1sgmprm - Metamath Proof Explorer

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Theorem 1sgmprm 26418

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.)

**Assertion**
Ref	Expression
1sgmprm	⊢ (𝑃 ∈ ℙ → (1 σ 𝑃) = (𝑃 + 1))

Proof of Theorem 1sgmprm Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-1cn 10999	. . 3 ⊢ 1 ∈ ℂ
2		1nn0 12319	. . 3 ⊢ 1 ∈ ℕ₀
3		sgmppw 26416	. . 3 ⊢ ((1 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 1 ∈ ℕ₀) → (1 σ (𝑃↑1)) = Σ𝑘 ∈ (0...1)((𝑃↑_𝑐1)↑𝑘))
4	1, 2, 3	mp3an13 1451	. 2 ⊢ (𝑃 ∈ ℙ → (1 σ (𝑃↑1)) = Σ𝑘 ∈ (0...1)((𝑃↑_𝑐1)↑𝑘))
5		prmnn 16446	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
6	5	nncnd 12059	. . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℂ)
7	6	exp1d 13929	. . 3 ⊢ (𝑃 ∈ ℙ → (𝑃↑1) = 𝑃)
8	7	oveq2d 7329	. 2 ⊢ (𝑃 ∈ ℙ → (1 σ (𝑃↑1)) = (1 σ 𝑃))
9	6	adantr 481	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → 𝑃 ∈ ℂ)
10	9	cxp1d 25932	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → (𝑃↑_𝑐1) = 𝑃)
11	10	oveq1d 7328	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → ((𝑃↑_𝑐1)↑𝑘) = (𝑃↑𝑘))
12	11	sumeq2dv 15484	. . 3 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)((𝑃↑_𝑐1)↑𝑘) = Σ𝑘 ∈ (0...1)(𝑃↑𝑘))
13		1m1e0 12115	. . . . . . . 8 ⊢ (1 − 1) = 0
14	13	oveq2i 7324	. . . . . . 7 ⊢ (0...(1 − 1)) = (0...0)
15	14	sumeq1i 15479	. . . . . 6 ⊢ Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) = Σ𝑘 ∈ (0...0)(𝑃↑𝑘)
16		0z 12400	. . . . . . . 8 ⊢ 0 ∈ ℤ
17	6	exp0d 13928	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (𝑃↑0) = 1)
18	17, 1	eqeltrdi 2846	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → (𝑃↑0) ∈ ℂ)
19		oveq2 7321	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑃↑𝑘) = (𝑃↑0))
20	19	fsum1 15528	. . . . . . . 8 ⊢ ((0 ∈ ℤ ∧ (𝑃↑0) ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝑃↑𝑘) = (𝑃↑0))
21	16, 18, 20	sylancr 587	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...0)(𝑃↑𝑘) = (𝑃↑0))
22	21, 17	eqtrd 2777	. . . . . 6 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...0)(𝑃↑𝑘) = 1)
23	15, 22	eqtrid 2789	. . . . 5 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) = 1)
24	23, 7	oveq12d 7331	. . . 4 ⊢ (𝑃 ∈ ℙ → (Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) + (𝑃↑1)) = (1 + 𝑃))
25	2	a1i 11	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 1 ∈ ℕ₀)
26		nn0uz 12690	. . . . . 6 ⊢ ℕ₀ = (ℤ_≥‘0)
27	25, 26	eleqtrdi 2848	. . . . 5 ⊢ (𝑃 ∈ ℙ → 1 ∈ (ℤ_≥‘0))
28		elfznn0 13419	. . . . . 6 ⊢ (𝑘 ∈ (0...1) → 𝑘 ∈ ℕ₀)
29		expcl 13870	. . . . . 6 ⊢ ((𝑃 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑃↑𝑘) ∈ ℂ)
30	6, 28, 29	syl2an 596	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → (𝑃↑𝑘) ∈ ℂ)
31		oveq2 7321	. . . . 5 ⊢ (𝑘 = 1 → (𝑃↑𝑘) = (𝑃↑1))
32	27, 30, 31	fsumm1 15532	. . . 4 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)(𝑃↑𝑘) = (Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) + (𝑃↑1)))
33		addcom 11231	. . . . 5 ⊢ ((𝑃 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑃 + 1) = (1 + 𝑃))
34	6, 1, 33	sylancl 586	. . . 4 ⊢ (𝑃 ∈ ℙ → (𝑃 + 1) = (1 + 𝑃))
35	24, 32, 34	3eqtr4d 2787	. . 3 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)(𝑃↑𝑘) = (𝑃 + 1))
36	12, 35	eqtrd 2777	. 2 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)((𝑃↑_𝑐1)↑𝑘) = (𝑃 + 1))
37	4, 8, 36	3eqtr3d 2785	1 ⊢ (𝑃 ∈ ℙ → (1 σ 𝑃) = (𝑃 + 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ‘cfv 6463 (class class class)co 7313 ℂcc 10939 0cc0 10941 1c1 10942 + caddc 10944 − cmin 11275 ℕ₀cn0 12303 ℤcz 12389 ℤ_≥cuz 12652 ...cfz 13309 ↑cexp 13852 Σcsu 15466 ℙcprime 16443 ↑_𝑐ccxp 25782 σ csgm 26316

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-rep 5222 ax-sep 5236 ax-nul 5243 ax-pow 5301 ax-pr 5365 ax-un 7626 ax-inf2 9467 ax-cnex 10997 ax-resscn 10998 ax-1cn 10999 ax-icn 11000 ax-addcl 11001 ax-addrcl 11002 ax-mulcl 11003 ax-mulrcl 11004 ax-mulcom 11005 ax-addass 11006 ax-mulass 11007 ax-distr 11008 ax-i2m1 11009 ax-1ne0 11010 ax-1rid 11011 ax-rnegex 11012 ax-rrecex 11013 ax-cnre 11014 ax-pre-lttri 11015 ax-pre-lttrn 11016 ax-pre-ltadd 11017 ax-pre-mulgt0 11018 ax-pre-sup 11019 ax-addf 11020 ax-mulf 11021

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4470 df-pw 4545 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4849 df-int 4891 df-iun 4937 df-iin 4938 df-br 5086 df-opab 5148 df-mpt 5169 df-tr 5203 df-id 5505 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5560 df-se 5561 df-we 5562 df-xp 5611 df-rel 5612 df-cnv 5613 df-co 5614 df-dm 5615 df-rn 5616 df-res 5617 df-ima 5618 df-pred 6222 df-ord 6289 df-on 6290 df-lim 6291 df-suc 6292 df-iota 6415 df-fun 6465 df-fn 6466 df-f 6467 df-f1 6468 df-fo 6469 df-f1o 6470 df-fv 6471 df-isom 6472 df-riota 7270 df-ov 7316 df-oprab 7317 df-mpo 7318 df-of 7571 df-om 7756 df-1st 7874 df-2nd 7875 df-supp 8023 df-frecs 8142 df-wrecs 8173 df-recs 8247 df-rdg 8286 df-1o 8342 df-2o 8343 df-er 8544 df-map 8663 df-pm 8664 df-ixp 8732 df-en 8780 df-dom 8781 df-sdom 8782 df-fin 8783 df-fsupp 9197 df-fi 9238 df-sup 9269 df-inf 9270 df-oi 9337 df-card 9765 df-pnf 11081 df-mnf 11082 df-xr 11083 df-ltxr 11084 df-le 11085 df-sub 11277 df-neg 11278 df-div 11703 df-nn 12044 df-2 12106 df-3 12107 df-4 12108 df-5 12109 df-6 12110 df-7 12111 df-8 12112 df-9 12113 df-n0 12304 df-z 12390 df-dec 12508 df-uz 12653 df-q 12759 df-rp 12801 df-xneg 12918 df-xadd 12919 df-xmul 12920 df-ioo 13153 df-ioc 13154 df-ico 13155 df-icc 13156 df-fz 13310 df-fzo 13453 df-fl 13582 df-mod 13660 df-seq 13792 df-exp 13853 df-fac 14058 df-bc 14087 df-hash 14115 df-shft 14847 df-cj 14879 df-re 14880 df-im 14881 df-sqrt 15015 df-abs 15016 df-limsup 15249 df-clim 15266 df-rlim 15267 df-sum 15467 df-ef 15846 df-sin 15848 df-cos 15849 df-pi 15851 df-dvds 16033 df-gcd 16271 df-prm 16444 df-pc 16605 df-struct 16915 df-sets 16932 df-slot 16950 df-ndx 16962 df-base 16980 df-ress 17009 df-plusg 17042 df-mulr 17043 df-starv 17044 df-sca 17045 df-vsca 17046 df-ip 17047 df-tset 17048 df-ple 17049 df-ds 17051 df-unif 17052 df-hom 17053 df-cco 17054 df-rest 17200 df-topn 17201 df-0g 17219 df-gsum 17220 df-topgen 17221 df-pt 17222 df-prds 17225 df-xrs 17280 df-qtop 17285 df-imas 17286 df-xps 17288 df-mre 17362 df-mrc 17363 df-acs 17365 df-mgm 18393 df-sgrp 18442 df-mnd 18453 df-submnd 18498 df-mulg 18768 df-cntz 18990 df-cmn 19455 df-psmet 20660 df-xmet 20661 df-met 20662 df-bl 20663 df-mopn 20664 df-fbas 20665 df-fg 20666 df-cnfld 20669 df-top 22114 df-topon 22131 df-topsp 22153 df-bases 22167 df-cld 22241 df-ntr 22242 df-cls 22243 df-nei 22320 df-lp 22358 df-perf 22359 df-cn 22449 df-cnp 22450 df-haus 22537 df-tx 22784 df-hmeo 22977 df-fil 23068 df-fm 23160 df-flim 23161 df-flf 23162 df-xms 23544 df-ms 23545 df-tms 23546 df-cncf 24112 df-limc 25101 df-dv 25102 df-log 25783 df-cxp 25784 df-sgm 26322

This theorem is referenced by: perfect1 26447

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