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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 13024 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin) | |
2 | dvdsssfz1 15376 | . . . . 5 ⊢ (𝑛 ∈ ℕ → {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} ⊆ (1...𝑛)) | |
3 | 2 | adantl 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} ⊆ (1...𝑛)) |
4 | ssfi 8421 | . . . 4 ⊢ (((1...𝑛) ∈ Fin ∧ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} ⊆ (1...𝑛)) → {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} ∈ Fin) | |
5 | 1, 3, 4 | syl2anc 580 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} ∈ Fin) |
6 | dchrisum0flb.r | . . . . 5 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ) | |
7 | 6 | ad2antrr 718 | . . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛}) → 𝑋:(Base‘𝑍)⟶ℝ) |
8 | rpvmasum.a | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
9 | 8 | nnnn0d 11637 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
10 | rpvmasum.z | . . . . . . . 8 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
11 | eqid 2798 | . . . . . . . 8 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
12 | rpvmasum.l | . . . . . . . 8 ⊢ 𝐿 = (ℤRHom‘𝑍) | |
13 | 10, 11, 12 | znzrhfo 20214 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝐿:ℤ–onto→(Base‘𝑍)) |
14 | fof 6330 | . . . . . . 7 ⊢ (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍)) | |
15 | 9, 13, 14 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍)) |
16 | 15 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐿:ℤ⟶(Base‘𝑍)) |
17 | elrabi 3550 | . . . . . 6 ⊢ (𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} → 𝑚 ∈ ℕ) | |
18 | 17 | nnzd 11768 | . . . . 5 ⊢ (𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} → 𝑚 ∈ ℤ) |
19 | ffvelrn 6582 | . . . . 5 ⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑚 ∈ ℤ) → (𝐿‘𝑚) ∈ (Base‘𝑍)) | |
20 | 16, 18, 19 | syl2an 590 | . . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛}) → (𝐿‘𝑚) ∈ (Base‘𝑍)) |
21 | 7, 20 | ffvelrnd 6585 | . . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛}) → (𝑋‘(𝐿‘𝑚)) ∈ ℝ) |
22 | 5, 21 | fsumrecl 14803 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑚)) ∈ ℝ) |
23 | dchrisum0f.f | . . 3 ⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣))) | |
24 | breq2 4846 | . . . . . . 7 ⊢ (𝑏 = 𝑛 → (𝑞 ∥ 𝑏 ↔ 𝑞 ∥ 𝑛)) | |
25 | 24 | rabbidv 3372 | . . . . . 6 ⊢ (𝑏 = 𝑛 → {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} = {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛}) |
26 | 25 | sumeq1d 14769 | . . . . 5 ⊢ (𝑏 = 𝑛 → Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣)) = Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑣))) |
27 | 2fveq3 6415 | . . . . . 6 ⊢ (𝑣 = 𝑚 → (𝑋‘(𝐿‘𝑣)) = (𝑋‘(𝐿‘𝑚))) | |
28 | 27 | cbvsumv 14764 | . . . . 5 ⊢ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑣)) = Σ𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑚)) |
29 | 26, 28 | syl6eq 2848 | . . . 4 ⊢ (𝑏 = 𝑛 → Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣)) = Σ𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑚))) |
30 | 29 | cbvmptv 4942 | . . 3 ⊢ (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣))) = (𝑛 ∈ ℕ ↦ Σ𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑚))) |
31 | 23, 30 | eqtri 2820 | . 2 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ Σ𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑚))) |
32 | 22, 31 | fmptd 6609 | 1 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 {crab 3092 ⊆ wss 3768 class class class wbr 4842 ↦ cmpt 4921 ⟶wf 6096 –onto→wfo 6098 ‘cfv 6100 (class class class)co 6877 Fincfn 8194 ℝcr 10222 1c1 10224 ℕcn 11311 ℕ0cn0 11577 ℤcz 11663 ...cfz 12577 Σcsu 14754 ∥ cdvds 15316 Basecbs 16181 0gc0g 16412 ℤRHomczrh 20167 ℤ/nℤczn 20170 DChrcdchr 25306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-rep 4963 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-inf2 8787 ax-cnex 10279 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 ax-pre-mulgt0 10300 ax-pre-sup 10301 ax-addf 10302 ax-mulf 10303 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-pss 3784 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-tp 4372 df-op 4374 df-uni 4628 df-int 4667 df-iun 4711 df-br 4843 df-opab 4905 df-mpt 4922 df-tr 4945 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-se 5271 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5897 df-ord 5943 df-on 5944 df-lim 5945 df-suc 5946 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-isom 6109 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-om 7299 df-1st 7400 df-2nd 7401 df-tpos 7589 df-wrecs 7644 df-recs 7706 df-rdg 7744 df-1o 7798 df-oadd 7802 df-er 7981 df-ec 7983 df-qs 7987 df-map 8096 df-en 8195 df-dom 8196 df-sdom 8197 df-fin 8198 df-sup 8589 df-inf 8590 df-oi 8656 df-card 9050 df-pnf 10364 df-mnf 10365 df-xr 10366 df-ltxr 10367 df-le 10368 df-sub 10557 df-neg 10558 df-div 10976 df-nn 11312 df-2 11373 df-3 11374 df-4 11375 df-5 11376 df-6 11377 df-7 11378 df-8 11379 df-9 11380 df-n0 11578 df-z 11664 df-dec 11781 df-uz 11928 df-rp 12072 df-fz 12578 df-fzo 12718 df-seq 13053 df-exp 13112 df-hash 13368 df-cj 14177 df-re 14178 df-im 14179 df-sqrt 14313 df-abs 14314 df-clim 14557 df-sum 14755 df-dvds 15317 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-ress 16189 df-plusg 16277 df-mulr 16278 df-starv 16279 df-sca 16280 df-vsca 16281 df-ip 16282 df-tset 16283 df-ple 16284 df-ds 16286 df-unif 16287 df-0g 16414 df-imas 16480 df-qus 16481 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-mhm 17647 df-grp 17738 df-minusg 17739 df-sbg 17740 df-mulg 17854 df-subg 17901 df-nsg 17902 df-eqg 17903 df-ghm 17968 df-cmn 18507 df-abl 18508 df-mgp 18803 df-ur 18815 df-ring 18862 df-cring 18863 df-oppr 18936 df-rnghom 19030 df-subrg 19093 df-lmod 19180 df-lss 19248 df-lsp 19290 df-sra 19492 df-rgmod 19493 df-lidl 19494 df-rsp 19495 df-2idl 19552 df-cnfld 20066 df-zring 20138 df-zrh 20171 df-zn 20174 |
This theorem is referenced by: dchrisum0flblem2 25547 dchrisum0fno1 25549 |
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