Theorem lincval0 42204

Description: The value of an empty linear combination. (Contributed by AV, 12-Apr-2019.)

Assertion

Ref	Expression
lincval0	|- ( M e. X -> ( (/) ( linC ` M ) (/) ) = ( 0g ` M ) )

Proof of Theorem lincval0

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4790	. . . . 5 |- (/) e. _V
2	1	snid 4208	. . . 4 |- (/) e. { (/) }
3		fvex 6201	. . . . . 6 |- ( Base ` (Scalar ` M ) ) e. _V
4		map0e 7895	. . . . . 6 |- ( ( Base ` (Scalar ` M ) ) e. _V -> ( ( Base ` (Scalar ` M ) ) ^m (/) ) = 1o )
5	3, 4	mp1i 13	. . . . 5 |- ( M e. X -> ( ( Base ` (Scalar ` M ) ) ^m (/) ) = 1o )
6		df1o2 7572	. . . . 5 |- 1o = { (/) }
7	5, 6	syl6eq 2672	. . . 4 |- ( M e. X -> ( ( Base ` (Scalar ` M ) ) ^m (/) ) = { (/) } )
8	2, 7	syl5eleqr 2708	. . 3 |- ( M e. X -> (/) e. ( ( Base ` (Scalar ` M ) ) ^m (/) ) )
9		0elpw 4834	. . . 4 |- (/) e. ~P ( Base ` M )
10	9	a1i 11	. . 3 |- ( M e. X -> (/) e. ~P ( Base ` M ) )
11		lincval 42198	. . 3 |- ( ( M e. X /\ (/) e. ( ( Base ` (Scalar ` M ) ) ^m (/) ) /\ (/) e. ~P ( Base ` M ) ) -> ( (/) ( linC ` M ) (/) ) = ( M gsumg ( v e. (/) |-> ( ( (/) ` v ) ( .s ` M ) v ) ) ) )
12	8, 10, 11	mpd3an23 1426	. 2 |- ( M e. X -> ( (/) ( linC ` M ) (/) ) = ( M gsumg ( v e. (/) |-> ( ( (/) ` v ) ( .s ` M ) v ) ) ) )
13		mpt0 6021	. . . . 5 |- ( v e. (/) |-> ( ( (/) ` v ) ( .s ` M ) v ) ) = (/)
14	13	a1i 11	. . . 4 |- ( M e. X -> ( v e. (/) |-> ( ( (/) ` v ) ( .s ` M ) v ) ) = (/) )
15	14	oveq2d 6666	. . 3 |- ( M e. X -> ( M gsumg ( v e. (/) |-> ( ( (/) ` v ) ( .s ` M ) v ) ) ) = ( M gsumg (/) ) )
16		eqid 2622	. . . 4 |- ( 0g ` M ) = ( 0g ` M )
17	16	gsum0 17278	. . 3 |- ( M gsumg (/) ) = ( 0g ` M )
18	15, 17	syl6eq 2672	. 2 |- ( M e. X -> ( M gsumg ( v e. (/) |-> ( ( (/) ` v ) ( .s ` M ) v ) ) ) = ( 0g ` M ) )
19	12, 18	eqtrd 2656	1 |- ( M e. X -> ( (/) ( linC ` M ) (/) ) = ( 0g ` M ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ~Pcpw 4158   {csn 4177   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   1oc1o 7553   ^m cmap 7857   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   gsum_g cgsu 16101   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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-map 7859  df-seq 12802  df-gsum 16103  df-linc 42195

This theorem is referenced by:  lco0  42216