1sgmprm - Intuitionistic Logic Explorer

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Theorem 1sgmprm 16091

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 8266	. . 3 ⊢ 1 ∈ ℂ
2		1nn0 9562	. . 3 ⊢ 1 ∈ ℕ₀
3		sgmppw 16089	. . 3 ⊢ ((1 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 1 ∈ ℕ₀) → (1 σ (𝑃↑1)) = Σ𝑘 ∈ (0...1)((𝑃↑_𝑐1)↑𝑘))
4	1, 2, 3	mp3an13 1369	. 2 ⊢ (𝑃 ∈ ℙ → (1 σ (𝑃↑1)) = Σ𝑘 ∈ (0...1)((𝑃↑_𝑐1)↑𝑘))
5		prmnn 12871	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
6	5	nncnd 9301	. . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℂ)
7	6	exp1d 11089	. . 3 ⊢ (𝑃 ∈ ℙ → (𝑃↑1) = 𝑃)
8	7	oveq2d 6095	. 2 ⊢ (𝑃 ∈ ℙ → (1 σ (𝑃↑1)) = (1 σ 𝑃))
9	5	adantr 276	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → 𝑃 ∈ ℕ)
10	9	nnrpd 10078	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → 𝑃 ∈ ℝ⁺)
11	10	rpcxp1d 16010	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → (𝑃↑_𝑐1) = 𝑃)
12	11	oveq1d 6094	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → ((𝑃↑_𝑐1)↑𝑘) = (𝑃↑𝑘))
13	12	sumeq2dv 12117	. . 3 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)((𝑃↑_𝑐1)↑𝑘) = Σ𝑘 ∈ (0...1)(𝑃↑𝑘))
14		1m1e0 9356	. . . . . . . 8 ⊢ (1 − 1) = 0
15	14	oveq2i 6090	. . . . . . 7 ⊢ (0...(1 − 1)) = (0...0)
16	15	sumeq1i 12112	. . . . . 6 ⊢ Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) = Σ𝑘 ∈ (0...0)(𝑃↑𝑘)
17		0z 9638	. . . . . . . 8 ⊢ 0 ∈ ℤ
18	6	exp0d 11088	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (𝑃↑0) = 1)
19	18, 1	eqeltrdi 2329	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → (𝑃↑0) ∈ ℂ)
20		oveq2 6087	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑃↑𝑘) = (𝑃↑0))
21	20	fsum1 12162	. . . . . . . 8 ⊢ ((0 ∈ ℤ ∧ (𝑃↑0) ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝑃↑𝑘) = (𝑃↑0))
22	17, 19, 21	sylancr 418	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...0)(𝑃↑𝑘) = (𝑃↑0))
23	22, 18	eqtrd 2271	. . . . . 6 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...0)(𝑃↑𝑘) = 1)
24	16, 23	eqtrid 2283	. . . . 5 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) = 1)
25	24, 7	oveq12d 6097	. . . 4 ⊢ (𝑃 ∈ ℙ → (Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) + (𝑃↑1)) = (1 + 𝑃))
26	2	a1i 9	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 1 ∈ ℕ₀)
27		nn0uz 9940	. . . . . 6 ⊢ ℕ₀ = (ℤ_≥‘0)
28	26, 27	eleqtrdi 2331	. . . . 5 ⊢ (𝑃 ∈ ℙ → 1 ∈ (ℤ_≥‘0))
29		elfznn0 10504	. . . . . 6 ⊢ (𝑘 ∈ (0...1) → 𝑘 ∈ ℕ₀)
30		expcl 10977	. . . . . 6 ⊢ ((𝑃 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑃↑𝑘) ∈ ℂ)
31	6, 29, 30	syl2an 289	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → (𝑃↑𝑘) ∈ ℂ)
32		oveq2 6087	. . . . 5 ⊢ (𝑘 = 1 → (𝑃↑𝑘) = (𝑃↑1))
33	28, 31, 32	fsumm1 12166	. . . 4 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)(𝑃↑𝑘) = (Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) + (𝑃↑1)))
34		addcom 8457	. . . . 5 ⊢ ((𝑃 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑃 + 1) = (1 + 𝑃))
35	6, 1, 34	sylancl 417	. . . 4 ⊢ (𝑃 ∈ ℙ → (𝑃 + 1) = (1 + 𝑃))
36	25, 33, 35	3eqtr4d 2281	. . 3 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)(𝑃↑𝑘) = (𝑃 + 1))
37	13, 36	eqtrd 2271	. 2 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)((𝑃↑_𝑐1)↑𝑘) = (𝑃 + 1))
38	4, 8, 37	3eqtr3d 2279	1 ⊢ (𝑃 ∈ ℙ → (1 σ 𝑃) = (𝑃 + 1))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1402 ∈ wcel 2209 ‘cfv 5375 (class class class)co 6079 ℂcc 8171 0cc0 8173 1c1 8174 + caddc 8176 − cmin 8491 ℕcn 9287 ℕ₀cn0 9546 ℤcz 9627 ℤ_≥cuz 9904 ...cfz 10394 ↑cexp 10958 Σcsu 12102 ℙcprime 12868 ↑_𝑐ccxp 15941 σ csgm 16078

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-mulrcl 8272 ax-addcom 8273 ax-mulcom 8274 ax-addass 8275 ax-mulass 8276 ax-distr 8277 ax-i2m1 8278 ax-0lt1 8279 ax-1rid 8280 ax-0id 8281 ax-rnegex 8282 ax-precex 8283 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltwlin 8286 ax-pre-lttrn 8287 ax-pre-apti 8288 ax-pre-ltadd 8289 ax-pre-mulgt0 8290 ax-pre-mulext 8291 ax-arch 8292 ax-caucvg 8293 ax-pre-suploc 8294 ax-addf 8295 ax-mulf 8296

This theorem depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-disj 4105 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-id 4436 df-po 4439 df-iso 4440 df-iord 4509 df-on 4511 df-ilim 4512 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-isom 5384 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-of 6296 df-1st 6368 df-2nd 6369 df-recs 6570 df-irdg 6635 df-frec 6656 df-1o 6681 df-2o 6682 df-oadd 6685 df-er 6801 df-map 6918 df-pm 6919 df-en 7017 df-dom 7018 df-fin 7019 df-sup 7318 df-inf 7319 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-le 8360 df-sub 8493 df-neg 8494 df-reap 8897 df-ap 8904 df-div 8997 df-inn 9288 df-2 9346 df-3 9347 df-4 9348 df-n0 9547 df-xnn0 9614 df-z 9628 df-uz 9905 df-q 10003 df-rp 10038 df-xneg 10157 df-xadd 10158 df-ioo 10277 df-ico 10279 df-icc 10280 df-fz 10395 df-fzo 10533 df-fl 10688 df-mod 10743 df-seqfrec 10868 df-exp 10959 df-fac 11147 df-bc 11169 df-ihash 11198 df-shft 11563 df-cj 11590 df-re 11591 df-im 11592 df-rsqrt 11747 df-abs 11748 df-clim 12028 df-sumdc 12103 df-ef 12398 df-e 12399 df-dvds 12538 df-gcd 12714 df-prm 12869 df-pc 13047 df-rest 13578 df-topgen 13597 df-psmet 14863 df-xmet 14864 df-met 14865 df-bl 14866 df-mopn 14867 df-top 15082 df-topon 15095 df-bases 15127 df-ntr 15180 df-cn 15272 df-cnp 15273 df-tx 15337 df-cncf 15655 df-limced 15740 df-dvap 15741 df-relog 15942 df-rpcxp 15943 df-sgm 16079

This theorem is referenced by: perfect1 16095

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