Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sgmppw | Structured version Visualization version GIF version |
Description: The value of the divisor function at a prime power. (Contributed by Mario Carneiro, 17-May-2016.) |
Ref | Expression |
---|---|
sgmppw | ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝐴 σ (𝑃↑𝑁)) = Σ𝑘 ∈ (0...𝑁)((𝑃↑𝑐𝐴)↑𝑘)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1128 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → 𝐴 ∈ ℂ) | |
2 | simp2 1129 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → 𝑃 ∈ ℙ) | |
3 | prmnn 16006 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → 𝑃 ∈ ℕ) |
5 | simp3 1130 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
6 | 4, 5 | nnexpcld 13594 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝑃↑𝑁) ∈ ℕ) |
7 | sgmval 25646 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑃↑𝑁) ∈ ℕ) → (𝐴 σ (𝑃↑𝑁)) = Σ𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} (𝑛↑𝑐𝐴)) | |
8 | 1, 6, 7 | syl2anc 584 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝐴 σ (𝑃↑𝑁)) = Σ𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} (𝑛↑𝑐𝐴)) |
9 | oveq1 7152 | . . 3 ⊢ (𝑛 = (𝑃↑𝑘) → (𝑛↑𝑐𝐴) = ((𝑃↑𝑘)↑𝑐𝐴)) | |
10 | fzfid 13329 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (0...𝑁) ∈ Fin) | |
11 | eqid 2818 | . . . . 5 ⊢ (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖)) = (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖)) | |
12 | 11 | dvdsppwf1o 25690 | . . . 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 584 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖)):(0...𝑁)–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)}) |
14 | oveq2 7153 | . . . . 5 ⊢ (𝑖 = 𝑘 → (𝑃↑𝑖) = (𝑃↑𝑘)) | |
15 | ovex 7178 | . . . . 5 ⊢ (𝑃↑𝑘) ∈ V | |
16 | 14, 11, 15 | fvmpt 6761 | . . . 4 ⊢ (𝑘 ∈ (0...𝑁) → ((𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖))‘𝑘) = (𝑃↑𝑘)) |
17 | 16 | adantl 482 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑖 ∈ (0...𝑁) ↦ (𝑃↑𝑖))‘𝑘) = (𝑃↑𝑘)) |
18 | elrabi 3672 | . . . . 5 ⊢ (𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} → 𝑛 ∈ ℕ) | |
19 | 18 | nncnd 11642 | . . . 4 ⊢ (𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} → 𝑛 ∈ ℂ) |
20 | cxpcl 25184 | . . . 4 ⊢ ((𝑛 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝑛↑𝑐𝐴) ∈ ℂ) | |
21 | 19, 1, 20 | syl2anr 596 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) ∧ 𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)}) → (𝑛↑𝑐𝐴) ∈ ℂ) |
22 | 9, 10, 13, 17, 21 | fsumf1o 15068 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → Σ𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝑁)} (𝑛↑𝑐𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑃↑𝑘)↑𝑐𝐴)) |
23 | elfznn0 12988 | . . . . . . . 8 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) | |
24 | 23 | adantl 482 | . . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0) |
25 | 24 | nn0cnd 11945 | . . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℂ) |
26 | 1 | adantr 481 | . . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ) |
27 | 25, 26 | mulcomd 10650 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → (𝑘 · 𝐴) = (𝐴 · 𝑘)) |
28 | 27 | oveq2d 7161 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑𝑐(𝑘 · 𝐴)) = (𝑃↑𝑐(𝐴 · 𝑘))) |
29 | 4 | adantr 481 | . . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑃 ∈ ℕ) |
30 | 29 | nnrpd 12417 | . . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑃 ∈ ℝ+) |
31 | 24 | nn0red 11944 | . . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℝ) |
32 | 30, 31, 26 | cxpmuld 25246 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑𝑐(𝑘 · 𝐴)) = ((𝑃↑𝑐𝑘)↑𝑐𝐴)) |
33 | 29 | nncnd 11642 | . . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑃 ∈ ℂ) |
34 | cxpexp 25178 | . . . . . . 7 ⊢ ((𝑃 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑐𝑘) = (𝑃↑𝑘)) | |
35 | 33, 24, 34 | syl2anc 584 | . . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑𝑐𝑘) = (𝑃↑𝑘)) |
36 | 35 | oveq1d 7160 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑃↑𝑐𝑘)↑𝑐𝐴) = ((𝑃↑𝑘)↑𝑐𝐴)) |
37 | 32, 36 | eqtrd 2853 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑𝑐(𝑘 · 𝐴)) = ((𝑃↑𝑘)↑𝑐𝐴)) |
38 | 33, 26, 24 | cxpmul2d 25219 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → (𝑃↑𝑐(𝐴 · 𝑘)) = ((𝑃↑𝑐𝐴)↑𝑘)) |
39 | 28, 37, 38 | 3eqtr3d 2861 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑃↑𝑘)↑𝑐𝐴) = ((𝑃↑𝑐𝐴)↑𝑘)) |
40 | 39 | sumeq2dv 15048 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑁)((𝑃↑𝑘)↑𝑐𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑃↑𝑐𝐴)↑𝑘)) |
41 | 8, 22, 40 | 3eqtrd 2857 | 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 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 {crab 3139 class class class wbr 5057 ↦ cmpt 5137 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 · cmul 10530 ℕcn 11626 ℕ0cn0 11885 ...cfz 12880 ↑cexp 13417 Σcsu 15030 ∥ cdvds 15595 ℙcprime 16003 ↑𝑐ccxp 25066 σ csgm 25600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-fac 13622 df-bc 13651 df-hash 13679 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-ef 15409 df-sin 15411 df-cos 15412 df-pi 15414 df-dvds 15596 df-gcd 15832 df-prm 16004 df-pc 16162 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-lp 21672 df-perf 21673 df-cn 21763 df-cnp 21764 df-haus 21851 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-xms 22857 df-ms 22858 df-tms 22859 df-cncf 23413 df-limc 24391 df-dv 24392 df-log 25067 df-cxp 25068 df-sgm 25606 |
This theorem is referenced by: 1sgmprm 25702 1sgm2ppw 25703 |
Copyright terms: Public domain | W3C validator |