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	⊢ (𝜑 → 𝐷 ∈ 𝑉)
mptscmfsupp0.q	⊢ (𝜑 → 𝑄 ∈ LMod)
mptscmfsupp0.r	⊢ (𝜑 → 𝑅 = (Scalar‘𝑄))
mptscmfsupp0.k	⊢ 𝐾 = (Base‘𝑄)
mptscmfsupp0.s	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑆 ∈ 𝐵)
mptscmfsupp0.w	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑊 ∈ 𝐾)
mptscmfsupp0.0	⊢ 0 = (0g‘𝑄)
mptscmfsupp0.z	⊢ 𝑍 = (0g‘𝑅)
mptscmfsupp0.m	⊢ ∗ = ( ·𝑠 ‘𝑄)
mptscmfsupp0.f	⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ 𝑆) finSupp 𝑍)

Assertion

Ref	Expression
mptscmfsupp0	⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) finSupp 0 )

Distinct variable groups:   𝐵,𝑘   𝐷,𝑘   𝑘,𝐾   𝜑,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 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
2		mptexg 6484	. . 3 ⊢ (𝐷 ∈ 𝑉 → (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) ∈ V)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) ∈ V)
4		funmpt 5926	. . 3 ⊢ Fun (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))
5	4	a1i 11	. 2 ⊢ (𝜑 → Fun (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)))
6		mptscmfsupp0.0	. . . 4 ⊢ 0 = (0g‘𝑄)
7		fvex 6201	. . . 4 ⊢ (0g‘𝑄) ∈ V
8	6, 7	eqeltri 2697	. . 3 ⊢ 0 ∈ V
9	8	a1i 11	. 2 ⊢ (𝜑 → 0 ∈ V)
10		mptscmfsupp0.f	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ 𝑆) finSupp 𝑍)
11	10	fsuppimpd 8282	. 2 ⊢ (𝜑 → ((𝑘 ∈ 𝐷 ↦ 𝑆) supp 𝑍) ∈ Fin)
12		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ 𝐷)
13		mptscmfsupp0.s	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑆 ∈ 𝐵)
14	13	ralrimiva 2966	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ 𝐷 𝑆 ∈ 𝐵)
15	14	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ∀𝑘 ∈ 𝐷 𝑆 ∈ 𝐵)
16		rspcsbela 4006	. . . . . . . . 9 ⊢ ((𝑑 ∈ 𝐷 ∧ ∀𝑘 ∈ 𝐷 𝑆 ∈ 𝐵) → ⦋𝑑 / 𝑘⦌𝑆 ∈ 𝐵)
17	12, 15, 16	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ⦋𝑑 / 𝑘⦌𝑆 ∈ 𝐵)
18		eqid 2622	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐷 ↦ 𝑆) = (𝑘 ∈ 𝐷 ↦ 𝑆)
19	18	fvmpts 6285	. . . . . . . 8 ⊢ ((𝑑 ∈ 𝐷 ∧ ⦋𝑑 / 𝑘⦌𝑆 ∈ 𝐵) → ((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) = ⦋𝑑 / 𝑘⦌𝑆)
20	12, 17, 19	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) = ⦋𝑑 / 𝑘⦌𝑆)
21	20	eqeq1d 2624	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (((𝑘 ∈ 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 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑄))
25	24	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑅 = (Scalar‘𝑄))
26	25	fveq2d 6195	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (0g‘𝑅) = (0g‘(Scalar‘𝑄)))
27	23, 26	syl5eq 2668	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑍 = (0g‘(Scalar‘𝑄)))
28	27	oveq1d 6665	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝑍 ∗ ⦋𝑑 / 𝑘⦌𝑊) = ((0g‘(Scalar‘𝑄)) ∗ ⦋𝑑 / 𝑘⦌𝑊))
29		mptscmfsupp0.q	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ LMod)
30	29	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑄 ∈ LMod)
31		mptscmfsupp0.w	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑊 ∈ 𝐾)
32	31	ralrimiva 2966	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ 𝐷 𝑊 ∈ 𝐾)
33	32	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ∀𝑘 ∈ 𝐷 𝑊 ∈ 𝐾)
34		rspcsbela 4006	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ 𝐷 ∧ ∀𝑘 ∈ 𝐷 𝑊 ∈ 𝐾) → ⦋𝑑 / 𝑘⦌𝑊 ∈ 𝐾)
35	12, 33, 34	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ⦋𝑑 / 𝑘⦌𝑊 ∈ 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 ∈ LMod ∧ ⦋𝑑 / 𝑘⦌𝑊 ∈ 𝐾) → ((0g‘(Scalar‘𝑄)) ∗ ⦋𝑑 / 𝑘⦌𝑊) = 0 )
41	30, 35, 40	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((0g‘(Scalar‘𝑄)) ∗ ⦋𝑑 / 𝑘⦌𝑊) = 0 )
42	28, 41	eqtrd 2656	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝑍 ∗ ⦋𝑑 / 𝑘⦌𝑊) = 0 )
43	22, 42	sylan9eqr 2678	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐷) ∧ ⦋𝑑 / 𝑘⦌𝑆 = 𝑍) → (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊) = 0 )
44		csbov12g 6689	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ 𝐷 → ⦋𝑑 / 𝑘⦌(𝑆 ∗ 𝑊) = (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊))
45	44	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ⦋𝑑 / 𝑘⦌(𝑆 ∗ 𝑊) = (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊))
46		ovex 6678	. . . . . . . . . . . . 13 ⊢ (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊) ∈ V
47	45, 46	syl6eqel 2709	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ⦋𝑑 / 𝑘⦌(𝑆 ∗ 𝑊) ∈ V)
48		eqid 2622	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) = (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))
49	48	fvmpts 6285	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ 𝐷 ∧ ⦋𝑑 / 𝑘⦌(𝑆 ∗ 𝑊) ∈ V) → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = ⦋𝑑 / 𝑘⦌(𝑆 ∗ 𝑊))
50	12, 47, 49	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = ⦋𝑑 / 𝑘⦌(𝑆 ∗ 𝑊))
51	50, 45	eqtrd 2656	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊))
52	51	eqeq1d 2624	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = 0 ↔ (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊) = 0 ))
53	52	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐷) ∧ ⦋𝑑 / 𝑘⦌𝑆 = 𝑍) → (((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = 0 ↔ (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊) = 0 ))
54	43, 53	mpbird 247	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐷) ∧ ⦋𝑑 / 𝑘⦌𝑆 = 𝑍) → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = 0 )
55	54	ex 450	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (⦋𝑑 / 𝑘⦌𝑆 = 𝑍 → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = 0 ))
56	21, 55	sylbid 230	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) = 𝑍 → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = 0 ))
57	56	necon3d 2815	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) ≠ 0 → ((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) ≠ 𝑍))
58	57	ss2rabdv 3683	. . 3 ⊢ (𝜑 → {𝑑 ∈ 𝐷 ∣ ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) ≠ 0 } ⊆ {𝑑 ∈ 𝐷 ∣ ((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) ≠ 𝑍})
59		ovex 6678	. . . . . 6 ⊢ (𝑆 ∗ 𝑊) ∈ V
60	59	rgenw 2924	. . . . 5 ⊢ ∀𝑘 ∈ 𝐷 (𝑆 ∗ 𝑊) ∈ V
61	48	fnmpt 6020	. . . . 5 ⊢ (∀𝑘 ∈ 𝐷 (𝑆 ∗ 𝑊) ∈ V → (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) Fn 𝐷)
62	60, 61	mp1i 13	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) Fn 𝐷)
63		suppvalfn 7302	. . . 4 ⊢ (((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) Fn 𝐷 ∧ 𝐷 ∈ 𝑉 ∧ 0 ∈ V) → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) supp 0 ) = {𝑑 ∈ 𝐷 ∣ ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) ≠ 0 })
64	62, 1, 9, 63	syl3anc 1326	. . 3 ⊢ (𝜑 → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) supp 0 ) = {𝑑 ∈ 𝐷 ∣ ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) ≠ 0 })
65	18	fnmpt 6020	. . . . 5 ⊢ (∀𝑘 ∈ 𝐷 𝑆 ∈ 𝐵 → (𝑘 ∈ 𝐷 ↦ 𝑆) Fn 𝐷)
66	14, 65	syl 17	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ 𝑆) Fn 𝐷)
67		fvex 6201	. . . . . 6 ⊢ (0g‘𝑅) ∈ V
68	23, 67	eqeltri 2697	. . . . 5 ⊢ 𝑍 ∈ V
69	68	a1i 11	. . . 4 ⊢ (𝜑 → 𝑍 ∈ V)
70		suppvalfn 7302	. . . 4 ⊢ (((𝑘 ∈ 𝐷 ↦ 𝑆) Fn 𝐷 ∧ 𝐷 ∈ 𝑉 ∧ 𝑍 ∈ V) → ((𝑘 ∈ 𝐷 ↦ 𝑆) supp 𝑍) = {𝑑 ∈ 𝐷 ∣ ((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) ≠ 𝑍})
71	66, 1, 69, 70	syl3anc 1326	. . 3 ⊢ (𝜑 → ((𝑘 ∈ 𝐷 ↦ 𝑆) supp 𝑍) = {𝑑 ∈ 𝐷 ∣ ((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) ≠ 𝑍})
72	58, 64, 71	3sstr4d 3648	. 2 ⊢ (𝜑 → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) supp 0 ) ⊆ ((𝑘 ∈ 𝐷 ↦ 𝑆) supp 𝑍))
73		suppssfifsupp 8290	. 2 ⊢ ((((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) ∈ V ∧ Fun (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) ∧ 0 ∈ V) ∧ (((𝑘 ∈ 𝐷 ↦ 𝑆) supp 𝑍) ∈ Fin ∧ ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) supp 0 ) ⊆ ((𝑘 ∈ 𝐷 ↦ 𝑆) supp 𝑍))) → (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) finSupp 0 )
74	3, 5, 9, 11, 72, 73	syl32anc 1334	1 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) finSupp 0 )

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  {crab 2916  Vcvv 3200  ⦋csb 3533   ⊆ 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   ·_𝑠 cvsca 15945  0_gc0g 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