1sgmprm - Intuitionistic Logic Explorer

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Theorem 1sgmprm 15676

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 8100	. . 3 ⊢ 1 ∈ ℂ
2		1nn0 9393	. . 3 ⊢ 1 ∈ ℕ₀
3		sgmppw 15674	. . 3 ⊢ ((1 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 1 ∈ ℕ₀) → (1 σ (𝑃↑1)) = Σ𝑘 ∈ (0...1)((𝑃↑_𝑐1)↑𝑘))
4	1, 2, 3	mp3an13 1362	. 2 ⊢ (𝑃 ∈ ℙ → (1 σ (𝑃↑1)) = Σ𝑘 ∈ (0...1)((𝑃↑_𝑐1)↑𝑘))
5		prmnn 12640	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
6	5	nncnd 9132	. . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℂ)
7	6	exp1d 10898	. . 3 ⊢ (𝑃 ∈ ℙ → (𝑃↑1) = 𝑃)
8	7	oveq2d 6023	. 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 9898	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → 𝑃 ∈ ℝ⁺)
11	10	rpcxp1d 15607	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → (𝑃↑_𝑐1) = 𝑃)
12	11	oveq1d 6022	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → ((𝑃↑_𝑐1)↑𝑘) = (𝑃↑𝑘))
13	12	sumeq2dv 11887	. . 3 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)((𝑃↑_𝑐1)↑𝑘) = Σ𝑘 ∈ (0...1)(𝑃↑𝑘))
14		1m1e0 9187	. . . . . . . 8 ⊢ (1 − 1) = 0
15	14	oveq2i 6018	. . . . . . 7 ⊢ (0...(1 − 1)) = (0...0)
16	15	sumeq1i 11882	. . . . . 6 ⊢ Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) = Σ𝑘 ∈ (0...0)(𝑃↑𝑘)
17		0z 9465	. . . . . . . 8 ⊢ 0 ∈ ℤ
18	6	exp0d 10897	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (𝑃↑0) = 1)
19	18, 1	eqeltrdi 2320	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → (𝑃↑0) ∈ ℂ)
20		oveq2 6015	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑃↑𝑘) = (𝑃↑0))
21	20	fsum1 11931	. . . . . . . 8 ⊢ ((0 ∈ ℤ ∧ (𝑃↑0) ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝑃↑𝑘) = (𝑃↑0))
22	17, 19, 21	sylancr 414	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...0)(𝑃↑𝑘) = (𝑃↑0))
23	22, 18	eqtrd 2262	. . . . . 6 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...0)(𝑃↑𝑘) = 1)
24	16, 23	eqtrid 2274	. . . . 5 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) = 1)
25	24, 7	oveq12d 6025	. . . 4 ⊢ (𝑃 ∈ ℙ → (Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) + (𝑃↑1)) = (1 + 𝑃))
26	2	a1i 9	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 1 ∈ ℕ₀)
27		nn0uz 9765	. . . . . 6 ⊢ ℕ₀ = (ℤ_≥‘0)
28	26, 27	eleqtrdi 2322	. . . . 5 ⊢ (𝑃 ∈ ℙ → 1 ∈ (ℤ_≥‘0))
29		elfznn0 10318	. . . . . 6 ⊢ (𝑘 ∈ (0...1) → 𝑘 ∈ ℕ₀)
30		expcl 10787	. . . . . 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 6015	. . . . 5 ⊢ (𝑘 = 1 → (𝑃↑𝑘) = (𝑃↑1))
33	28, 31, 32	fsumm1 11935	. . . 4 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)(𝑃↑𝑘) = (Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) + (𝑃↑1)))
34		addcom 8291	. . . . 5 ⊢ ((𝑃 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑃 + 1) = (1 + 𝑃))
35	6, 1, 34	sylancl 413	. . . 4 ⊢ (𝑃 ∈ ℙ → (𝑃 + 1) = (1 + 𝑃))
36	25, 33, 35	3eqtr4d 2272	. . 3 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)(𝑃↑𝑘) = (𝑃 + 1))
37	13, 36	eqtrd 2262	. 2 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)((𝑃↑_𝑐1)↑𝑘) = (𝑃 + 1))
38	4, 8, 37	3eqtr3d 2270	1 ⊢ (𝑃 ∈ ℙ → (1 σ 𝑃) = (𝑃 + 1))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1395 ∈ wcel 2200 ‘cfv 5318 (class class class)co 6007 ℂcc 8005 0cc0 8007 1c1 8008 + caddc 8010 − cmin 8325 ℕcn 9118 ℕ₀cn0 9377 ℤcz 9454 ℤ_≥cuz 9730 ...cfz 10212 ↑cexp 10768 Σcsu 11872 ℙcprime 12637 ↑_𝑐ccxp 15539 σ csgm 15663

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-mulrcl 8106 ax-addcom 8107 ax-mulcom 8108 ax-addass 8109 ax-mulass 8110 ax-distr 8111 ax-i2m1 8112 ax-0lt1 8113 ax-1rid 8114 ax-0id 8115 ax-rnegex 8116 ax-precex 8117 ax-cnre 8118 ax-pre-ltirr 8119 ax-pre-ltwlin 8120 ax-pre-lttrn 8121 ax-pre-apti 8122 ax-pre-ltadd 8123 ax-pre-mulgt0 8124 ax-pre-mulext 8125 ax-arch 8126 ax-caucvg 8127 ax-pre-suploc 8128 ax-addf 8129 ax-mulf 8130

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-disj 4060 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-of 6224 df-1st 6292 df-2nd 6293 df-recs 6457 df-irdg 6522 df-frec 6543 df-1o 6568 df-2o 6569 df-oadd 6572 df-er 6688 df-map 6805 df-pm 6806 df-en 6896 df-dom 6897 df-fin 6898 df-sup 7159 df-inf 7160 df-pnf 8191 df-mnf 8192 df-xr 8193 df-ltxr 8194 df-le 8195 df-sub 8327 df-neg 8328 df-reap 8730 df-ap 8737 df-div 8828 df-inn 9119 df-2 9177 df-3 9178 df-4 9179 df-n0 9378 df-xnn0 9441 df-z 9455 df-uz 9731 df-q 9823 df-rp 9858 df-xneg 9976 df-xadd 9977 df-ioo 10096 df-ico 10098 df-icc 10099 df-fz 10213 df-fzo 10347 df-fl 10498 df-mod 10553 df-seqfrec 10678 df-exp 10769 df-fac 10956 df-bc 10978 df-ihash 11006 df-shft 11334 df-cj 11361 df-re 11362 df-im 11363 df-rsqrt 11517 df-abs 11518 df-clim 11798 df-sumdc 11873 df-ef 12167 df-e 12168 df-dvds 12307 df-gcd 12483 df-prm 12638 df-pc 12816 df-rest 13282 df-topgen 13301 df-psmet 14515 df-xmet 14516 df-met 14517 df-bl 14518 df-mopn 14519 df-top 14680 df-topon 14693 df-bases 14725 df-ntr 14778 df-cn 14870 df-cnp 14871 df-tx 14935 df-cncf 15253 df-limced 15338 df-dvap 15339 df-relog 15540 df-rpcxp 15541 df-sgm 15664

This theorem is referenced by: perfect1 15680

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