ( ( M e. LMod /\ V e. ~P B ) -> F finSupp .0. )
37		lincvalsc0.b	. . 3 |- B = ( Base ` M )
38		lincvalsc0.z	. . 3 |- Z = ( 0g ` M )
39	37, 1, 3, 38, 6	lincvalsc0 42210	. 2 |- ( ( M e. LMod /\ V e. ~P B ) -> ( F ( linC ` M ) V ) = Z )
40	13, 36, 39	3jca 1242	1 |- ( ( M e. LMod /\ V e. ~P B ) -> ( F e. ( R ^m V ) /\ F finSupp .0. /\ ( F ( linC ` M ) V ) = Z ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   supp csupp 7295   ^m cmap 7857   Fincfn 7955   finSupp cfsupp 8275   Basecbs 15857  Scalarcsca 15944   0gc0g 16100   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

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-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

This theorem is referenced by:  lcoel0  42217