Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > vtsval | Structured version Visualization version GIF version |
Description: Value of the Vinogradov trigonometric sums (Contributed by Thierry Arnoux, 1-Dec-2021.) |
Ref | Expression |
---|---|
vtsval.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
vtsval.x | ⊢ (𝜑 → 𝑋 ∈ ℂ) |
vtsval.l | ⊢ (𝜑 → 𝐿:ℕ⟶ℂ) |
Ref | Expression |
---|---|
vtsval | ⊢ (𝜑 → ((𝐿vts𝑁)‘𝑋) = Σ𝑎 ∈ (1...𝑁)((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑋))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtsval.l | . . . 4 ⊢ (𝜑 → 𝐿:ℕ⟶ℂ) | |
2 | cnex 10017 | . . . . 5 ⊢ ℂ ∈ V | |
3 | nnex 11026 | . . . . 5 ⊢ ℕ ∈ V | |
4 | 2, 3 | elmap 7886 | . . . 4 ⊢ (𝐿 ∈ (ℂ ↑𝑚 ℕ) ↔ 𝐿:ℕ⟶ℂ) |
5 | 1, 4 | sylibr 224 | . . 3 ⊢ (𝜑 → 𝐿 ∈ (ℂ ↑𝑚 ℕ)) |
6 | vtsval.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
7 | fveq1 6190 | . . . . . . 7 ⊢ (𝑙 = 𝐿 → (𝑙‘𝑎) = (𝐿‘𝑎)) | |
8 | 7 | oveq1d 6665 | . . . . . 6 ⊢ (𝑙 = 𝐿 → ((𝑙‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥)))) = ((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥))))) |
9 | 8 | sumeq2sdv 14435 | . . . . 5 ⊢ (𝑙 = 𝐿 → Σ𝑎 ∈ (1...𝑛)((𝑙‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥)))) = Σ𝑎 ∈ (1...𝑛)((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥))))) |
10 | 9 | mpteq2dv 4745 | . . . 4 ⊢ (𝑙 = 𝐿 → (𝑥 ∈ ℂ ↦ Σ𝑎 ∈ (1...𝑛)((𝑙‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥))))) = (𝑥 ∈ ℂ ↦ Σ𝑎 ∈ (1...𝑛)((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥)))))) |
11 | oveq2 6658 | . . . . . 6 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁)) | |
12 | 11 | sumeq1d 14431 | . . . . 5 ⊢ (𝑛 = 𝑁 → Σ𝑎 ∈ (1...𝑛)((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥)))) = Σ𝑎 ∈ (1...𝑁)((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥))))) |
13 | 12 | mpteq2dv 4745 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ ℂ ↦ Σ𝑎 ∈ (1...𝑛)((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥))))) = (𝑥 ∈ ℂ ↦ Σ𝑎 ∈ (1...𝑁)((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥)))))) |
14 | df-vts 30714 | . . . 4 ⊢ vts = (𝑙 ∈ (ℂ ↑𝑚 ℕ), 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ ℂ ↦ Σ𝑎 ∈ (1...𝑛)((𝑙‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥)))))) | |
15 | 2 | mptex 6486 | . . . 4 ⊢ (𝑥 ∈ ℂ ↦ Σ𝑎 ∈ (1...𝑁)((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥))))) ∈ V |
16 | 10, 13, 14, 15 | ovmpt2 6796 | . . 3 ⊢ ((𝐿 ∈ (ℂ ↑𝑚 ℕ) ∧ 𝑁 ∈ ℕ0) → (𝐿vts𝑁) = (𝑥 ∈ ℂ ↦ Σ𝑎 ∈ (1...𝑁)((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥)))))) |
17 | 5, 6, 16 | syl2anc 693 | . 2 ⊢ (𝜑 → (𝐿vts𝑁) = (𝑥 ∈ ℂ ↦ Σ𝑎 ∈ (1...𝑁)((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥)))))) |
18 | oveq2 6658 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑎 · 𝑥) = (𝑎 · 𝑋)) | |
19 | 18 | oveq2d 6666 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((i · (2 · π)) · (𝑎 · 𝑥)) = ((i · (2 · π)) · (𝑎 · 𝑋))) |
20 | 19 | fveq2d 6195 | . . . . 5 ⊢ (𝑥 = 𝑋 → (exp‘((i · (2 · π)) · (𝑎 · 𝑥))) = (exp‘((i · (2 · π)) · (𝑎 · 𝑋)))) |
21 | 20 | oveq2d 6666 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥)))) = ((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑋))))) |
22 | 21 | sumeq2sdv 14435 | . . 3 ⊢ (𝑥 = 𝑋 → Σ𝑎 ∈ (1...𝑁)((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥)))) = Σ𝑎 ∈ (1...𝑁)((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑋))))) |
23 | 22 | adantl 482 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → Σ𝑎 ∈ (1...𝑁)((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥)))) = Σ𝑎 ∈ (1...𝑁)((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑋))))) |
24 | vtsval.x | . 2 ⊢ (𝜑 → 𝑋 ∈ ℂ) | |
25 | sumex 14418 | . . 3 ⊢ Σ𝑎 ∈ (1...𝑁)((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑋)))) ∈ V | |
26 | 25 | a1i 11 | . 2 ⊢ (𝜑 → Σ𝑎 ∈ (1...𝑁)((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑋)))) ∈ V) |
27 | 17, 23, 24, 26 | fvmptd 6288 | 1 ⊢ (𝜑 → ((𝐿vts𝑁)‘𝑋) = Σ𝑎 ∈ (1...𝑁)((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑋))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 ℂcc 9934 1c1 9937 ici 9938 · cmul 9941 ℕcn 11020 2c2 11070 ℕ0cn0 11292 ...cfz 12326 Σcsu 14416 expce 14792 πcpi 14797 vtscvts 30713 |
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 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
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-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 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-pred 5680 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-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-seq 12802 df-sum 14417 df-vts 30714 |
This theorem is referenced by: vtscl 30716 vtsprod 30717 |
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