sgmppw - Intuitionistic Logic Explorer

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Theorem sgmppw 15860

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 1024	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ∈ ℂ)
2		simp2 1025	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → 𝑃 ∈ ℙ)
3		prmnn 12807	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	2, 3	syl 14	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → 𝑃 ∈ ℕ)
5		simp3 1026	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
6	4, 5	nnexpcld 11057	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝑃↑𝑁) ∈ ℕ)
7		sgmval 15851	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑃↑𝑁) ∈ ℕ) → (𝐴 σ (𝑃↑𝑁)) = Σ𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} (𝑛↑_𝑐𝐴))
8	1, 6, 7	syl2anc 411	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝐴 σ (𝑃↑𝑁)) = Σ𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} (𝑛↑_𝑐𝐴))
9		oveq1 6057	. . 3 ⊢ (𝑛 = (𝑃↑𝑘) → (𝑛↑_𝑐𝐴) = ((𝑃↑𝑘)↑_𝑐𝐴))
10		0zd 9589	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → 0 ∈ ℤ)
11	5	nn0zd 9698	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℤ)
12	10, 11	fzfigd 10793	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (0...𝑁) ∈ Fin)
13		eqid 2232	. . . . 5 ⊢ (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖)) = (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖))
14	13	dvdsppwf1o 15857	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖)):(0...𝑁)–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)})
15	2, 5, 14	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖)):(0...𝑁)–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)})
16		oveq2 6058	. . . 4 ⊢ (𝑖 = 𝑘 → (𝑃↑𝑖) = (𝑃↑𝑘))
17		simpr 110	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...𝑁))
18	4	adantr 276	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑃 ∈ ℕ)
19		elfznn0 10448	. . . . . 6 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
20	19	adantl 277	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
21	18, 20	nnexpcld 11057	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑𝑘) ∈ ℕ)
22	13, 16, 17, 21	fvmptd3 5771	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖))‘𝑘) = (𝑃↑𝑘))
23		elrabi 2970	. . . . . 6 ⊢ (𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} → 𝑛 ∈ ℕ)
24	23	adantl 277	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)}) → 𝑛 ∈ ℕ)
25	24	nnrpd 10027	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)}) → 𝑛 ∈ ℝ⁺)
26	1	adantr 276	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)}) → 𝐴 ∈ ℂ)
27	25, 26	rpcncxpcld 15792	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)}) → (𝑛↑_𝑐𝐴) ∈ ℂ)
28	9, 12, 15, 22, 27	fsumf1o 12076	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → Σ𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} (𝑛↑_𝑐𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑃↑𝑘)↑_𝑐𝐴))
29	20	nn0cnd 9555	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℂ)
30	1	adantr 276	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ)
31	29, 30	mulcomd 8295	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑘 · 𝐴) = (𝐴 · 𝑘))
32	31	oveq2d 6066	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝑘 · 𝐴)) = (𝑃↑_𝑐(𝐴 · 𝑘)))
33	18	nnrpd 10027	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑃 ∈ ℝ⁺)
34	20	nn0red 9554	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℝ)
35	33, 34, 30	cxpmuld 15802	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝑘 · 𝐴)) = ((𝑃↑_𝑐𝑘)↑_𝑐𝐴))
36	20	nn0zd 9698	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℤ)
37		cxpexpnn 15761	. . . . . . 7 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℤ) → (𝑃↑_𝑐𝑘) = (𝑃↑𝑘))
38	18, 36, 37	syl2anc 411	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐𝑘) = (𝑃↑𝑘))
39	38	oveq1d 6065	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑃↑_𝑐𝑘)↑_𝑐𝐴) = ((𝑃↑𝑘)↑_𝑐𝐴))
40	35, 39	eqtrd 2265	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝑘 · 𝐴)) = ((𝑃↑𝑘)↑_𝑐𝐴))
41	33, 30, 20	rpcxpmul2d 15797	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝐴 · 𝑘)) = ((𝑃↑_𝑐𝐴)↑𝑘))
42	32, 40, 41	3eqtr3d 2273	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑃↑𝑘)↑_𝑐𝐴) = ((𝑃↑_𝑐𝐴)↑𝑘))
43	42	sumeq2dv 12053	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → Σ𝑘 ∈ (0...𝑁)((𝑃↑𝑘)↑_𝑐𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑃↑_𝑐𝐴)↑𝑘))
44	8, 28, 43	3eqtrd 2269	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝐴 σ (𝑃↑𝑁)) = Σ𝑘 ∈ (0...𝑁)((𝑃↑_𝑐𝐴)↑𝑘))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 1005 = wceq 1398 ∈ wcel 2203 {crab 2524 class class class wbr 4109 ↦ cmpt 4171 –1-1-onto→wf1o 5351 (class class class)co 6050 ℂcc 8125 0cc0 8127 · cmul 8132 ℕcn 9237 ℕ₀cn0 9496 ℤcz 9577 ...cfz 10342 ↑cexp 10900 Σcsu 12038 ∥ cdvds 12473 ℙcprime 12804 ↑_𝑐ccxp 15722 σ csgm 15849

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245 ax-arch 8246 ax-caucvg 8247 ax-pre-suploc 8248 ax-addf 8249 ax-mulf 8250

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-disj 4086 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-isom 5361 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-of 6266 df-1st 6334 df-2nd 6335 df-recs 6536 df-irdg 6601 df-frec 6622 df-1o 6647 df-2o 6648 df-oadd 6651 df-er 6767 df-map 6884 df-pm 6885 df-en 6976 df-dom 6977 df-fin 6978 df-sup 7275 df-inf 7276 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-2 9296 df-3 9297 df-4 9298 df-n0 9497 df-xnn0 9564 df-z 9578 df-uz 9854 df-q 9952 df-rp 9987 df-xneg 10105 df-xadd 10106 df-ioo 10225 df-ico 10227 df-icc 10228 df-fz 10343 df-fzo 10477 df-fl 10630 df-mod 10685 df-seqfrec 10810 df-exp 10901 df-fac 11088 df-bc 11110 df-ihash 11139 df-shft 11500 df-cj 11527 df-re 11528 df-im 11529 df-rsqrt 11683 df-abs 11684 df-clim 11964 df-sumdc 12039 df-ef 12334 df-e 12335 df-dvds 12474 df-gcd 12650 df-prm 12805 df-pc 12983 df-rest 13454 df-topgen 13473 df-psmet 14691 df-xmet 14692 df-met 14693 df-bl 14694 df-mopn 14695 df-top 14863 df-topon 14876 df-bases 14908 df-ntr 14961 df-cn 15053 df-cnp 15054 df-tx 15118 df-cncf 15436 df-limced 15521 df-dvap 15522 df-relog 15723 df-rpcxp 15724 df-sgm 15850

This theorem is referenced by: 1sgmprm 15862 1sgm2ppw 15863

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