Theorem mptscmfsupp0 18928

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 6484	. . 3 |- ( D e. V -> ( k e. D |-> ( S .* W ) ) e. _V )
3	1, 2	syl 17	. 2 |- ( ph -> ( k e. D |-> ( S .* W ) ) e. _V )
4		funmpt 5926	. . 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 6201	. . . 4 |- ( 0g ` Q ) e. _V
8	6, 7	eqeltri 2697	. . 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 8282	. 2 |- ( ph -> ( ( k e. D |-> S ) supp Z ) e. Fin )
12		simpr 477	. . . . . . . 8 |- ( ( ph /\ d e. D ) -> d e. D )
13		mptscmfsupp0.s	. . . . . . . . . . 11 |- ( ( ph /\ k e. D ) -> S e. B )
14	13	ralrimiva 2966	. . . . . . . . . 10 |- ( ph -> A. k e. D S e. B )
15	14	adantr 481	. . . . . . . . 9 |- ( ( ph /\ d e. D ) -> A. k e. D S e. B )
16		rspcsbela 4006	. . . . . . . . 9 |- ( ( d e. D /\ A. k e. D S e. B ) -> [_ d / k ]_ S e. B )
17	12, 15, 16	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ d e. D ) -> [_ d / k ]_ S e. B )
18		eqid 2622	. . . . . . . . 9 |- ( k e. D |-> S ) = ( k e. D |-> S )
19	18	fvmpts 6285	. . . . . . . 8 |- ( ( d e. D /\ [_ d / k ]_ S e. B ) -> ( ( k e. D |-> S ) ` d ) = [_ d / k ]_ S )
20	12, 17, 19	syl2anc 693	. . . . . . 7 |- ( ( ph /\ d e. D ) -> ( ( k e. D |-> S ) ` d ) = [_ d / k ]_ S )
21	20	eqeq1d 2624	. . . . . 6 |- ( ( ph /\ d e. D ) -> ( ( ( k e. D |-> S ) ` d ) = Z <-> [_ d / k ]_ S = Z ) )
22		oveq1 6657	. . . . . . . . 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 481	. . . . . . . . . . . . 13 |- ( ( ph /\ d e. D ) -> R = (Scalar ` Q ) )
26	25	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ph /\ d e. D ) -> ( 0g ` R ) = ( 0g ` (Scalar ` Q ) ) )
27	23, 26	syl5eq 2668	. . . . . . . . . . 11 |- ( ( ph /\ d e. D ) -> Z = ( 0g ` (Scalar ` Q ) ) )
28	27	oveq1d 6665	. . . . . . . . . 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 481	. . . . . . . . . . 11 |- ( ( ph /\ d e. D ) -> Q e. LMod )
31		mptscmfsupp0.w	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. D ) -> W e. K )
32	31	ralrimiva 2966	. . . . . . . . . . . . 13 |- ( ph -> A. k e. D W e. K )
33	32	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ d e. D ) -> A. k e. D W e. K )
34		rspcsbela 4006	. . . . . . . . . . . 12 |- ( ( d e. D /\ A. k e. D W e. K ) -> [_ d / k ]_ W e. K )
35	12, 33, 34	syl2anc 693	. . . . . . . . . . 11 |- ( ( ph /\ d e. D ) -> [_ d / k ]_ W e. K )
36		mptscmfsupp0.k	. . . . . . . . . . . 12 |- K = ( Base ` Q )
37		eqid 2622	. . . . . . . . . . . 12 |- (Scalar ` Q ) = (Scalar ` Q )
38		mptscmfsupp0.m	. . . . . . . . . . . 12 |- .* = ( .s ` Q )
39		eqid 2622	. . . . . . . . . . . 12 |- ( 0g ` (Scalar ` Q ) ) = ( 0g ` (Scalar ` Q ) )
40	36, 37, 38, 39, 6	lmod0vs 18896	. . . . . . . . . . 11 |- ( ( Q e. LMod /\ [_ d / k ]_ W e. K ) -> ( ( 0g ` (Scalar ` Q ) ) .* [_ d / k ]_ W ) = .0. )
41	30, 35, 40	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ d e. D ) -> ( ( 0g ` (Scalar ` Q ) ) .* [_ d / k ]_ W ) = .0. )
42	28, 41	eqtrd 2656	. . . . . . . . 9 |- ( ( ph /\ d e. D ) -> ( Z .* [_ d / k ]_ W ) = .0. )
43	22, 42	sylan9eqr 2678	. . . . . . . 8 |- ( ( ( ph /\ d e. D ) /\ [_ d / k ]_ S = Z ) -> ( [_ d / k ]_ S .* [_ d / k ]_ W ) = .0. )
44		csbov12g 6689	. . . . . . . . . . . . . 14 |- ( d e. D -> [_ d / k ]_ ( S .* W ) = ( [_ d / k ]_ S .* [_ d / k ]_ W ) )
45	44	adantl 482	. . . . . . . . . . . . 13 |- ( ( ph /\ d e. D ) -> [_ d / k ]_ ( S .* W ) = ( [_ d / k ]_ S .* [_ d / k ]_ W ) )
46		ovex 6678	. . . . . . . . . . . . 13 |- ( [_ d / k ]_ S .* [_ d / k ]_ W ) e. _V
47	45, 46	syl6eqel 2709	. . . . . . . . . . . 12 |- ( ( ph /\ d e. D ) -> [_ d / k ]_ ( S .* W ) e. _V )
48		eqid 2622	. . . . . . . . . . . . 13 |- ( k e. D |-> ( S .* W ) ) = ( k e. D |-> ( S .* W ) )
49	48	fvmpts 6285	. . . . . . . . . . . 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 693	. . . . . . . . . . 11 |- ( ( ph /\ d e. D ) -> ( ( k e. D |-> ( S .* W ) ) ` d ) = [_ d / k ]_ ( S .* W ) )
51	50, 45	eqtrd 2656	. . . . . . . . . 10 |- ( ( ph /\ d e. D ) -> ( ( k e. D |-> ( S .* W ) ) ` d ) = ( [_ d / k ]_ S .* [_ d / k ]_ W ) )
52	51	eqeq1d 2624	. . . . . . . . 9 |- ( ( ph /\ d e. D ) -> ( ( ( k e. D |-> ( S .* W ) ) ` d ) = .0. <-> ( [_ d / k ]_ S .* [_ d / k ]_ W ) = .0. ) )
53	52	adantr 481	. . . . . . . 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 247	. . . . . . 7 |- ( ( ( ph /\ d e. D ) /\ [_ d / k ]_ S = Z ) -> ( ( k e. D |-> ( S .* W ) ) ` d ) = .0. )
55	54	ex 450	. . . . . 6 |- ( ( ph /\ d e. D ) -> ( [_ d / k ]_ S = Z -> ( ( k e. D |-> ( S .* W ) ) ` d ) = .0. ) )
56	21, 55	sylbid 230	. . . . 5 |- ( ( ph /\ d e. D ) -> ( ( ( k e. D |-> S ) ` d ) = Z -> ( ( k e. D |-> ( S .* W ) ) ` d ) = .0. ) )
57	56	necon3d 2815	. . . 4 |- ( ( ph /\ d e. D ) -> ( ( ( k e. D |-> ( S .* W ) ) ` d ) =/= .0. -> ( ( k e. D |-> S ) ` d ) =/= Z ) )
58	57	ss2rabdv 3683	. . 3 |- ( ph -> { d e. D | ( ( k e. D |-> ( S .* W ) ) ` d ) =/= .0. } C_ { d e. D | ( ( k e. D |-> S ) ` d ) =/= Z } )
59		ovex 6678	. . . . . 6 |- ( S .* W ) e. _V
60	59	rgenw 2924	. . . . 5 |- A. k e. D ( S .* W ) e. _V
61	48	fnmpt 6020	. . . . 5 |- ( A. k e. D ( S .* W ) e. _V -> ( k e. D |-> ( S .* W ) ) Fn D )
62	60, 61	mp1i 13	. . . 4 |- ( ph -> ( k e. D |-> ( S .* W ) ) Fn D )
63		suppvalfn 7302	. . . 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 1326	. . 3 |- ( ph -> ( ( k e. D |-> ( S .* W ) ) supp .0. ) = { d e. D | ( ( k e. D |-> ( S .* W ) ) ` d ) =/= .0. } )
65	18	fnmpt 6020	. . . . 5 |- ( A. k e. D S e. B -> ( k e. D |-> S ) Fn D )
66	14, 65	syl 17	. . . 4 |- ( ph -> ( k e. D |-> S ) Fn D )
67		fvex 6201	. . . . . 6 |- ( 0g ` R ) e. _V
68	23, 67	eqeltri 2697	. . . . 5 |- Z e. _V
69	68	a1i 11	. . . 4 |- ( ph -> Z e. _V )
70		suppvalfn 7302	. . . 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 1326	. . 3 |- ( ph -> ( ( k e. D |-> S ) supp Z ) = { d e. D | ( ( k e. D |-> S ) ` d ) =/= Z } )
72	58, 64, 71	3sstr4d 3648	. 2 |- ( ph -> ( ( k e. D |-> ( S .* W ) ) supp .0. ) C_ ( ( k e. D |-> S ) supp Z ) )
73		suppssfifsupp 8290	. 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 1334	1 |- ( ph -> ( k e. D |-> ( S .* W ) ) finSupp .0. )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   [_csb 3533   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   supp csupp 7295   Fincfn 7955   finSupp cfsupp 8275   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   LModclmod 18863

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-fal 1489  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-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-supp 7296  df-er 7742  df-en 7956  df-fin 7959  df-fsupp 8276  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ring 18549  df-lmod 18865

This theorem is referenced by:  mptscmfsuppd  18929  gsumsmonply1  19673  pm2mpcl  20602  mply1topmatcllem  20608  mp2pm2mplem5  20615  pm2mpghmlem2  20617  chcoeffeqlem  20690