sgmppw - Metamath Proof Explorer

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Theorem sgmppw 27581

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 1154	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ∈ ℂ)
2		simp2 1155	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → 𝑃 ∈ ℙ)
3		prmnn 16893	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	2, 3	syl 18	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → 𝑃 ∈ ℕ)
5		simp3 1156	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
6	4, 5	nnexpcld 14430	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝑃↑𝑁) ∈ ℕ)
7		sgmval 27526	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑃↑𝑁) ∈ ℕ) → (𝐴 σ (𝑃↑𝑁)) = Σ𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} (𝑛↑_𝑐𝐴))
8	1, 6, 7	syl2anc 596	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝐴 σ (𝑃↑𝑁)) = Σ𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} (𝑛↑_𝑐𝐴))
9		oveq1 7458	. . 3 ⊢ (𝑛 = (𝑃↑𝑘) → (𝑛↑_𝑐𝐴) = ((𝑃↑𝑘)↑_𝑐𝐴))
10		fzfid 14156	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (0...𝑁) ∈ Fin)
11		eqid 2763	. . . . 5 ⊢ (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖)) = (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖))
12	11	dvdsppwf1o 27570	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖)):(0...𝑁)⟶1-1onto{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)})
13	2, 5, 12	syl2anc 596	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖)):(0...𝑁)⟶1-1onto{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)})
14		oveq2 7459	. . . . 5 ⊢ (𝑖 = 𝑘 → (𝑃↑𝑖) = (𝑃↑𝑘))
15		ovex 7484	. . . . 5 ⊢ (𝑃↑𝑘) ∈ V
16	14, 11, 15	fvmpt 7024	. . . 4 ⊢ (𝑘 ∈ (0...𝑁) → ((𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖))‘𝑘) = (𝑃↑𝑘))
17	16	adantl 487	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖))‘𝑘) = (𝑃↑𝑘))
18		elrabi 3644	. . . . 5 ⊢ (𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} → 𝑛 ∈ ℕ)
19	18	nncnd 12391	. . . 4 ⊢ (𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} → 𝑛 ∈ ℂ)
20		cxpcl 27059	. . . 4 ⊢ ((𝑛 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝑛↑_𝑐𝐴) ∈ ℂ)
21	19, 1, 20	syl2anr 609	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)}) → (𝑛↑_𝑐𝐴) ∈ ℂ)
22	9, 10, 13, 17, 21	fsumf1o 15933	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → Σ𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} (𝑛↑_𝑐𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑃↑𝑘)↑_𝑐𝐴))
23		elfznn0 13794	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
24	23	adantl 487	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
25	24	nn0cnd 12709	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℂ)
26	1	adantr 486	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ)
27	25, 26	mulcomd 11369	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑘 · 𝐴) = (𝐴 · 𝑘))
28	27	oveq2d 7467	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝑘 · 𝐴)) = (𝑃↑_𝑐(𝐴 · 𝑘)))
29	4	adantr 486	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑃 ∈ ℕ)
30	29	nnrpd 13202	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑃 ∈ ℝ⁺)
31	24	nn0red 12708	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℝ)
32	30, 31, 26	cxpmuld 27122	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝑘 · 𝐴)) = ((𝑃↑_𝑐𝑘)↑_𝑐𝐴))
33	29	nncnd 12391	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑃 ∈ ℂ)
34		cxpexp 27053	. . . . . . 7 ⊢ ((𝑃 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑃↑_𝑐𝑘) = (𝑃↑𝑘))
35	33, 24, 34	syl2anc 596	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐𝑘) = (𝑃↑𝑘))
36	35	oveq1d 7466	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑃↑_𝑐𝑘)↑_𝑐𝐴) = ((𝑃↑𝑘)↑_𝑐𝐴))
37	32, 36	eqtrd 2798	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝑘 · 𝐴)) = ((𝑃↑𝑘)↑_𝑐𝐴))
38	33, 26, 24	cxpmul2d 27094	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝐴 · 𝑘)) = ((𝑃↑_𝑐𝐴)↑𝑘))
39	28, 37, 38	3eqtr3d 2806	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑃↑𝑘)↑_𝑐𝐴) = ((𝑃↑_𝑐𝐴)↑𝑘))
40	39	sumeq2dv 15913	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → Σ𝑘 ∈ (0...𝑁)((𝑃↑𝑘)↑_𝑐𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑃↑_𝑐𝐴)↑𝑘))
41	8, 22, 40	3eqtrd 2802	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝐴 σ (𝑃↑𝑁)) = Σ𝑘 ∈ (0...𝑁)((𝑃↑_𝑐𝐴)↑𝑘))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 {crab 3416 class class class wbr 5108 ↦ cmpt 5192 ⟶1-1ontowf1o 6565 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 0cc0 11239 · cmul 11244 ℕcn 12375 ℕ₀cn0 12646 ...cfz 13679 ↑cexp 14244 Σcsu 15897 ∥ cdvds 16466 ℙcprime 16890 ↑_𝑐ccxp 26940 σ csgm 27480

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ioc 13521 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-fac 14459 df-bc 14488 df-hash 14516 df-shft 15263 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-limsup 15682 df-clim 15699 df-rlim 15700 df-sum 15898 df-ef 16277 df-sin 16279 df-cos 16280 df-pi 16282 df-dvds 16467 df-gcd 16709 df-prm 16891 df-pc 17059 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-xrs 17724 df-qtop 17729 df-imas 17730 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-mulg 19328 df-cntz 19582 df-cmn 20047 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-haus 23688 df-tx 23936 df-hmeo 24129 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-tms 24696 df-cncf 25254 df-limc 26241 df-dv 26242 df-log 26941 df-cxp 26942 df-sgm 27486

This theorem is used by: 1sgmprm 27583 1sgm2ppw 27584

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