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Theorem dchrisum0ff 27488

Description: The function 𝐹 is a real function. (Contributed by Mario Carneiro, 5-May-2016.)

**Hypotheses**
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 = (0_g‘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)

**Assertion**
Ref	Expression
dchrisum0ff	⊢ (𝜑 → 𝐹:ℕ⟶ℝ)

Distinct variable groups: 𝑞,𝑏,𝑣 𝑁,𝑞 𝐿,𝑏,𝑣 𝑋,𝑏,𝑣 Allowed substitution hints: 𝜑(𝑣,𝑞,𝑏) 𝐷(𝑣,𝑞,𝑏) 1 (𝑣,𝑞,𝑏) 𝐹(𝑣,𝑞,𝑏) 𝐺(𝑣,𝑞,𝑏) 𝐿(𝑞) 𝑁(𝑣,𝑏) 𝑋(𝑞) 𝑍(𝑣,𝑞,𝑏)

Proof of Theorem dchrisum0ff Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 13930	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
2		dvdsssfz1 16282	. . . . 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 9174	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} ∈ Fin)
5		dchrisum0flb.r	. . . . 5 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)
6	5	ad2antrr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛}) → 𝑋:(Base‘𝑍)⟶ℝ)
7		rpvmasum.a	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	nnnn0d 12493	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
9		rpvmasum.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
10		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑍) = (Base‘𝑍)
11		rpvmasum.l	. . . . . . . 8 ⊢ 𝐿 = (ℤRHom‘𝑍)
12	9, 10, 11	znzrhfo 21541	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
13		fof 6748	. . . . . . 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 3631	. . . . . 6 ⊢ (𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} → 𝑚 ∈ ℕ)
17	16	nnzd 12545	. . . . 5 ⊢ (𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} → 𝑚 ∈ ℤ)
18		ffvelcdm 7029	. . . . 5 ⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑚 ∈ ℤ) → (𝐿‘𝑚) ∈ (Base‘𝑍))
19	15, 17, 18	syl2an 597	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛}) → (𝐿‘𝑚) ∈ (Base‘𝑍))
20	6, 19	ffvelcdmd 7033	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛}) → (𝑋‘(𝐿‘𝑚)) ∈ ℝ)
21	4, 20	fsumrecl 15691	. 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 5090	. . . . . . 7 ⊢ (𝑏 = 𝑛 → (𝑞 ∥ 𝑏 ↔ 𝑞 ∥ 𝑛))
24	23	rabbidv 3397	. . . . . 6 ⊢ (𝑏 = 𝑛 → {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} = {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛})
25	24	sumeq1d 15657	. . . . 5 ⊢ (𝑏 = 𝑛 → Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣)) = Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑣)))
26		2fveq3 6841	. . . . . 6 ⊢ (𝑣 = 𝑚 → (𝑋‘(𝐿‘𝑣)) = (𝑋‘(𝐿‘𝑚)))
27	26	cbvsumv 15653	. . . . 5 ⊢ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑣)) = Σ𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑚))
28	25, 27	eqtrdi 2788	. . . 4 ⊢ (𝑏 = 𝑛 → Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣)) = Σ𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑚)))
29	28	cbvmptv 5190	. . 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 2760	. 2 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ Σ𝑚 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑛} (𝑋‘(𝐿‘𝑚)))
31	21, 30	fmptd 7062	1 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {crab 3390 ⊆ wss 3890 class class class wbr 5086 ↦ cmpt 5167 ⟶wf 6490 –onto→wfo 6492 ‘cfv 6494 (class class class)co 7362 ℝcr 11032 1c1 11034 ℕcn 12169 ℕ₀cn0 12432 ℤcz 12519 ...cfz 13456 Σcsu 15643 ∥ cdvds 16216 Basecbs 17174 0_gc0g 17397 ℤRHomczrh 21493 ℤ/nℤczn 21496 DChrcdchr 27213

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-inf2 9557 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112 ax-mulf 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-se 5580 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-tpos 8171 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-er 8638 df-ec 8640 df-qs 8644 df-map 8770 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-sup 9350 df-inf 9351 df-oi 9420 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-rp 12938 df-fz 13457 df-fzo 13604 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-clim 15445 df-sum 15644 df-dvds 16217 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-0g 17399 df-imas 17467 df-qus 17468 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19039 df-subg 19094 df-nsg 19095 df-eqg 19096 df-ghm 19183 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-cring 20212 df-oppr 20312 df-rhm 20447 df-subrng 20518 df-subrg 20542 df-lmod 20852 df-lss 20922 df-lsp 20962 df-sra 21164 df-rgmod 21165 df-lidl 21202 df-rsp 21203 df-2idl 21244 df-cnfld 21349 df-zring 21441 df-zrh 21497 df-zn 21500

This theorem is referenced by: dchrisum0flblem2 27490 dchrisum0fno1 27492

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