sgmppw - Metamath Proof Explorer

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Theorem sgmppw 27524

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 16849	. . . . 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 14389	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝑃↑𝑁) ∈ ℕ)
7		sgmval 27469	. . 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 7427	. . 3 ⊢ (𝑛 = (𝑃↑𝑘) → (𝑛↑_𝑐𝐴) = ((𝑃↑𝑘)↑_𝑐𝐴))
10		fzfid 14116	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (0...𝑁) ∈ Fin)
11		eqid 2761	. . . . 5 ⊢ (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖)) = (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖))
12	11	dvdsppwf1o 27513	. . . 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 596	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖)):(0...𝑁)–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)})
14		oveq2 7428	. . . . 5 ⊢ (𝑖 = 𝑘 → (𝑃↑𝑖) = (𝑃↑𝑘))
15		ovex 7453	. . . . 5 ⊢ (𝑃↑𝑘) ∈ V
16	14, 11, 15	fvmpt 6993	. . . 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 3641	. . . . 5 ⊢ (𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} → 𝑛 ∈ ℕ)
19	18	nncnd 12351	. . . 4 ⊢ (𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} → 𝑛 ∈ ℂ)
20		cxpcl 27002	. . . 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 15889	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → Σ𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} (𝑛↑_𝑐𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑃↑𝑘)↑_𝑐𝐴))
23		elfznn0 13754	. . . . . . . 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 12669	. . . . . 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 11330	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑘 · 𝐴) = (𝐴 · 𝑘))
28	27	oveq2d 7436	. . . 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 13162	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑃 ∈ ℝ⁺)
31	24	nn0red 12668	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℝ)
32	30, 31, 26	cxpmuld 27065	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝑘 · 𝐴)) = ((𝑃↑_𝑐𝑘)↑_𝑐𝐴))
33	29	nncnd 12351	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑃 ∈ ℂ)
34		cxpexp 26996	. . . . . . 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 7435	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑃↑_𝑐𝑘)↑_𝑐𝐴) = ((𝑃↑𝑘)↑_𝑐𝐴))
37	32, 36	eqtrd 2796	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝑘 · 𝐴)) = ((𝑃↑𝑘)↑_𝑐𝐴))
38	33, 26, 24	cxpmul2d 27037	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝐴 · 𝑘)) = ((𝑃↑_𝑐𝐴)↑𝑘))
39	28, 37, 38	3eqtr3d 2804	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑃↑𝑘)↑_𝑐𝐴) = ((𝑃↑_𝑐𝐴)↑𝑘))
40	39	sumeq2dv 15869	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → Σ𝑘 ∈ (0...𝑁)((𝑃↑𝑘)↑_𝑐𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑃↑_𝑐𝐴)↑𝑘))
41	8, 22, 40	3eqtrd 2800	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 2145 {crab 3413 class class class wbr 5103 ↦ cmpt 5186 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 0cc0 11200 · cmul 11205 ℕcn 12335 ℕ₀cn0 12606 ...cfz 13639 ↑cexp 14204 Σcsu 15853 ∥ cdvds 16422 ℙcprime 16846 ↑_𝑐ccxp 26883 σ csgm 27423

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ioc 13481 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-fac 14418 df-bc 14447 df-hash 14475 df-shft 15220 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-limsup 15638 df-clim 15655 df-rlim 15656 df-sum 15854 df-ef 16233 df-sin 16235 df-cos 16236 df-pi 16238 df-dvds 16423 df-gcd 16665 df-prm 16847 df-pc 17015 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-haus 23633 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-limc 26186 df-dv 26187 df-log 26884 df-cxp 26885 df-sgm 27429

This theorem is used by: 1sgmprm 27526 1sgm2ppw 27527

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