Theorem mptscmfsupp0 18164

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 6121	. . 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 5597	. . 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 5858	. . . 4 |- ( 0g ` Q ) e. _V
8	6, 7	eqeltri 2526	. . 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 7877	. 2 |- ( ph -> ( ( k e. D |-> S ) supp Z ) e. Fin )
12		simpr 467	. . . . . . . 8 |- ( ( ph /\ d e. D ) -> d e. D )
13		mptscmfsupp0.s	. . . . . . . . . . 11 |- ( ( ph /\ k e. D ) -> S e. B )
14	13	ralrimiva 2790	. . . . . . . . . 10 |- ( ph -> A. k e. D S e. B )
15	14	adantr 471	. . . . . . . . 9 |- ( ( ph /\ d e. D ) -> A. k e. D S e. B )
16		rspcsbela 3763	. . . . . . . . 9 |- ( ( d e. D /\ A. k e. D S e. B ) -> [_ d / k ]_ S e. B )
17	12, 15, 16	syl2anc 671	. . . . . . . 8 |- ( ( ph /\ d e. D ) -> [_ d / k ]_ S e. B )
18		eqid 2452	. . . . . . . . 9 |- ( k e. D |-> S ) = ( k e. D |-> S )
19	18	fvmpts 5935	. . . . . . . 8 |- ( ( d e. D /\ [_ d / k ]_ S e. B ) -> ( ( k e. D |-> S ) ` d ) = [_ d / k ]_ S )
20	12, 17, 19	syl2anc 671	. . . . . . 7 |- ( ( ph /\ d e. D ) -> ( ( k e. D |-> S ) ` d ) = [_ d / k ]_ S )
21	20	eqeq1d 2454	. . . . . 6 |- ( ( ph /\ d e. D ) -> ( ( ( k e. D |-> S ) ` d ) = Z <-> [_ d / k ]_ S = Z ) )
22		oveq1 6283	. . . . . . . . 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 471	. . . . . . . . . . . . 13 |- ( ( ph /\ d e. D ) -> R = (Scalar ` Q ) )
26	25	fveq2d 5852	. . . . . . . . . . . 12 |- ( ( ph /\ d e. D ) -> ( 0g ` R ) = ( 0g ` (Scalar ` Q ) ) )
27	23, 26	syl5eq 2498	. . . . . . . . . . 11 |- ( ( ph /\ d e. D ) -> Z = ( 0g ` (Scalar ` Q ) ) )
28	27	oveq1d 6291	. . . . . . . . . 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 471	. . . . . . . . . . 11 |- ( ( ph /\ d e. D ) -> Q e. LMod )
31		mptscmfsupp0.w	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. D ) -> W e. K )
32	31	ralrimiva 2790	. . . . . . . . . . . . 13 |- ( ph -> A. k e. D W e. K )
33	32	adantr 471	. . . . . . . . . . . 12 |- ( ( ph /\ d e. D ) -> A. k e. D W e. K )
34		rspcsbela 3763	. . . . . . . . . . . 12 |- ( ( d e. D /\ A. k e. D W e. K ) -> [_ d / k ]_ W e. K )
35	12, 33, 34	syl2anc 671	. . . . . . . . . . 11 |- ( ( ph /\ d e. D ) -> [_ d / k ]_ W e. K )
36		mptscmfsupp0.k	. . . . . . . . . . . 12 |- K = ( Base ` Q )
37		eqid 2452	. . . . . . . . . . . 12 |- (Scalar ` Q ) = (Scalar ` Q )
38		mptscmfsupp0.m	. . . . . . . . . . . 12 |- .* = ( .s ` Q )
39		eqid 2452	. . . . . . . . . . . 12 |- ( 0g ` (Scalar ` Q ) ) = ( 0g ` (Scalar ` Q ) )
40	36, 37, 38, 39, 6	lmod0vs 18135	. . . . . . . . . . 11 |- ( ( Q e. LMod /\ [_ d / k ]_ W e. K ) -> ( ( 0g ` (Scalar ` Q ) ) .* [_ d / k ]_ W ) = .0. )
41	30, 35, 40	syl2anc 671	. . . . . . . . . 10 |- ( ( ph /\ d e. D ) -> ( ( 0g ` (Scalar ` Q ) ) .* [_ d / k ]_ W ) = .0. )
42	28, 41	eqtrd 2486	. . . . . . . . 9 |- ( ( ph /\ d e. D ) -> ( Z .* [_ d / k ]_ W ) = .0. )
43	22, 42	sylan9eqr 2508	. . . . . . . 8 |- ( ( ( ph /\ d e. D ) /\ [_ d / k ]_ S = Z ) -> ( [_ d / k ]_ S .* [_ d / k ]_ W ) = .0. )
44		csbov12g 6312	. . . . . . . . . . . . . 14 |- ( d e. D -> [_ d / k ]_ ( S .* W ) = ( [_ d / k ]_ S .* [_ d / k ]_ W ) )
45	44	adantl 472	. . . . . . . . . . . . 13 |- ( ( ph /\ d e. D ) -> [_ d / k ]_ ( S .* W ) = ( [_ d / k ]_ S .* [_ d / k ]_ W ) )
46		ovex 6304	. . . . . . . . . . . . 13 |- ( [_ d / k ]_ S .* [_ d / k ]_ W ) e. _V
47	45, 46	syl6eqel 2538	. . . . . . . . . . . 12 |- ( ( ph /\ d e. D ) -> [_ d / k ]_ ( S .* W ) e. _V )
48		eqid 2452	. . . . . . . . . . . . 13 |- ( k e. D |-> ( S .* W ) ) = ( k e. D |-> ( S .* W ) )
49	48	fvmpts 5935	. . . . . . . . . . . 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 671	. . . . . . . . . . 11 |- ( ( ph /\ d e. D ) -> ( ( k e. D |-> ( S .* W ) ) ` d ) = [_ d / k ]_ ( S .* W ) )
51	50, 45	eqtrd 2486	. . . . . . . . . 10 |- ( ( ph /\ d e. D ) -> ( ( k e. D |-> ( S .* W ) ) ` d ) = ( [_ d / k ]_ S .* [_ d / k ]_ W ) )
52	51	eqeq1d 2454	. . . . . . . . 9 |- ( ( ph /\ d e. D ) -> ( ( ( k e. D |-> ( S .* W ) ) ` d ) = .0. <-> ( [_ d / k ]_ S .* [_ d / k ]_ W ) = .0. ) )
53	52	adantr 471	. . . . . . . 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 240	. . . . . . 7 |- ( ( ( ph /\ d e. D ) /\ [_ d / k ]_ S = Z ) -> ( ( k e. D |-> ( S .* W ) ) ` d ) = .0. )
55	54	ex 440	. . . . . 6 |- ( ( ph /\ d e. D ) -> ( [_ d / k ]_ S = Z -> ( ( k e. D |-> ( S .* W ) ) ` d ) = .0. ) )
56	21, 55	sylbid 223	. . . . 5 |- ( ( ph /\ d e. D ) -> ( ( ( k e. D |-> S ) ` d ) = Z -> ( ( k e. D |-> ( S .* W ) ) ` d ) = .0. ) )
57	56	necon3d 2645	. . . 4 |- ( ( ph /\ d e. D ) -> ( ( ( k e. D |-> ( S .* W ) ) ` d ) =/= .0. -> ( ( k e. D |-> S ) ` d ) =/= Z ) )
58	57	ss2rabdv 3478	. . 3 |- ( ph -> { d e. D | ( ( k e. D |-> ( S .* W ) ) ` d ) =/= .0. } C_ { d e. D | ( ( k e. D |-> S ) ` d ) =/= Z } )
59		ovex 6304	. . . . . 6 |- ( S .* W ) e. _V
60	59	rgenw 2749	. . . . 5 |- A. k e. D ( S .* W ) e. _V
61	48	fnmpt 5686	. . . . 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 6909	. . . 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 1271	. . 3 |- ( ph -> ( ( k e. D |-> ( S .* W ) ) supp .0. ) = { d e. D | ( ( k e. D |-> ( S .* W ) ) ` d ) =/= .0. } )
65	18	fnmpt 5686	. . . . 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 5858	. . . . . 6 |- ( 0g ` R ) e. _V
68	23, 67	eqeltri 2526	. . . . 5 |- Z e. _V
69	68	a1i 11	. . . 4 |- ( ph -> Z e. _V )
70		suppvalfn 6909	. . . 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 1271	. . 3 |- ( ph -> ( ( k e. D |-> S ) supp Z ) = { d e. D | ( ( k e. D |-> S ) ` d ) =/= Z } )
72	58, 64, 71	3sstr4d 3443	. 2 |- ( ph -> ( ( k e. D |-> ( S .* W ) ) supp .0. ) C_ ( ( k e. D |-> S ) supp Z ) )
73		suppssfifsupp 7885	. 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 1279	1 |- ( ph -> ( k e. D |-> ( S .* W ) ) finSupp .0. )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1448   e. wcel 1891   =/= wne 2622   A.wral 2737   {crab 2741   _Vcvv 3013   [_csb 3331   C_ wss 3372   class class class wbr 4374   |-> cmpt 4433   Fun wfun 5555   Fn wfn 5556   ` cfv 5561  (class class class)co 6276   supp csupp 6902   Fincfn 7556   finSupp cfsupp 7870   Basecbs 15132  Scalarcsca 15204   .scvsca 15205   0gc0g 15349   LModclmod 18102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-rep 4487  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1451  df-fal 1454  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4169  df-iun 4250  df-br 4375  df-opab 4434  df-mpt 4435  df-tr 4470  df-eprel 4723  df-id 4727  df-po 4733  df-so 4734  df-fr 4771  df-we 4773  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-ord 5405  df-on 5406  df-lim 5407  df-suc 5408  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569  df-riota 6238  df-ov 6279  df-oprab 6280  df-mpt2 6281  df-om 6681  df-supp 6903  df-er 7350  df-en 7557  df-fin 7560  df-fsupp 7871  df-0g 15351  df-mgm 16499  df-sgrp 16538  df-mnd 16548  df-grp 16684  df-ring 17793  df-lmod 18104

This theorem is referenced by:  mptscmfsuppd  18165  gsumsmonply1  18908  pm2mpcl  19832  mply1topmatcllem  19838  mp2pm2mplem5  19845  pm2mpghmlem2  19847  chcoeffeqlem  19920