Theorem lincvalsc0 42210

Description: The linear combination where all scalars are 0. (Contributed by AV, 12-Apr-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. )

Assertion

Ref	Expression
lincvalsc0	|- ( ( M e. LMod /\ V e. ~P B ) -> ( F ( linC ` M ) V ) = Z )

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

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

Proof of Theorem lincvalsc0

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		lincvalsc0.s	. . . . . . . 8 |- S = (Scalar ` M )
3	2	eqcomi 2631	. . . . . . . . 9 |- (Scalar ` M ) = S
4	3	fveq2i 6194	. . . . . . . 8 |- ( Base ` (Scalar ` M ) ) = ( Base ` S )
5		lincvalsc0.0	. . . . . . . 8 |- .0. = ( 0g ` S )
6	2, 4, 5	lmod0cl 18889	. . . . . . 7 |- ( M e. LMod -> .0. e. ( Base ` (Scalar ` M ) ) )
7	6	adantr 481	. . . . . 6 |- ( ( M e. LMod /\ V e. ~P B ) -> .0. e. ( Base ` (Scalar ` M ) ) )
8	7	adantr 481	. . . . 5 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ x e. V ) -> .0. e. ( Base ` (Scalar ` M ) ) )
9		lincvalsc0.f	. . . . 5 |- F = ( x e. V |-> .0. )
10	8, 9	fmptd 6385	. . . 4 |- ( ( M e. LMod /\ V e. ~P B ) -> F : V --> ( Base ` (Scalar ` M ) ) )
11		fvexd 6203	. . . . 5 |- ( M e. LMod -> ( Base ` (Scalar ` M ) ) e. _V )
12		elmapg 7870	. . . . 5 |- ( ( ( Base ` (Scalar ` M ) ) e. _V /\ V e. ~P B ) -> ( F e. ( ( Base ` (Scalar ` M ) ) ^m V ) <-> F : V --> ( Base ` (Scalar ` M ) ) ) )
13	11, 12	sylan 488	. . . 4 |- ( ( M e. LMod /\ V e. ~P B ) -> ( F e. ( ( Base ` (Scalar ` M ) ) ^m V ) <-> F : V --> ( Base ` (Scalar ` M ) ) ) )
14	10, 13	mpbird 247	. . 3 |- ( ( M e. LMod /\ V e. ~P B ) -> F e. ( ( Base ` (Scalar ` M ) ) ^m V ) )
15		lincvalsc0.b	. . . . . . 7 |- B = ( Base ` M )
16	15	pweqi 4162	. . . . . 6 |- ~P B = ~P ( Base ` M )
17	16	eleq2i 2693	. . . . 5 |- ( V e. ~P B <-> V e. ~P ( Base ` M ) )
18	17	biimpi 206	. . . 4 |- ( V e. ~P B -> V e. ~P ( Base ` M ) )
19	18	adantl 482	. . 3 |- ( ( M e. LMod /\ V e. ~P B ) -> V e. ~P ( Base ` M ) )
20		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 ) ) ) )
21	1, 14, 19, 20	syl3anc 1326	. 2 |- ( ( M e. LMod /\ V e. ~P B ) -> ( F ( linC ` M ) V ) = ( M gsumg ( v e. V |-> ( ( F ` v ) ( .s ` M ) v ) ) ) )
22		simpr 477	. . . . . . 7 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) -> v e. V )
23		fvex 6201	. . . . . . . 8 |- ( 0g ` S ) e. _V
24	5, 23	eqeltri 2697	. . . . . . 7 |- .0. e. _V
25		eqidd 2623	. . . . . . . 8 |- ( x = v -> .0. = .0. )
26	25, 9	fvmptg 6280	. . . . . . 7 |- ( ( v e. V /\ .0. e. _V ) -> ( F ` v ) = .0. )
27	22, 24, 26	sylancl 694	. . . . . 6 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) -> ( F ` v ) = .0. )
28	27	oveq1d 6665	. . . . 5 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) -> ( ( F ` v ) ( .s ` M ) v ) = ( .0. ( .s ` M ) v ) )
29	1	adantr 481	. . . . . 6 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) -> M e. LMod )
30		elelpwi 4171	. . . . . . . . 9 |- ( ( v e. V /\ V e. ~P B ) -> v e. B )
31	30	expcom 451	. . . . . . . 8 |- ( V e. ~P B -> ( v e. V -> v e. B ) )
32	31	adantl 482	. . . . . . 7 |- ( ( M e. LMod /\ V e. ~P B ) -> ( v e. V -> v e. B ) )
33	32	imp 445	. . . . . 6 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) -> v e. B )
34		eqid 2622	. . . . . . 7 |- ( .s ` M ) = ( .s ` M )
35		lincvalsc0.z	. . . . . . 7 |- Z = ( 0g ` M )
36	15, 2, 34, 5, 35	lmod0vs 18896	. . . . . 6 |- ( ( M e. LMod /\ v e. B ) -> ( .0. ( .s ` M ) v ) = Z )
37	29, 33, 36	syl2anc 693	. . . . 5 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) -> ( .0. ( .s ` M ) v ) = Z )
38	28, 37	eqtrd 2656	. . . 4 |- ( ( ( M e. LMod /\ V e. ~P B ) /\ v e. V ) -> ( ( F ` v ) ( .s ` M ) v ) = Z )
39	38	mpteq2dva 4744	. . 3 |- ( ( M e. LMod /\ V e. ~P B ) -> ( v e. V |-> ( ( F ` v ) ( .s ` M ) v ) ) = ( v e. V |-> Z ) )
40	39	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 ) ) )
41		lmodgrp 18870	. . . 4 |- ( M e. LMod -> M e. Grp )
42		grpmnd 17429	. . . 4 |- ( M e. Grp -> M e. Mnd )
43	41, 42	syl 17	. . 3 |- ( M e. LMod -> M e. Mnd )
44	35	gsumz 17374	. . 3 |- ( ( M e. Mnd /\ V e. ~P B ) -> ( M gsumg ( v e. V |-> Z ) ) = Z )
45	43, 44	sylan 488	. 2 |- ( ( M e. LMod /\ V e. ~P B ) -> ( M gsumg ( v e. V |-> Z ) ) = Z )
46	21, 40, 45	3eqtrd 2660	1 |- ( ( M e. LMod /\ V e. ~P B ) -> ( F ( linC ` M ) V ) = Z )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   ~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   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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-map 7859  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:  lcoc0  42211