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Theorem 1sgmprm 27143

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 11070	. . 3 ⊢ 1 ∈ ℂ
2		1nn0 12403	. . 3 ⊢ 1 ∈ ℕ₀
3		sgmppw 27141	. . 3 ⊢ ((1 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 1 ∈ ℕ₀) → (1 σ (𝑃↑1)) = Σ𝑘 ∈ (0...1)((𝑃↑_𝑐1)↑𝑘))
4	1, 2, 3	mp3an13 1454	. 2 ⊢ (𝑃 ∈ ℙ → (1 σ (𝑃↑1)) = Σ𝑘 ∈ (0...1)((𝑃↑_𝑐1)↑𝑘))
5		prmnn 16591	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
6	5	nncnd 12147	. . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℂ)
7	6	exp1d 14054	. . 3 ⊢ (𝑃 ∈ ℙ → (𝑃↑1) = 𝑃)
8	7	oveq2d 7368	. 2 ⊢ (𝑃 ∈ ℙ → (1 σ (𝑃↑1)) = (1 σ 𝑃))
9	6	adantr 480	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → 𝑃 ∈ ℂ)
10	9	cxp1d 26648	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → (𝑃↑_𝑐1) = 𝑃)
11	10	oveq1d 7367	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → ((𝑃↑_𝑐1)↑𝑘) = (𝑃↑𝑘))
12	11	sumeq2dv 15615	. . 3 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)((𝑃↑_𝑐1)↑𝑘) = Σ𝑘 ∈ (0...1)(𝑃↑𝑘))
13		1m1e0 12203	. . . . . . . 8 ⊢ (1 − 1) = 0
14	13	oveq2i 7363	. . . . . . 7 ⊢ (0...(1 − 1)) = (0...0)
15	14	sumeq1i 15610	. . . . . 6 ⊢ Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) = Σ𝑘 ∈ (0...0)(𝑃↑𝑘)
16		0z 12485	. . . . . . . 8 ⊢ 0 ∈ ℤ
17	6	exp0d 14053	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (𝑃↑0) = 1)
18	17, 1	eqeltrdi 2839	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → (𝑃↑0) ∈ ℂ)
19		oveq2 7360	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑃↑𝑘) = (𝑃↑0))
20	19	fsum1 15660	. . . . . . . 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 2766	. . . . . 6 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...0)(𝑃↑𝑘) = 1)
23	15, 22	eqtrid 2778	. . . . 5 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) = 1)
24	23, 7	oveq12d 7370	. . . 4 ⊢ (𝑃 ∈ ℙ → (Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) + (𝑃↑1)) = (1 + 𝑃))
25	2	a1i 11	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 1 ∈ ℕ₀)
26		nn0uz 12780	. . . . . 6 ⊢ ℕ₀ = (ℤ_≥‘0)
27	25, 26	eleqtrdi 2841	. . . . 5 ⊢ (𝑃 ∈ ℙ → 1 ∈ (ℤ_≥‘0))
28		elfznn0 13526	. . . . . 6 ⊢ (𝑘 ∈ (0...1) → 𝑘 ∈ ℕ₀)
29		expcl 13992	. . . . . 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 7360	. . . . 5 ⊢ (𝑘 = 1 → (𝑃↑𝑘) = (𝑃↑1))
32	27, 30, 31	fsumm1 15664	. . . 4 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)(𝑃↑𝑘) = (Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) + (𝑃↑1)))
33		addcom 11305	. . . . 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 2776	. . 3 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)(𝑃↑𝑘) = (𝑃 + 1))
36	12, 35	eqtrd 2766	. 2 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)((𝑃↑_𝑐1)↑𝑘) = (𝑃 + 1))
37	4, 8, 36	3eqtr3d 2774	1 ⊢ (𝑃 ∈ ℙ → (1 σ 𝑃) = (𝑃 + 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ‘cfv 6487 (class class class)co 7352 ℂcc 11010 0cc0 11012 1c1 11013 + caddc 11015 − cmin 11350 ℕ₀cn0 12387 ℤcz 12474 ℤ_≥cuz 12738 ...cfz 13413 ↑cexp 13974 Σcsu 15599 ℙcprime 16588 ↑_𝑐ccxp 26497 σ csgm 27039

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 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-inf2 9537 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089 ax-pre-sup 11090 ax-addf 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-pm 8759 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9838 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-div 11781 df-nn 12132 df-2 12194 df-3 12195 df-4 12196 df-5 12197 df-6 12198 df-7 12199 df-8 12200 df-9 12201 df-n0 12388 df-z 12475 df-dec 12595 df-uz 12739 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-ioo 13255 df-ioc 13256 df-ico 13257 df-icc 13258 df-fz 13414 df-fzo 13561 df-fl 13702 df-mod 13780 df-seq 13915 df-exp 13975 df-fac 14187 df-bc 14216 df-hash 14244 df-shft 14980 df-cj 15012 df-re 15013 df-im 15014 df-sqrt 15148 df-abs 15149 df-limsup 15384 df-clim 15401 df-rlim 15402 df-sum 15600 df-ef 15980 df-sin 15982 df-cos 15983 df-pi 15985 df-dvds 16170 df-gcd 16412 df-prm 16589 df-pc 16755 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17127 df-ress 17148 df-plusg 17180 df-mulr 17181 df-starv 17182 df-sca 17183 df-vsca 17184 df-ip 17185 df-tset 17186 df-ple 17187 df-ds 17189 df-unif 17190 df-hom 17191 df-cco 17192 df-rest 17332 df-topn 17333 df-0g 17351 df-gsum 17352 df-topgen 17353 df-pt 17354 df-prds 17357 df-xrs 17412 df-qtop 17417 df-imas 17418 df-xps 17420 df-mre 17494 df-mrc 17495 df-acs 17497 df-mgm 18554 df-sgrp 18633 df-mnd 18649 df-submnd 18698 df-mulg 18987 df-cntz 19235 df-cmn 19700 df-psmet 21289 df-xmet 21290 df-met 21291 df-bl 21292 df-mopn 21293 df-fbas 21294 df-fg 21295 df-cnfld 21298 df-top 22815 df-topon 22832 df-topsp 22854 df-bases 22867 df-cld 22940 df-ntr 22941 df-cls 22942 df-nei 23019 df-lp 23057 df-perf 23058 df-cn 23148 df-cnp 23149 df-haus 23236 df-tx 23483 df-hmeo 23676 df-fil 23767 df-fm 23859 df-flim 23860 df-flf 23861 df-xms 24241 df-ms 24242 df-tms 24243 df-cncf 24804 df-limc 25800 df-dv 25801 df-log 26498 df-cxp 26499 df-sgm 27045

This theorem is referenced by: perfect1 27172

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