Theorem mptscmfsupp0 21235

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	1	mptexd 7256	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) ∈ V)
3		funmpt 6601	. . 3 ⊢ Fun (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))
4	3	a1i 11	. 2 ⊢ (𝜑 → Fun (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)))
5		mptscmfsupp0.0	. . . 4 ⊢ 0 = (0g‘𝑄)
6	5	fvexi 6925	. . 3 ⊢ 0 ∈ V
7	6	a1i 11	. 2 ⊢ (𝜑 → 0 ∈ V)
8		mptscmfsupp0.f	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ 𝑆) finSupp 𝑍)
9	8	fsuppimpd 9392	. 2 ⊢ (𝜑 → ((𝑘 ∈ 𝐷 ↦ 𝑆) supp 𝑍) ∈ Fin)
10		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ 𝐷)
11		mptscmfsupp0.s	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑆 ∈ 𝐵)
12	11	ralrimiva 3158	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ 𝐷 𝑆 ∈ 𝐵)
13	12	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ∀𝑘 ∈ 𝐷 𝑆 ∈ 𝐵)
14		rspcsbela 4399	. . . . . . . . 9 ⊢ ((𝑑 ∈ 𝐷 ∧ ∀𝑘 ∈ 𝐷 𝑆 ∈ 𝐵) → ⦋𝑑 / 𝑘⦌𝑆 ∈ 𝐵)
15	10, 13, 14	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ⦋𝑑 / 𝑘⦌𝑆 ∈ 𝐵)
16		eqid 2763	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐷 ↦ 𝑆) = (𝑘 ∈ 𝐷 ↦ 𝑆)
17	16	fvmpts 7023	. . . . . . . 8 ⊢ ((𝑑 ∈ 𝐷 ∧ ⦋𝑑 / 𝑘⦌𝑆 ∈ 𝐵) → ((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) = ⦋𝑑 / 𝑘⦌𝑆)
18	10, 15, 17	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) = ⦋𝑑 / 𝑘⦌𝑆)
19	18	eqeq1d 2765	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) = 𝑍 ↔ ⦋𝑑 / 𝑘⦌𝑆 = 𝑍))
20		oveq1 7453	. . . . . . . . 9 ⊢ (⦋𝑑 / 𝑘⦌𝑆 = 𝑍 → (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊) = (𝑍 ∗ ⦋𝑑 / 𝑘⦌𝑊))
21		mptscmfsupp0.z	. . . . . . . . . . . 12 ⊢ 𝑍 = (0g‘𝑅)
22		mptscmfsupp0.r	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑄))
23	22	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑅 = (Scalar‘𝑄))
24	23	fveq2d 6915	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (0g‘𝑅) = (0g‘(Scalar‘𝑄)))
25	21, 24	eqtrid 2810	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑍 = (0g‘(Scalar‘𝑄)))
26	25	oveq1d 7461	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝑍 ∗ ⦋𝑑 / 𝑘⦌𝑊) = ((0g‘(Scalar‘𝑄)) ∗ ⦋𝑑 / 𝑘⦌𝑊))
27		mptscmfsupp0.q	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ LMod)
28	27	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑄 ∈ LMod)
29		mptscmfsupp0.w	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑊 ∈ 𝐾)
30	29	ralrimiva 3158	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ 𝐷 𝑊 ∈ 𝐾)
31	30	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ∀𝑘 ∈ 𝐷 𝑊 ∈ 𝐾)
32		rspcsbela 4399	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ 𝐷 ∧ ∀𝑘 ∈ 𝐷 𝑊 ∈ 𝐾) → ⦋𝑑 / 𝑘⦌𝑊 ∈ 𝐾)
33	10, 31, 32	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ⦋𝑑 / 𝑘⦌𝑊 ∈ 𝐾)
34		mptscmfsupp0.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (Base‘𝑄)
35		eqid 2763	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑄) = (Scalar‘𝑄)
36		mptscmfsupp0.m	. . . . . . . . . . . 12 ⊢ ∗ = ( ·𝑠 ‘𝑄)
37		eqid 2763	. . . . . . . . . . . 12 ⊢ (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄))
38	34, 35, 36, 37, 5	lmod0vs 21203	. . . . . . . . . . 11 ⊢ ((𝑄 ∈ LMod ∧ ⦋𝑑 / 𝑘⦌𝑊 ∈ 𝐾) → ((0g‘(Scalar‘𝑄)) ∗ ⦋𝑑 / 𝑘⦌𝑊) = 0 )
39	28, 33, 38	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((0g‘(Scalar‘𝑄)) ∗ ⦋𝑑 / 𝑘⦌𝑊) = 0 )
40	26, 39	eqtrd 2798	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝑍 ∗ ⦋𝑑 / 𝑘⦌𝑊) = 0 )
41	20, 40	sylan9eqr 2820	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐷) ∧ ⦋𝑑 / 𝑘⦌𝑆 = 𝑍) → (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊) = 0 )
42		csbov12g 7492	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ 𝐷 → ⦋𝑑 / 𝑘⦌(𝑆 ∗ 𝑊) = (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊))
43	42	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ⦋𝑑 / 𝑘⦌(𝑆 ∗ 𝑊) = (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊))
44		ovex 7479	. . . . . . . . . . . . 13 ⊢ (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊) ∈ V
45	43, 44	eqeltrdi 2872	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ⦋𝑑 / 𝑘⦌(𝑆 ∗ 𝑊) ∈ V)
46		eqid 2763	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) = (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))
47	46	fvmpts 7023	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ 𝐷 ∧ ⦋𝑑 / 𝑘⦌(𝑆 ∗ 𝑊) ∈ V) → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = ⦋𝑑 / 𝑘⦌(𝑆 ∗ 𝑊))
48	10, 45, 47	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = ⦋𝑑 / 𝑘⦌(𝑆 ∗ 𝑊))
49	48, 43	eqtrd 2798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊))
50	49	eqeq1d 2765	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = 0 ↔ (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊) = 0 ))
51	50	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐷) ∧ ⦋𝑑 / 𝑘⦌𝑆 = 𝑍) → (((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = 0 ↔ (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊) = 0 ))
52	41, 51	mpbird 260	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐷) ∧ ⦋𝑑 / 𝑘⦌𝑆 = 𝑍) → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = 0 )
53	52	ex 418	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (⦋𝑑 / 𝑘⦌𝑆 = 𝑍 → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = 0 ))
54	19, 53	sylbid 243	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) = 𝑍 → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = 0 ))
55	54	necon3d 2980	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) ≠ 0 → ((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) ≠ 𝑍))
56	55	ss2rabdv 4003	. . 3 ⊢ (𝜑 → {𝑑 ∈ 𝐷 ∣ ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) ≠ 0 } ⊆ {𝑑 ∈ 𝐷 ∣ ((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) ≠ 𝑍})
57		ovex 7479	. . . . . 6 ⊢ (𝑆 ∗ 𝑊) ∈ V
58	57	rgenw 3084	. . . . 5 ⊢ ∀𝑘 ∈ 𝐷 (𝑆 ∗ 𝑊) ∈ V
59	46	fnmpt 6704	. . . . 5 ⊢ (∀𝑘 ∈ 𝐷 (𝑆 ∗ 𝑊) ∈ V → (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) Fn 𝐷)
60	58, 59	mp1i 14	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) Fn 𝐷)
61		suppvalfn 8216	. . . 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 })
62	60, 1, 7, 61	syl3anc 1399	. . 3 ⊢ (𝜑 → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) supp 0 ) = {𝑑 ∈ 𝐷 ∣ ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) ≠ 0 })
63	16	fnmpt 6704	. . . . 5 ⊢ (∀𝑘 ∈ 𝐷 𝑆 ∈ 𝐵 → (𝑘 ∈ 𝐷 ↦ 𝑆) Fn 𝐷)
64	12, 63	syl 18	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ 𝑆) Fn 𝐷)
65	21	fvexi 6925	. . . . 5 ⊢ 𝑍 ∈ V
66	65	a1i 11	. . . 4 ⊢ (𝜑 → 𝑍 ∈ V)
67		suppvalfn 8216	. . . 4 ⊢ (((𝑘 ∈ 𝐷 ↦ 𝑆) Fn 𝐷 ∧ 𝐷 ∈ 𝑉 ∧ 𝑍 ∈ V) → ((𝑘 ∈ 𝐷 ↦ 𝑆) supp 𝑍) = {𝑑 ∈ 𝐷 ∣ ((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) ≠ 𝑍})
68	64, 1, 66, 67	syl3anc 1399	. . 3 ⊢ (𝜑 → ((𝑘 ∈ 𝐷 ↦ 𝑆) supp 𝑍) = {𝑑 ∈ 𝐷 ∣ ((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) ≠ 𝑍})
69	56, 62, 68	3sstr4d 3968	. 2 ⊢ (𝜑 → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) supp 0 ) ⊆ ((𝑘 ∈ 𝐷 ↦ 𝑆) supp 𝑍))
70		suppssfifsupp 9402	. 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 )
71	2, 4, 7, 9, 69, 70	syl32anc 1406	1 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) finSupp 0 )

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  {crab 3416  Vcvv 3454  ⦋csb 3851   ⊆ wss 3900   class class class wbr 5106   ↦ cmpt 5190  Fun wfun 6555   Fn wfn 6556  ‘cfv 6561  (class class class)co 7446   supp csupp 8207  Fincfn 9004   finSupp cfsupp 9384  Basecbs 17419  Scalarcsca 17463   ·_𝑠 cvsca 17464  0_gc0g 17642  LModclmod 21168

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-supp 8208  df-1o 8507  df-en 9005  df-fin 9008  df-fsupp 9385  df-0g 17644  df-mgm 18848  df-sgrp 18940  df-mnd 18956  df-grp 19179  df-ring 20494  df-lmod 21170

This theorem is used by:  mptscmfsuppd  21236  gsumsmonply1  22658  pm2mpcl  23148  mply1topmatcllem  23154  mp2pm2mplem5  23161  pm2mpghmlem2  23163  chcoeffeqlem  23236  lbsdiflsp0  34341  fedgmullem2  34345