Theorem mptscmfsupp0 17137

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 6059	. . 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 5565	. . 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 5812	. . . 4 |- ( 0g ` Q ) e. _V
8	6, 7	eqeltri 2538	. . 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 7741	. 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 2830	. . . . . . . . . 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 3816	. . . . . . . . 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 2454	. . . . . . . . 9 |- ( k e. D |-> S ) = ( k e. D |-> S )
19	18	fvmpts 5888	. . . . . . . 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 2456	. . . . . 6 |- ( ( ph /\ d e. D ) -> ( ( ( k e. D |-> S ) ` d ) = Z <-> [_ d / k ]_ S = Z ) )
22		oveq1 6210	. . . . . . . . 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 5806	. . . . . . . . . . . 12 |- ( ( ph /\ d e. D ) -> ( 0g ` R ) = ( 0g ` (Scalar ` Q ) ) )
27	23, 26	syl5eq 2507	. . . . . . . . . . 11 |- ( ( ph /\ d e. D ) -> Z = ( 0g ` (Scalar ` Q ) ) )
28	27	oveq1d 6218	. . . . . . . . . 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 2830	. . . . . . . . . . . . 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 3816	. . . . . . . . . . . 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 2454	. . . . . . . . . . . 12 |- (Scalar ` Q ) = (Scalar ` Q )
38		mptscmfsupp0.m	. . . . . . . . . . . 12 |- .* = ( .s ` Q )
39		eqid 2454	. . . . . . . . . . . 12 |- ( 0g ` (Scalar ` Q ) ) = ( 0g ` (Scalar ` Q ) )
40	36, 37, 38, 39, 6	lmod0vs 17107	. . . . . . . . . . 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 2495	. . . . . . . . 9 |- ( ( ph /\ d e. D ) -> ( Z .* [_ d / k ]_ W ) = .0. )
43	22, 42	sylan9eqr 2517	. . . . . . . 8 |- ( ( ( ph /\ d e. D ) /\ [_ d / k ]_ S = Z ) -> ( [_ d / k ]_ S .* [_ d / k ]_ W ) = .0. )
44		csbov12g 6237	. . . . . . . . . . . . . 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 6228	. . . . . . . . . . . . 13 |- ( [_ d / k ]_ S .* [_ d / k ]_ W ) e. _V
47	45, 46	syl6eqel 2550	. . . . . . . . . . . 12 |- ( ( ph /\ d e. D ) -> [_ d / k ]_ ( S .* W ) e. _V )
48		eqid 2454	. . . . . . . . . . . . 13 |- ( k e. D |-> ( S .* W ) ) = ( k e. D |-> ( S .* W ) )
49	48	fvmpts 5888	. . . . . . . . . . . 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 2495	. . . . . . . . . 10 |- ( ( ph /\ d e. D ) -> ( ( k e. D |-> ( S .* W ) ) ` d ) = ( [_ d / k ]_ S .* [_ d / k ]_ W ) )
52	51	eqeq1d 2456	. . . . . . . . 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 2676	. . . 4 |- ( ( ph /\ d e. D ) -> ( ( ( k e. D |-> ( S .* W ) ) ` d ) =/= .0. -> ( ( k e. D |-> S ) ` d ) =/= Z ) )
58	57	ss2rabdv 3544	. . 3 |- ( ph -> { d e. D | ( ( k e. D |-> ( S .* W ) ) ` d ) =/= .0. } C_ { d e. D | ( ( k e. D |-> S ) ` d ) =/= Z } )
59		ovex 6228	. . . . . 6 |- ( S .* W ) e. _V
60	59	rgenw 2901	. . . . 5 |- A. k e. D ( S .* W ) e. _V
61	48	fnmpt 5648	. . . . 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 6810	. . . 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 1219	. . 3 |- ( ph -> ( ( k e. D |-> ( S .* W ) ) supp .0. ) = { d e. D | ( ( k e. D |-> ( S .* W ) ) ` d ) =/= .0. } )
65	18	fnmpt 5648	. . . . 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 5812	. . . . . 6 |- ( 0g ` R ) e. _V
68	23, 67	eqeltri 2538	. . . . 5 |- Z e. _V
69	68	a1i 11	. . . 4 |- ( ph -> Z e. _V )
70		suppvalfn 6810	. . . 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 1219	. . 3 |- ( ph -> ( ( k e. D |-> S ) supp Z ) = { d e. D | ( ( k e. D |-> S ) ` d ) =/= Z } )
72	58, 64, 71	3sstr4d 3510	. 2 |- ( ph -> ( ( k e. D |-> ( S .* W ) ) supp .0. ) C_ ( ( k e. D |-> S ) supp Z ) )
73		suppssfifsupp 7749	. 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 1227	1 |- ( ph -> ( k e. D |-> ( S .* W ) ) finSupp .0. )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   =/= wne 2648   A.wral 2799   {crab 2803   _Vcvv 3078   [_csb 3398   C_ wss 3439   class class class wbr 4403   |-> cmpt 4461   Fun wfun 5523   Fn wfn 5524   ` cfv 5529  (class class class)co 6203   supp csupp 6803   Fincfn 7423   finSupp cfsupp 7734   Basecbs 14295  Scalarcsca 14363   .scvsca 14364   0gc0g 14500   LModclmod 17074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-supp 6804  df-er 7214  df-en 7424  df-fin 7427  df-fsupp 7735  df-0g 14502  df-mnd 15537  df-grp 15667  df-rng 16773  df-lmod 17076

This theorem is referenced by:  mptscmfsuppd  17138  gsumsmonply1  31015  pm2mpcl  31304  mply1topmatcllem  31310  mp2pm2mplem2  31314  mp2pm2mplem5  31317  pm2mpghmlem2  31319  chcoeffeqlem  31392