sgmppw - Metamath Proof Explorer

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Theorem sgmppw 25142

Description: The value of the divisor function at a prime power. (Contributed by Mario Carneiro, 17-May-2016.)

**Assertion**
Ref	Expression
sgmppw	⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝐴 σ (𝑃↑𝑁)) = Σ𝑘 ∈ (0...𝑁)((𝑃↑_𝑐𝐴)↑𝑘))

Distinct variable groups: 𝐴,𝑘 𝑘,𝑁 𝑃,𝑘

Proof of Theorem sgmppw Dummy variables 𝑖 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1131	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ∈ ℂ)
2		simp2 1132	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → 𝑃 ∈ ℙ)
3		prmnn 15610	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	2, 3	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → 𝑃 ∈ ℕ)
5		simp3 1133	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
6	4, 5	nnexpcld 13244	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝑃↑𝑁) ∈ ℕ)
7		sgmval 25088	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑃↑𝑁) ∈ ℕ) → (𝐴 σ (𝑃↑𝑁)) = Σ𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} (𝑛↑_𝑐𝐴))
8	1, 6, 7	syl2anc 696	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝐴 σ (𝑃↑𝑁)) = Σ𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} (𝑛↑_𝑐𝐴))
9		oveq1 6821	. . 3 ⊢ (𝑛 = (𝑃↑𝑘) → (𝑛↑_𝑐𝐴) = ((𝑃↑𝑘)↑_𝑐𝐴))
10		fzfid 12986	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (0...𝑁) ∈ Fin)
11		eqid 2760	. . . . 5 ⊢ (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖)) = (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖))
12	11	dvdsppwf1o 25132	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖)):(0...𝑁)–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)})
13	2, 5, 12	syl2anc 696	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖)):(0...𝑁)–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)})
14		oveq2 6822	. . . . 5 ⊢ (𝑖 = 𝑘 → (𝑃↑𝑖) = (𝑃↑𝑘))
15		ovex 6842	. . . . 5 ⊢ (𝑃↑𝑘) ∈ V
16	14, 11, 15	fvmpt 6445	. . . 4 ⊢ (𝑘 ∈ (0...𝑁) → ((𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖))‘𝑘) = (𝑃↑𝑘))
17	16	adantl 473	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖))‘𝑘) = (𝑃↑𝑘))
18		elrabi 3499	. . . . 5 ⊢ (𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} → 𝑛 ∈ ℕ)
19	18	nncnd 11248	. . . 4 ⊢ (𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} → 𝑛 ∈ ℂ)
20		cxpcl 24640	. . . 4 ⊢ ((𝑛 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝑛↑_𝑐𝐴) ∈ ℂ)
21	19, 1, 20	syl2anr 496	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)}) → (𝑛↑_𝑐𝐴) ∈ ℂ)
22	9, 10, 13, 17, 21	fsumf1o 14673	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → Σ𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} (𝑛↑_𝑐𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑃↑𝑘)↑_𝑐𝐴))
23		elfznn0 12646	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
24	23	adantl 473	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
25	24	nn0cnd 11565	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℂ)
26	1	adantr 472	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ)
27	25, 26	mulcomd 10273	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑘 · 𝐴) = (𝐴 · 𝑘))
28	27	oveq2d 6830	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝑘 · 𝐴)) = (𝑃↑_𝑐(𝐴 · 𝑘)))
29	4	adantr 472	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑃 ∈ ℕ)
30	29	nnrpd 12083	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑃 ∈ ℝ⁺)
31	24	nn0red 11564	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℝ)
32	30, 31, 26	cxpmuld 24700	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝑘 · 𝐴)) = ((𝑃↑_𝑐𝑘)↑_𝑐𝐴))
33	29	nncnd 11248	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑃 ∈ ℂ)
34		cxpexp 24634	. . . . . . 7 ⊢ ((𝑃 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑃↑_𝑐𝑘) = (𝑃↑𝑘))
35	33, 24, 34	syl2anc 696	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐𝑘) = (𝑃↑𝑘))
36	35	oveq1d 6829	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑃↑_𝑐𝑘)↑_𝑐𝐴) = ((𝑃↑𝑘)↑_𝑐𝐴))
37	32, 36	eqtrd 2794	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝑘 · 𝐴)) = ((𝑃↑𝑘)↑_𝑐𝐴))
38	33, 26, 24	cxpmul2d 24675	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝐴 · 𝑘)) = ((𝑃↑_𝑐𝐴)↑𝑘))
39	28, 37, 38	3eqtr3d 2802	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑃↑𝑘)↑_𝑐𝐴) = ((𝑃↑_𝑐𝐴)↑𝑘))
40	39	sumeq2dv 14652	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → Σ𝑘 ∈ (0...𝑁)((𝑃↑𝑘)↑_𝑐𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑃↑_𝑐𝐴)↑𝑘))
41	8, 22, 40	3eqtrd 2798	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝐴 σ (𝑃↑𝑁)) = Σ𝑘 ∈ (0...𝑁)((𝑃↑_𝑐𝐴)↑𝑘))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 {crab 3054 class class class wbr 4804 ↦ cmpt 4881 –1-1-onto→wf1o 6048 ‘cfv 6049 (class class class)co 6814 ℂcc 10146 0cc0 10148 · cmul 10153 ℕcn 11232 ℕ₀cn0 11504 ...cfz 12539 ↑cexp 13074 Σcsu 14635 ∥ cdvds 15202 ℙcprime 15607 ↑_𝑐ccxp 24522 σ csgm 25042

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-inf2 8713 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-pre-sup 10226 ax-addf 10227 ax-mulf 10228

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-of 7063 df-om 7232 df-1st 7334 df-2nd 7335 df-supp 7465 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-2o 7731 df-oadd 7734 df-er 7913 df-map 8027 df-pm 8028 df-ixp 8077 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-fsupp 8443 df-fi 8484 df-sup 8515 df-inf 8516 df-oi 8582 df-card 8975 df-cda 9202 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-uz 11900 df-q 12002 df-rp 12046 df-xneg 12159 df-xadd 12160 df-xmul 12161 df-ioo 12392 df-ioc 12393 df-ico 12394 df-icc 12395 df-fz 12540 df-fzo 12680 df-fl 12807 df-mod 12883 df-seq 13016 df-exp 13075 df-fac 13275 df-bc 13304 df-hash 13332 df-shft 14026 df-cj 14058 df-re 14059 df-im 14060 df-sqrt 14194 df-abs 14195 df-limsup 14421 df-clim 14438 df-rlim 14439 df-sum 14636 df-ef 15017 df-sin 15019 df-cos 15020 df-pi 15022 df-dvds 15203 df-gcd 15439 df-prm 15608 df-pc 15764 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-starv 16178 df-sca 16179 df-vsca 16180 df-ip 16181 df-tset 16182 df-ple 16183 df-ds 16186 df-unif 16187 df-hom 16188 df-cco 16189 df-rest 16305 df-topn 16306 df-0g 16324 df-gsum 16325 df-topgen 16326 df-pt 16327 df-prds 16330 df-xrs 16384 df-qtop 16389 df-imas 16390 df-xps 16392 df-mre 16468 df-mrc 16469 df-acs 16471 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-submnd 17557 df-mulg 17762 df-cntz 17970 df-cmn 18415 df-psmet 19960 df-xmet 19961 df-met 19962 df-bl 19963 df-mopn 19964 df-fbas 19965 df-fg 19966 df-cnfld 19969 df-top 20921 df-topon 20938 df-topsp 20959 df-bases 20972 df-cld 21045 df-ntr 21046 df-cls 21047 df-nei 21124 df-lp 21162 df-perf 21163 df-cn 21253 df-cnp 21254 df-haus 21341 df-tx 21587 df-hmeo 21780 df-fil 21871 df-fm 21963 df-flim 21964 df-flf 21965 df-xms 22346 df-ms 22347 df-tms 22348 df-cncf 22902 df-limc 23849 df-dv 23850 df-log 24523 df-cxp 24524 df-sgm 25048

This theorem is referenced by: 1sgmprm 25144 1sgm2ppw 25145

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