Theorem mptscmfsupp0 17555

Description: A mapping to a scalar product is finitely supported if the mapping to the scalar is finitely supported. (Contributed by AV, 5-Oct-2019.)

Hypotheses

Ref	Expression
mptscmfsupp0.d	|- ( ph -> D e. V )
mptscmfsupp0.q	|- ( ph -> Q e. LMod )
mptscmfsupp0.r	|- ( ph -> R = (Scalar ` Q ) )
mptscmfsupp0.k	|- K = ( Base ` Q )
mptscmfsupp0.s	|- ( ( ph /\ k e. D ) -> S e. B )
mptscmfsupp0.w	|- ( ( ph /\ k e. D ) -> W e. K )
mptscmfsupp0.0	|- .0. = ( 0g ` Q )
mptscmfsupp0.z	|- Z = ( 0g ` R )
mptscmfsupp0.m	|- .* = ( .s ` Q )
mptscmfsupp0.f	|- ( ph -> ( k e. D |-> S ) finSupp Z )

Assertion

Ref	Expression
mptscmfsupp0	|- ( ph -> ( k e. D |-> ( S .* W ) ) finSupp .0. )

Distinct variable groups:   B, k   D, k   k, K   ph, k   .* , k

Allowed substitution hints:   Q( k)   R( k)   S( k)   V( k)   W( k)   .0. ( k)   Z( k)

Proof of Theorem mptscmfsupp0

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mptscmfsupp0.d	. . 3 |- ( ph -> D e. V )
2		mptexg 6127	. . 3 |- ( D e. V -> ( k e. D |-> ( S .* W ) ) e. _V )
3	1, 2	syl 16	. 2 |- ( ph -> ( k e. D |-> ( S .* W ) ) e. _V )
4		funmpt 5614	. . 3 |- Fun ( k e. D |-> ( S .* W ) )
5	4	a1i 11	. 2 |- ( ph -> Fun ( k e. D |-> ( S .* W ) ) )
6		mptscmfsupp0.0	. . . 4 |- .0. = ( 0g ` Q )
7		fvex 5866	. . . 4 |- ( 0g ` Q ) e. _V
8	6, 7	eqeltri 2527	. . 3 |- .0. e. _V
9	8	a1i 11	. 2 |- ( ph -> .0. e. _V )
10		mptscmfsupp0.f	. . 3 |- ( ph -> ( k e. D |-> S ) finSupp Z )
11	10	fsuppimpd 7838	. 2 |- ( ph -> ( ( k e. D |-> S ) supp Z ) e. Fin )
12		simpr 461	. . . . . . . 8 |- ( ( ph /\ d e. D ) -> d e. D )
13		mptscmfsupp0.s	. . . . . . . . . . 11 |- ( ( ph /\ k e. D ) -> S e. B )
14	13	ralrimiva 2857	. . . . . . . . . 10 |- ( ph -> A. k e. D S e. B )
15	14	adantr 465	. . . . . . . . 9 |- ( ( ph /\ d e. D ) -> A. k e. D S e. B )
16		rspcsbela 3839	. . . . . . . . 9 |- ( ( d e. D /\ A. k e. D S e. B ) -> [_ d / k ]_ S e. B )
17	12, 15, 16	syl2anc 661	. . . . . . . 8 |- ( ( ph /\ d e. D ) -> [_ d / k ]_ S e. B )
18		eqid 2443	. . . . . . . . 9 |- ( k e. D |-> S ) = ( k e. D |-> S )
19	18	fvmpts 5943	. . . . . . . 8 |- ( ( d e. D /\ [_ d / k ]_ S e. B ) -> ( ( k e. D |-> S ) ` d ) = [_ d / k ]_ S )
20	12, 17, 19	syl2anc 661	. . . . . . 7 |- ( ( ph /\ d e. D ) -> ( ( k e. D |-> S ) ` d ) = [_ d / k ]_ S )
21	20	eqeq1d 2445	. . . . . 6 |- ( ( ph /\ d e. D ) -> ( ( ( k e. D |-> S ) ` d ) = Z <-> [_ d / k ]_ S = Z ) )
22		oveq1 6288	. . . . . . . . 9 |- ( [_ d / k ]_ S = Z -> ( [_ d / k ]_ S .* [_ d / k ]_ W ) = ( Z .* [_ d / k ]_ W ) )
23		mptscmfsupp0.z	. . . . . . . . . . . 12 |- Z = ( 0g ` R )
24		mptscmfsupp0.r	. . . . . . . . . . . . . 14 |- ( ph -> R = (Scalar ` Q ) )
25	24	adantr 465	. . . . . . . . . . . . 13 |- ( ( ph /\ d e. D ) -> R = (Scalar ` Q ) )
26	25	fveq2d 5860	. . . . . . . . . . . 12 |- ( ( ph /\ d e. D ) -> ( 0g ` R ) = ( 0g ` (Scalar ` Q ) ) )
27	23, 26	syl5eq 2496	. . . . . . . . . . 11 |- ( ( ph /\ d e. D ) -> Z = ( 0g ` (Scalar ` Q ) ) )
28	27	oveq1d 6296	. . . . . . . . . 10 |- ( ( ph /\ d e. D ) -> ( Z .* [_ d / k ]_ W ) = ( ( 0g ` (Scalar ` Q ) ) .* [_ d / k ]_ W ) )
29		mptscmfsupp0.q	. . . . . . . . . . . 12 |- ( ph -> Q e. LMod )
30	29	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ d e. D ) -> Q e. LMod )
31		mptscmfsupp0.w	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. D ) -> W e. K )
32	31	ralrimiva 2857	. . . . . . . . . . . . 13 |- ( ph -> A. k e. D W e. K )
33	32	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ d e. D ) -> A. k e. D W e. K )
34		rspcsbela 3839	. . . . . . . . . . . 12 |- ( ( d e. D /\ A. k e. D W e. K ) -> [_ d / k ]_ W e. K )
35	12, 33, 34	syl2anc 661	. . . . . . . . . . 11 |- ( ( ph /\ d e. D ) -> [_ d / k ]_ W e. K )
36		mptscmfsupp0.k	. . . . . . . . . . . 12 |- K = ( Base ` Q )
37		eqid 2443	. . . . . . . . . . . 12 |- (Scalar ` Q ) = (Scalar ` Q )
38		mptscmfsupp0.m	. . . . . . . . . . . 12 |- .* = ( .s ` Q )
39		eqid 2443	. . . . . . . . . . . 12 |- ( 0g ` (Scalar ` Q ) ) = ( 0g ` (Scalar ` Q ) )
40	36, 37, 38, 39, 6	lmod0vs 17524	. . . . . . . . . . 11 |- ( ( Q e. LMod /\ [_ d / k ]_ W e. K ) -> ( ( 0g ` (Scalar ` Q ) ) .* [_ d / k ]_ W ) = .0. )
41	30, 35, 40	syl2anc 661	. . . . . . . . . 10 |- ( ( ph /\ d e. D ) -> ( ( 0g ` (Scalar ` Q ) ) .* [_ d / k ]_ W ) = .0. )
42	28, 41	eqtrd 2484	. . . . . . . . 9 |- ( ( ph /\ d e. D ) -> ( Z .* [_ d / k ]_ W ) = .0. )
43	22, 42	sylan9eqr 2506	. . . . . . . 8 |- ( ( ( ph /\ d e. D ) /\ [_ d / k ]_ S = Z ) -> ( [_ d / k ]_ S .* [_ d / k ]_ W ) = .0. )
44		csbov12g 6318	. . . . . . . . . . . . . 14 |- ( d e. D -> [_ d / k ]_ ( S .* W ) = ( [_ d / k ]_ S .* [_ d / k ]_ W ) )
45	44	adantl 466	. . . . . . . . . . . . 13 |- ( ( ph /\ d e. D ) -> [_ d / k ]_ ( S .* W ) = ( [_ d / k ]_ S .* [_ d / k ]_ W ) )
46		ovex 6309	. . . . . . . . . . . . 13 |- ( [_ d / k ]_ S .* [_ d / k ]_ W ) e. _V
47	45, 46	syl6eqel 2539	. . . . . . . . . . . 12 |- ( ( ph /\ d e. D ) -> [_ d / k ]_ ( S .* W ) e. _V )
48		eqid 2443	. . . . . . . . . . . . 13 |- ( k e. D |-> ( S .* W ) ) = ( k e. D |-> ( S .* W ) )
49	48	fvmpts 5943	. . . . . . . . . . . 12 |- ( ( d e. D /\ [_ d / k ]_ ( S .* W ) e. _V ) -> ( ( k e. D |-> ( S .* W ) ) ` d ) = [_ d / k ]_ ( S .* W ) )
50	12, 47, 49	syl2anc 661	. . . . . . . . . . 11 |- ( ( ph /\ d e. D ) -> ( ( k e. D |-> ( S .* W ) ) ` d ) = [_ d / k ]_ ( S .* W ) )
51	50, 45	eqtrd 2484	. . . . . . . . . 10 |- ( ( ph /\ d e. D ) -> ( ( k e. D |-> ( S .* W ) ) ` d ) = ( [_ d / k ]_ S .* [_ d / k ]_ W ) )
52	51	eqeq1d 2445	. . . . . . . . 9 |- ( ( ph /\ d e. D ) -> ( ( ( k e. D |-> ( S .* W ) ) ` d ) = .0. <-> ( [_ d / k ]_ S .* [_ d / k ]_ W ) = .0. ) )
53	52	adantr 465	. . . . . . . 8 |- ( ( ( ph /\ d e. D ) /\ [_ d / k ]_ S = Z ) -> ( ( ( k e. D |-> ( S .* W ) ) ` d ) = .0. <-> ( [_ d / k ]_ S .* [_ d / k ]_ W ) = .0. ) )
54	43, 53	mpbird 232	. . . . . . 7 |- ( ( ( ph /\ d e. D ) /\ [_ d / k ]_ S = Z ) -> ( ( k e. D |-> ( S .* W ) ) ` d ) = .0. )
55	54	ex 434	. . . . . 6 |- ( ( ph /\ d e. D ) -> ( [_ d / k ]_ S = Z -> ( ( k e. D |-> ( S .* W ) ) ` d ) = .0. ) )
56	21, 55	sylbid 215	. . . . 5 |- ( ( ph /\ d e. D ) -> ( ( ( k e. D |-> S ) ` d ) = Z -> ( ( k e. D |-> ( S .* W ) ) ` d ) = .0. ) )
57	56	necon3d 2667	. . . 4 |- ( ( ph /\ d e. D ) -> ( ( ( k e. D |-> ( S .* W ) ) ` d ) =/= .0. -> ( ( k e. D |-> S ) ` d ) =/= Z ) )
58	57	ss2rabdv 3566	. . 3 |- ( ph -> { d e. D | ( ( k e. D |-> ( S .* W ) ) ` d ) =/= .0. } C_ { d e. D | ( ( k e. D |-> S ) ` d ) =/= Z } )
59		ovex 6309	. . . . . 6 |- ( S .* W ) e. _V
60	59	rgenw 2804	. . . . 5 |- A. k e. D ( S .* W ) e. _V
61	48	fnmpt 5697	. . . . 5 |- ( A. k e. D ( S .* W ) e. _V -> ( k e. D |-> ( S .* W ) ) Fn D )
62	60, 61	mp1i 12	. . . 4 |- ( ph -> ( k e. D |-> ( S .* W ) ) Fn D )
63		suppvalfn 6910	. . . 4 |- ( ( ( k e. D |-> ( S .* W ) ) Fn D /\ D e. V /\ .0. e. _V ) -> ( ( k e. D |-> ( S .* W ) ) supp .0. ) = { d e. D | ( ( k e. D |-> ( S .* W ) ) ` d ) =/= .0. } )
64	62, 1, 9, 63	syl3anc 1229	. . 3 |- ( ph -> ( ( k e. D |-> ( S .* W ) ) supp .0. ) = { d e. D | ( ( k e. D |-> ( S .* W ) ) ` d ) =/= .0. } )
65	18	fnmpt 5697	. . . . 5 |- ( A. k e. D S e. B -> ( k e. D |-> S ) Fn D )
66	14, 65	syl 16	. . . 4 |- ( ph -> ( k e. D |-> S ) Fn D )
67		fvex 5866	. . . . . 6 |- ( 0g ` R ) e. _V
68	23, 67	eqeltri 2527	. . . . 5 |- Z e. _V
69	68	a1i 11	. . . 4 |- ( ph -> Z e. _V )
70		suppvalfn 6910	. . . 4 |- ( ( ( k e. D |-> S ) Fn D /\ D e. V /\ Z e. _V ) -> ( ( k e. D |-> S ) supp Z ) = { d e. D | ( ( k e. D |-> S ) ` d ) =/= Z } )
71	66, 1, 69, 70	syl3anc 1229	. . 3 |- ( ph -> ( ( k e. D |-> S ) supp Z ) = { d e. D | ( ( k e. D |-> S ) ` d ) =/= Z } )
72	58, 64, 71	3sstr4d 3532	. 2 |- ( ph -> ( ( k e. D |-> ( S .* W ) ) supp .0. ) C_ ( ( k e. D |-> S ) supp Z ) )
73		suppssfifsupp 7846	. 2 |- ( ( ( ( k e. D |-> ( S .* W ) ) e. _V /\ Fun ( k e. D |-> ( S .* W ) ) /\ .0. e. _V ) /\ ( ( ( k e. D |-> S ) supp Z ) e. Fin /\ ( ( k e. D |-> ( S .* W ) ) supp .0. ) C_ ( ( k e. D |-> S ) supp Z ) ) ) -> ( k e. D |-> ( S .* W ) ) finSupp .0. )
74	3, 5, 9, 11, 72, 73	syl32anc 1237	1 |- ( ph -> ( k e. D |-> ( S .* W ) ) finSupp .0. )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1383   e. wcel 1804   =/= wne 2638   A.wral 2793   {crab 2797   _Vcvv 3095   [_csb 3420   C_ wss 3461   class class class wbr 4437   |-> cmpt 4495   Fun wfun 5572   Fn wfn 5573   ` cfv 5578  (class class class)co 6281   supp csupp 6903   Fincfn 7518   finSupp cfsupp 7831   Basecbs 14614  Scalarcsca 14682   .scvsca 14683   0gc0g 14819   LModclmod 17491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-supp 6904  df-er 7313  df-en 7519  df-fin 7522  df-fsupp 7832  df-0g 14821  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-grp 16036  df-ring 17179  df-lmod 17493

This theorem is referenced by:  mptscmfsuppd  17556  gsumsmonply1  18324  pm2mpcl  19276  mply1topmatcllem  19282  mp2pm2mplem5  19289  pm2mpghmlem2  19291  chcoeffeqlem  19364