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Mirrors > Home > MPE Home > Th. List > dchrisum0ff | Structured version Visualization version GIF version |
Description: The function 𝐹 is a real function. (Contributed by Mario Carneiro, 5-May-2016.) |
Ref | Expression |
---|---|
rpvmasum.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
rpvmasum.l | ⊢ 𝐿 = (ℤRHom‘𝑍) |
rpvmasum.a | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
rpvmasum2.g | ⊢ 𝐺 = (DChr‘𝑁) |
rpvmasum2.d | ⊢ 𝐷 = (Base‘𝐺) |
rpvmasum2.1 | ⊢ 1 = (0g‘𝐺) |
dchrisum0f.f | ⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣))) |
dchrisum0f.x | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
dchrisum0flb.r | ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ) |
Ref | Expression |
---|---|
dchrisum0ff | ⊢ (𝜑 → 𝐹:ℕ⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzfid 14026 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin) | |
2 | dvdsssfz1 16368 | . . . . 5 ⊢ (𝑛 ∈ ℕ → {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} ⊆ (1...𝑛)) | |
3 | 2 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} ⊆ (1...𝑛)) |
4 | 1, 3 | ssfid 9331 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} ∈ Fin) |
5 | dchrisum0flb.r | . . . . 5 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ) | |
6 | 5 | ad2antrr 725 | . . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛}) → 𝑋:(Base‘𝑍)⟶ℝ) |
7 | rpvmasum.a | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
8 | 7 | nnnn0d 12615 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
9 | rpvmasum.z | . . . . . . . 8 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
10 | eqid 2740 | . . . . . . . 8 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
11 | rpvmasum.l | . . . . . . . 8 ⊢ 𝐿 = (ℤRHom‘𝑍) | |
12 | 9, 10, 11 | znzrhfo 21591 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝐿:ℤ–onto→(Base‘𝑍)) |
13 | fof 6836 | . . . . . . 7 ⊢ (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍)) | |
14 | 8, 12, 13 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍)) |
15 | 14 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐿:ℤ⟶(Base‘𝑍)) |
16 | elrabi 3703 | . . . . . 6 ⊢ (𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} → 𝑚 ∈ ℕ) | |
17 | 16 | nnzd 12668 | . . . . 5 ⊢ (𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} → 𝑚 ∈ ℤ) |
18 | ffvelcdm 7117 | . . . . 5 ⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑚 ∈ ℤ) → (𝐿‘𝑚) ∈ (Base‘𝑍)) | |
19 | 15, 17, 18 | syl2an 595 | . . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛}) → (𝐿‘𝑚) ∈ (Base‘𝑍)) |
20 | 6, 19 | ffvelcdmd 7121 | . . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛}) → (𝑋‘(𝐿‘𝑚)) ∈ ℝ) |
21 | 4, 20 | fsumrecl 15784 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑚)) ∈ ℝ) |
22 | dchrisum0f.f | . . 3 ⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣))) | |
23 | breq2 5170 | . . . . . . 7 ⊢ (𝑏 = 𝑛 → (𝑞 ∥ 𝑏 ↔ 𝑞 ∥ 𝑛)) | |
24 | 23 | rabbidv 3451 | . . . . . 6 ⊢ (𝑏 = 𝑛 → {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} = {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛}) |
25 | 24 | sumeq1d 15750 | . . . . 5 ⊢ (𝑏 = 𝑛 → Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣)) = Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑣))) |
26 | 2fveq3 6927 | . . . . . 6 ⊢ (𝑣 = 𝑚 → (𝑋‘(𝐿‘𝑣)) = (𝑋‘(𝐿‘𝑚))) | |
27 | 26 | cbvsumv 15746 | . . . . 5 ⊢ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑣)) = Σ𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑚)) |
28 | 25, 27 | eqtrdi 2796 | . . . 4 ⊢ (𝑏 = 𝑛 → Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣)) = Σ𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑚))) |
29 | 28 | cbvmptv 5279 | . . 3 ⊢ (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣))) = (𝑛 ∈ ℕ ↦ Σ𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑚))) |
30 | 22, 29 | eqtri 2768 | . 2 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ Σ𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑚))) |
31 | 21, 30 | fmptd 7150 | 1 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 {crab 3443 ⊆ wss 3976 class class class wbr 5166 ↦ cmpt 5249 ⟶wf 6571 –onto→wfo 6573 ‘cfv 6575 (class class class)co 7450 ℝcr 11185 1c1 11187 ℕcn 12295 ℕ0cn0 12555 ℤcz 12641 ...cfz 13569 Σcsu 15736 ∥ cdvds 16304 Basecbs 17260 0gc0g 17501 ℤRHomczrh 21535 ℤ/nℤczn 21538 DChrcdchr 27296 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-inf2 9712 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-pre-sup 11264 ax-addf 11265 ax-mulf 11266 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-isom 6584 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-tpos 8269 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-er 8765 df-ec 8767 df-qs 8771 df-map 8888 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-sup 9513 df-inf 9514 df-oi 9581 df-card 10010 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-div 11950 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-7 12363 df-8 12364 df-9 12365 df-n0 12556 df-z 12642 df-dec 12761 df-uz 12906 df-rp 13060 df-fz 13570 df-fzo 13714 df-seq 14055 df-exp 14115 df-hash 14382 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15536 df-sum 15737 df-dvds 16305 df-struct 17196 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-ress 17290 df-plusg 17326 df-mulr 17327 df-starv 17328 df-sca 17329 df-vsca 17330 df-ip 17331 df-tset 17332 df-ple 17333 df-ds 17335 df-unif 17336 df-0g 17503 df-imas 17570 df-qus 17571 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-mhm 18820 df-grp 18978 df-minusg 18979 df-sbg 18980 df-mulg 19110 df-subg 19165 df-nsg 19166 df-eqg 19167 df-ghm 19255 df-cmn 19826 df-abl 19827 df-mgp 20164 df-rng 20182 df-ur 20211 df-ring 20264 df-cring 20265 df-oppr 20362 df-rhm 20500 df-subrng 20574 df-subrg 20599 df-lmod 20884 df-lss 20955 df-lsp 20995 df-sra 21197 df-rgmod 21198 df-lidl 21243 df-rsp 21244 df-2idl 21285 df-cnfld 21390 df-zring 21483 df-zrh 21539 df-zn 21542 |
This theorem is referenced by: dchrisum0flblem2 27573 dchrisum0fno1 27575 |
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