sgmppw - Intuitionistic Logic Explorer

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Theorem sgmppw 15877

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 12811	. . . . 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 11061	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝑃↑𝑁) ∈ ℕ)
7		sgmval 15868	. . 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 6059	. . 3 ⊢ (𝑛 = (𝑃↑𝑘) → (𝑛↑_𝑐𝐴) = ((𝑃↑𝑘)↑_𝑐𝐴))
10		0zd 9591	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → 0 ∈ ℤ)
11	5	nn0zd 9701	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℤ)
12	10, 11	fzfigd 10797	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (0...𝑁) ∈ Fin)
13		eqid 2234	. . . . 5 ⊢ (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖)) = (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖))
14	13	dvdsppwf1o 15874	. . . 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 6060	. . . 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 10452	. . . . . 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 11061	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑𝑘) ∈ ℕ)
22	13, 16, 17, 21	fvmptd3 5773	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖))‘𝑘) = (𝑃↑𝑘))
23		elrabi 2972	. . . . . 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 10030	. . . 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 15809	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)}) → (𝑛↑_𝑐𝐴) ∈ ℂ)
28	9, 12, 15, 22, 27	fsumf1o 12080	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → Σ𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} (𝑛↑_𝑐𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑃↑𝑘)↑_𝑐𝐴))
29	20	nn0cnd 9557	. . . . . 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 8297	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑘 · 𝐴) = (𝐴 · 𝑘))
32	31	oveq2d 6068	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝑘 · 𝐴)) = (𝑃↑_𝑐(𝐴 · 𝑘)))
33	18	nnrpd 10030	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑃 ∈ ℝ⁺)
34	20	nn0red 9556	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℝ)
35	33, 34, 30	cxpmuld 15819	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝑘 · 𝐴)) = ((𝑃↑_𝑐𝑘)↑_𝑐𝐴))
36	20	nn0zd 9701	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℤ)
37		cxpexpnn 15778	. . . . . . 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 6067	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑃↑_𝑐𝑘)↑_𝑐𝐴) = ((𝑃↑𝑘)↑_𝑐𝐴))
40	35, 39	eqtrd 2267	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝑘 · 𝐴)) = ((𝑃↑𝑘)↑_𝑐𝐴))
41	33, 30, 20	rpcxpmul2d 15814	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑_𝑐(𝐴 · 𝑘)) = ((𝑃↑_𝑐𝐴)↑𝑘))
42	32, 40, 41	3eqtr3d 2275	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑃↑𝑘)↑_𝑐𝐴) = ((𝑃↑_𝑐𝐴)↑𝑘))
43	42	sumeq2dv 12057	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → Σ𝑘 ∈ (0...𝑁)((𝑃↑𝑘)↑_𝑐𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑃↑_𝑐𝐴)↑𝑘))
44	8, 28, 43	3eqtrd 2271	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 2205 {crab 2526 class class class wbr 4111 ↦ cmpt 4173 –1-1-onto→wf1o 5353 (class class class)co 6052 ℂcc 8127 0cc0 8129 · cmul 8134 ℕcn 9239 ℕ₀cn0 9498 ℤcz 9579 ...cfz 10345 ↑cexp 10904 Σcsu 12042 ∥ cdvds 12477 ℙcprime 12808 ↑_𝑐ccxp 15739 σ csgm 15866

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 2207 ax-14 2208 ax-ext 2216 ax-coll 4227 ax-sep 4230 ax-nul 4238 ax-pow 4289 ax-pr 4324 ax-un 4556 ax-setind 4661 ax-iinf 4712 ax-cnex 8220 ax-resscn 8221 ax-1cn 8222 ax-1re 8223 ax-icn 8224 ax-addcl 8225 ax-addrcl 8226 ax-mulcl 8227 ax-mulrcl 8228 ax-addcom 8229 ax-mulcom 8230 ax-addass 8231 ax-mulass 8232 ax-distr 8233 ax-i2m1 8234 ax-0lt1 8235 ax-1rid 8236 ax-0id 8237 ax-rnegex 8238 ax-precex 8239 ax-cnre 8240 ax-pre-ltirr 8241 ax-pre-ltwlin 8242 ax-pre-lttrn 8243 ax-pre-apti 8244 ax-pre-ltadd 8245 ax-pre-mulgt0 8246 ax-pre-mulext 8247 ax-arch 8248 ax-caucvg 8249 ax-pre-suploc 8250 ax-addf 8251 ax-mulf 8252

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 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3045 df-csb 3141 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-nul 3511 df-if 3623 df-pw 3673 df-sn 3697 df-pr 3698 df-op 3700 df-uni 3917 df-int 3952 df-iun 3995 df-disj 4088 df-br 4112 df-opab 4174 df-mpt 4175 df-tr 4211 df-id 4416 df-po 4419 df-iso 4420 df-iord 4489 df-on 4491 df-ilim 4492 df-suc 4494 df-iom 4715 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-rn 4762 df-res 4763 df-ima 4764 df-iota 5314 df-fun 5356 df-fn 5357 df-f 5358 df-f1 5359 df-fo 5360 df-f1o 5361 df-fv 5362 df-isom 5363 df-riota 6005 df-ov 6055 df-oprab 6056 df-mpo 6057 df-of 6268 df-1st 6336 df-2nd 6337 df-recs 6538 df-irdg 6603 df-frec 6624 df-1o 6649 df-2o 6650 df-oadd 6653 df-er 6769 df-map 6886 df-pm 6887 df-en 6978 df-dom 6979 df-fin 6980 df-sup 7277 df-inf 7278 df-pnf 8312 df-mnf 8313 df-xr 8314 df-ltxr 8315 df-le 8316 df-sub 8448 df-neg 8449 df-reap 8851 df-ap 8858 df-div 8949 df-inn 9240 df-2 9298 df-3 9299 df-4 9300 df-n0 9499 df-xnn0 9566 df-z 9580 df-uz 9857 df-q 9955 df-rp 9990 df-xneg 10108 df-xadd 10109 df-ioo 10228 df-ico 10230 df-icc 10231 df-fz 10346 df-fzo 10481 df-fl 10634 df-mod 10689 df-seqfrec 10814 df-exp 10905 df-fac 11092 df-bc 11114 df-ihash 11143 df-shft 11504 df-cj 11531 df-re 11532 df-im 11533 df-rsqrt 11687 df-abs 11688 df-clim 11968 df-sumdc 12043 df-ef 12338 df-e 12339 df-dvds 12478 df-gcd 12654 df-prm 12809 df-pc 12987 df-rest 13471 df-topgen 13490 df-psmet 14708 df-xmet 14709 df-met 14710 df-bl 14711 df-mopn 14712 df-top 14880 df-topon 14893 df-bases 14925 df-ntr 14978 df-cn 15070 df-cnp 15071 df-tx 15135 df-cncf 15453 df-limced 15538 df-dvap 15539 df-relog 15740 df-rpcxp 15741 df-sgm 15867

This theorem is referenced by: 1sgmprm 15879 1sgm2ppw 15880

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