Theorem mptscmfsupp0 20188

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 7100	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) ∈ V)
3		funmpt 6472	. . 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 6788	. . 3 ⊢ 0 ∈ V
7	6	a1i 11	. 2 ⊢ (𝜑 → 0 ∈ V)
8		mptscmfsupp0.f	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ 𝑆) finSupp 𝑍)
9	8	fsuppimpd 9135	. 2 ⊢ (𝜑 → ((𝑘 ∈ 𝐷 ↦ 𝑆) supp 𝑍) ∈ Fin)
10		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ 𝐷)
11		mptscmfsupp0.s	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑆 ∈ 𝐵)
12	11	ralrimiva 3103	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ 𝐷 𝑆 ∈ 𝐵)
13	12	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ∀𝑘 ∈ 𝐷 𝑆 ∈ 𝐵)
14		rspcsbela 4369	. . . . . . . . 9 ⊢ ((𝑑 ∈ 𝐷 ∧ ∀𝑘 ∈ 𝐷 𝑆 ∈ 𝐵) → ⦋𝑑 / 𝑘⦌𝑆 ∈ 𝐵)
15	10, 13, 14	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ⦋𝑑 / 𝑘⦌𝑆 ∈ 𝐵)
16		eqid 2738	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐷 ↦ 𝑆) = (𝑘 ∈ 𝐷 ↦ 𝑆)
17	16	fvmpts 6878	. . . . . . . 8 ⊢ ((𝑑 ∈ 𝐷 ∧ ⦋𝑑 / 𝑘⦌𝑆 ∈ 𝐵) → ((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) = ⦋𝑑 / 𝑘⦌𝑆)
18	10, 15, 17	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) = ⦋𝑑 / 𝑘⦌𝑆)
19	18	eqeq1d 2740	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) = 𝑍 ↔ ⦋𝑑 / 𝑘⦌𝑆 = 𝑍))
20		oveq1 7282	. . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑅 = (Scalar‘𝑄))
24	23	fveq2d 6778	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (0g‘𝑅) = (0g‘(Scalar‘𝑄)))
25	21, 24	eqtrid 2790	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑍 = (0g‘(Scalar‘𝑄)))
26	25	oveq1d 7290	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝑍 ∗ ⦋𝑑 / 𝑘⦌𝑊) = ((0g‘(Scalar‘𝑄)) ∗ ⦋𝑑 / 𝑘⦌𝑊))
27		mptscmfsupp0.q	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ LMod)
28	27	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑄 ∈ LMod)
29		mptscmfsupp0.w	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑊 ∈ 𝐾)
30	29	ralrimiva 3103	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ 𝐷 𝑊 ∈ 𝐾)
31	30	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ∀𝑘 ∈ 𝐷 𝑊 ∈ 𝐾)
32		rspcsbela 4369	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ 𝐷 ∧ ∀𝑘 ∈ 𝐷 𝑊 ∈ 𝐾) → ⦋𝑑 / 𝑘⦌𝑊 ∈ 𝐾)
33	10, 31, 32	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ⦋𝑑 / 𝑘⦌𝑊 ∈ 𝐾)
34		mptscmfsupp0.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (Base‘𝑄)
35		eqid 2738	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑄) = (Scalar‘𝑄)
36		mptscmfsupp0.m	. . . . . . . . . . . 12 ⊢ ∗ = ( ·𝑠 ‘𝑄)
37		eqid 2738	. . . . . . . . . . . 12 ⊢ (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄))
38	34, 35, 36, 37, 5	lmod0vs 20156	. . . . . . . . . . 11 ⊢ ((𝑄 ∈ LMod ∧ ⦋𝑑 / 𝑘⦌𝑊 ∈ 𝐾) → ((0g‘(Scalar‘𝑄)) ∗ ⦋𝑑 / 𝑘⦌𝑊) = 0 )
39	28, 33, 38	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((0g‘(Scalar‘𝑄)) ∗ ⦋𝑑 / 𝑘⦌𝑊) = 0 )
40	26, 39	eqtrd 2778	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝑍 ∗ ⦋𝑑 / 𝑘⦌𝑊) = 0 )
41	20, 40	sylan9eqr 2800	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐷) ∧ ⦋𝑑 / 𝑘⦌𝑆 = 𝑍) → (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊) = 0 )
42		csbov12g 7319	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ 𝐷 → ⦋𝑑 / 𝑘⦌(𝑆 ∗ 𝑊) = (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊))
43	42	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ⦋𝑑 / 𝑘⦌(𝑆 ∗ 𝑊) = (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊))
44		ovex 7308	. . . . . . . . . . . . 13 ⊢ (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊) ∈ V
45	43, 44	eqeltrdi 2847	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ⦋𝑑 / 𝑘⦌(𝑆 ∗ 𝑊) ∈ V)
46		eqid 2738	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) = (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))
47	46	fvmpts 6878	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ 𝐷 ∧ ⦋𝑑 / 𝑘⦌(𝑆 ∗ 𝑊) ∈ V) → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = ⦋𝑑 / 𝑘⦌(𝑆 ∗ 𝑊))
48	10, 45, 47	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = ⦋𝑑 / 𝑘⦌(𝑆 ∗ 𝑊))
49	48, 43	eqtrd 2778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊))
50	49	eqeq1d 2740	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = 0 ↔ (⦋𝑑 / 𝑘⦌𝑆 ∗ ⦋𝑑 / 𝑘⦌𝑊) = 0 ))
51	50	adantr 481	. . . . . . . 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 256	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐷) ∧ ⦋𝑑 / 𝑘⦌𝑆 = 𝑍) → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = 0 )
53	52	ex 413	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (⦋𝑑 / 𝑘⦌𝑆 = 𝑍 → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = 0 ))
54	19, 53	sylbid 239	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) = 𝑍 → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) = 0 ))
55	54	necon3d 2964	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) ≠ 0 → ((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) ≠ 𝑍))
56	55	ss2rabdv 4009	. . 3 ⊢ (𝜑 → {𝑑 ∈ 𝐷 ∣ ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) ≠ 0 } ⊆ {𝑑 ∈ 𝐷 ∣ ((𝑘 ∈ 𝐷 ↦ 𝑆)‘𝑑) ≠ 𝑍})
57		ovex 7308	. . . . . 6 ⊢ (𝑆 ∗ 𝑊) ∈ V
58	57	rgenw 3076	. . . . 5 ⊢ ∀𝑘 ∈ 𝐷 (𝑆 ∗ 𝑊) ∈ V
59	46	fnmpt 6573	. . . . 5 ⊢ (∀𝑘 ∈ 𝐷 (𝑆 ∗ 𝑊) ∈ V → (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) Fn 𝐷)
60	58, 59	mp1i 13	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) Fn 𝐷)
61		suppvalfn 7985	. . . 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 1370	. . 3 ⊢ (𝜑 → ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) supp 0 ) = {𝑑 ∈ 𝐷 ∣ ((𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊))‘𝑑) ≠ 0 })
63	16	fnmpt 6573	. . . . 5 ⊢ (∀𝑘 ∈ 𝐷 𝑆 ∈ 𝐵 → (𝑘 ∈ 𝐷 ↦ 𝑆) Fn 𝐷)
64	12, 63	syl 17	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ 𝑆) Fn 𝐷)
65	21	fvexi 6788	. . . . 5 ⊢ 𝑍 ∈ V
66	65	a1i 11	. . . 4 ⊢ (𝜑 → 𝑍 ∈ V)
67		suppvalfn 7985	. . . 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 1370	. . 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 9143	. 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 1377	1 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) finSupp 0 )

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  {crab 3068  Vcvv 3432  ⦋csb 3832   ⊆ wss 3887   class class class wbr 5074   ↦ cmpt 5157  Fun wfun 6427   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275   supp csupp 7977  Fincfn 8733   finSupp cfsupp 9128  Basecbs 16912  Scalarcsca 16965   ·_𝑠 cvsca 16966  0_gc0g 17150  LModclmod 20123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-supp 7978  df-1o 8297  df-en 8734  df-fin 8737  df-fsupp 9129  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-ring 19785  df-lmod 20125

This theorem is referenced by:  mptscmfsuppd  20189  gsumsmonply1  21474  pm2mpcl  21946  mply1topmatcllem  21952  mp2pm2mplem5  21959  pm2mpghmlem2  21961  chcoeffeqlem  22034  lbsdiflsp0  31707  fedgmullem2  31711