![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 1sgmprm | Structured version Visualization version GIF version |
Description: The sum of divisors for a prime is 𝑃 + 1 because the only divisors are 1 and 𝑃. (Contributed by Mario Carneiro, 17-May-2016.) |
Ref | Expression |
---|---|
1sgmprm | ⊢ (𝑃 ∈ ℙ → (1 σ 𝑃) = (𝑃 + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 10618 | . . 3 ⊢ 1 ∈ ℂ | |
2 | 1nn0 11935 | . . 3 ⊢ 1 ∈ ℕ0 | |
3 | sgmppw 25865 | . . 3 ⊢ ((1 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 1 ∈ ℕ0) → (1 σ (𝑃↑1)) = Σ𝑘 ∈ (0...1)((𝑃↑𝑐1)↑𝑘)) | |
4 | 1, 2, 3 | mp3an13 1450 | . 2 ⊢ (𝑃 ∈ ℙ → (1 σ (𝑃↑1)) = Σ𝑘 ∈ (0...1)((𝑃↑𝑐1)↑𝑘)) |
5 | prmnn 16055 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
6 | 5 | nncnd 11675 | . . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℂ) |
7 | 6 | exp1d 13540 | . . 3 ⊢ (𝑃 ∈ ℙ → (𝑃↑1) = 𝑃) |
8 | 7 | oveq2d 7159 | . 2 ⊢ (𝑃 ∈ ℙ → (1 σ (𝑃↑1)) = (1 σ 𝑃)) |
9 | 6 | adantr 485 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → 𝑃 ∈ ℂ) |
10 | 9 | cxp1d 25381 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → (𝑃↑𝑐1) = 𝑃) |
11 | 10 | oveq1d 7158 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → ((𝑃↑𝑐1)↑𝑘) = (𝑃↑𝑘)) |
12 | 11 | sumeq2dv 15093 | . . 3 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)((𝑃↑𝑐1)↑𝑘) = Σ𝑘 ∈ (0...1)(𝑃↑𝑘)) |
13 | 1m1e0 11731 | . . . . . . . 8 ⊢ (1 − 1) = 0 | |
14 | 13 | oveq2i 7154 | . . . . . . 7 ⊢ (0...(1 − 1)) = (0...0) |
15 | 14 | sumeq1i 15088 | . . . . . 6 ⊢ Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) = Σ𝑘 ∈ (0...0)(𝑃↑𝑘) |
16 | 0z 12016 | . . . . . . . 8 ⊢ 0 ∈ ℤ | |
17 | 6 | exp0d 13539 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (𝑃↑0) = 1) |
18 | 17, 1 | eqeltrdi 2859 | . . . . . . . 8 ⊢ (𝑃 ∈ ℙ → (𝑃↑0) ∈ ℂ) |
19 | oveq2 7151 | . . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑃↑𝑘) = (𝑃↑0)) | |
20 | 19 | fsum1 15135 | . . . . . . . 8 ⊢ ((0 ∈ ℤ ∧ (𝑃↑0) ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝑃↑𝑘) = (𝑃↑0)) |
21 | 16, 18, 20 | sylancr 591 | . . . . . . 7 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...0)(𝑃↑𝑘) = (𝑃↑0)) |
22 | 21, 17 | eqtrd 2794 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...0)(𝑃↑𝑘) = 1) |
23 | 15, 22 | syl5eq 2806 | . . . . 5 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) = 1) |
24 | 23, 7 | oveq12d 7161 | . . . 4 ⊢ (𝑃 ∈ ℙ → (Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) + (𝑃↑1)) = (1 + 𝑃)) |
25 | 2 | a1i 11 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 1 ∈ ℕ0) |
26 | nn0uz 12305 | . . . . . 6 ⊢ ℕ0 = (ℤ≥‘0) | |
27 | 25, 26 | eleqtrdi 2861 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 1 ∈ (ℤ≥‘0)) |
28 | elfznn0 13034 | . . . . . 6 ⊢ (𝑘 ∈ (0...1) → 𝑘 ∈ ℕ0) | |
29 | expcl 13482 | . . . . . 6 ⊢ ((𝑃 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℂ) | |
30 | 6, 28, 29 | syl2an 599 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → (𝑃↑𝑘) ∈ ℂ) |
31 | oveq2 7151 | . . . . 5 ⊢ (𝑘 = 1 → (𝑃↑𝑘) = (𝑃↑1)) | |
32 | 27, 30, 31 | fsumm1 15139 | . . . 4 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)(𝑃↑𝑘) = (Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) + (𝑃↑1))) |
33 | addcom 10849 | . . . . 5 ⊢ ((𝑃 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑃 + 1) = (1 + 𝑃)) | |
34 | 6, 1, 33 | sylancl 590 | . . . 4 ⊢ (𝑃 ∈ ℙ → (𝑃 + 1) = (1 + 𝑃)) |
35 | 24, 32, 34 | 3eqtr4d 2804 | . . 3 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)(𝑃↑𝑘) = (𝑃 + 1)) |
36 | 12, 35 | eqtrd 2794 | . 2 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)((𝑃↑𝑐1)↑𝑘) = (𝑃 + 1)) |
37 | 4, 8, 36 | 3eqtr3d 2802 | 1 ⊢ (𝑃 ∈ ℙ → (1 σ 𝑃) = (𝑃 + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 ‘cfv 6328 (class class class)co 7143 ℂcc 10558 0cc0 10560 1c1 10561 + caddc 10563 − cmin 10893 ℕ0cn0 11919 ℤcz 12005 ℤ≥cuz 12267 ...cfz 12924 ↑cexp 13464 Σcsu 15075 ℙcprime 16052 ↑𝑐ccxp 25231 σ csgm 25765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-inf2 9122 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 ax-pre-sup 10638 ax-addf 10639 ax-mulf 10640 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-iin 4879 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-se 5477 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-isom 6337 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-of 7398 df-om 7573 df-1st 7686 df-2nd 7687 df-supp 7829 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8473 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-fsupp 8852 df-fi 8893 df-sup 8924 df-inf 8925 df-oi 8992 df-card 9386 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-div 11321 df-nn 11660 df-2 11722 df-3 11723 df-4 11724 df-5 11725 df-6 11726 df-7 11727 df-8 11728 df-9 11729 df-n0 11920 df-z 12006 df-dec 12123 df-uz 12268 df-q 12374 df-rp 12416 df-xneg 12533 df-xadd 12534 df-xmul 12535 df-ioo 12768 df-ioc 12769 df-ico 12770 df-icc 12771 df-fz 12925 df-fzo 13068 df-fl 13196 df-mod 13272 df-seq 13404 df-exp 13465 df-fac 13669 df-bc 13698 df-hash 13726 df-shft 14459 df-cj 14491 df-re 14492 df-im 14493 df-sqrt 14627 df-abs 14628 df-limsup 14861 df-clim 14878 df-rlim 14879 df-sum 15076 df-ef 15454 df-sin 15456 df-cos 15457 df-pi 15459 df-dvds 15641 df-gcd 15879 df-prm 16053 df-pc 16214 df-struct 16528 df-ndx 16529 df-slot 16530 df-base 16532 df-sets 16533 df-ress 16534 df-plusg 16621 df-mulr 16622 df-starv 16623 df-sca 16624 df-vsca 16625 df-ip 16626 df-tset 16627 df-ple 16628 df-ds 16630 df-unif 16631 df-hom 16632 df-cco 16633 df-rest 16739 df-topn 16740 df-0g 16758 df-gsum 16759 df-topgen 16760 df-pt 16761 df-prds 16764 df-xrs 16818 df-qtop 16823 df-imas 16824 df-xps 16826 df-mre 16900 df-mrc 16901 df-acs 16903 df-mgm 17903 df-sgrp 17952 df-mnd 17963 df-submnd 18008 df-mulg 18277 df-cntz 18499 df-cmn 18960 df-psmet 20143 df-xmet 20144 df-met 20145 df-bl 20146 df-mopn 20147 df-fbas 20148 df-fg 20149 df-cnfld 20152 df-top 21579 df-topon 21596 df-topsp 21618 df-bases 21631 df-cld 21704 df-ntr 21705 df-cls 21706 df-nei 21783 df-lp 21821 df-perf 21822 df-cn 21912 df-cnp 21913 df-haus 22000 df-tx 22247 df-hmeo 22440 df-fil 22531 df-fm 22623 df-flim 22624 df-flf 22625 df-xms 23007 df-ms 23008 df-tms 23009 df-cncf 23564 df-limc 24550 df-dv 24551 df-log 25232 df-cxp 25233 df-sgm 25771 |
This theorem is referenced by: perfect1 25896 |
Copyright terms: Public domain | W3C validator |