1sgmprm - Intuitionistic Logic Explorer

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Theorem 1sgmprm 15711

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 8118	. . 3 ⊢ 1 ∈ ℂ
2		1nn0 9411	. . 3 ⊢ 1 ∈ ℕ₀
3		sgmppw 15709	. . 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 12675	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
6	5	nncnd 9150	. . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℂ)
7	6	exp1d 10923	. . 3 ⊢ (𝑃 ∈ ℙ → (𝑃↑1) = 𝑃)
8	7	oveq2d 6029	. 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 9922	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → 𝑃 ∈ ℝ⁺)
11	10	rpcxp1d 15642	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → (𝑃↑_𝑐1) = 𝑃)
12	11	oveq1d 6028	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (0...1)) → ((𝑃↑_𝑐1)↑𝑘) = (𝑃↑𝑘))
13	12	sumeq2dv 11922	. . 3 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)((𝑃↑_𝑐1)↑𝑘) = Σ𝑘 ∈ (0...1)(𝑃↑𝑘))
14		1m1e0 9205	. . . . . . . 8 ⊢ (1 − 1) = 0
15	14	oveq2i 6024	. . . . . . 7 ⊢ (0...(1 − 1)) = (0...0)
16	15	sumeq1i 11917	. . . . . 6 ⊢ Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) = Σ𝑘 ∈ (0...0)(𝑃↑𝑘)
17		0z 9483	. . . . . . . 8 ⊢ 0 ∈ ℤ
18	6	exp0d 10922	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (𝑃↑0) = 1)
19	18, 1	eqeltrdi 2320	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → (𝑃↑0) ∈ ℂ)
20		oveq2 6021	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑃↑𝑘) = (𝑃↑0))
21	20	fsum1 11966	. . . . . . . 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 6031	. . . 4 ⊢ (𝑃 ∈ ℙ → (Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) + (𝑃↑1)) = (1 + 𝑃))
26	2	a1i 9	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 1 ∈ ℕ₀)
27		nn0uz 9784	. . . . . 6 ⊢ ℕ₀ = (ℤ_≥‘0)
28	26, 27	eleqtrdi 2322	. . . . 5 ⊢ (𝑃 ∈ ℙ → 1 ∈ (ℤ_≥‘0))
29		elfznn0 10342	. . . . . 6 ⊢ (𝑘 ∈ (0...1) → 𝑘 ∈ ℕ₀)
30		expcl 10812	. . . . . 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 6021	. . . . 5 ⊢ (𝑘 = 1 → (𝑃↑𝑘) = (𝑃↑1))
33	28, 31, 32	fsumm1 11970	. . . 4 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ (0...1)(𝑃↑𝑘) = (Σ𝑘 ∈ (0...(1 − 1))(𝑃↑𝑘) + (𝑃↑1)))
34		addcom 8309	. . . . 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 5324 (class class class)co 6013 ℂcc 8023 0cc0 8025 1c1 8026 + caddc 8028 − cmin 8343 ℕcn 9136 ℕ₀cn0 9395 ℤcz 9472 ℤ_≥cuz 9748 ...cfz 10236 ↑cexp 10793 Σcsu 11907 ℙcprime 12672 ↑_𝑐ccxp 15574 σ csgm 15698

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 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-mulrcl 8124 ax-addcom 8125 ax-mulcom 8126 ax-addass 8127 ax-mulass 8128 ax-distr 8129 ax-i2m1 8130 ax-0lt1 8131 ax-1rid 8132 ax-0id 8133 ax-rnegex 8134 ax-precex 8135 ax-cnre 8136 ax-pre-ltirr 8137 ax-pre-ltwlin 8138 ax-pre-lttrn 8139 ax-pre-apti 8140 ax-pre-ltadd 8141 ax-pre-mulgt0 8142 ax-pre-mulext 8143 ax-arch 8144 ax-caucvg 8145 ax-pre-suploc 8146 ax-addf 8147 ax-mulf 8148

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 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-disj 4063 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-isom 5333 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-of 6230 df-1st 6298 df-2nd 6299 df-recs 6466 df-irdg 6531 df-frec 6552 df-1o 6577 df-2o 6578 df-oadd 6581 df-er 6697 df-map 6814 df-pm 6815 df-en 6905 df-dom 6906 df-fin 6907 df-sup 7177 df-inf 7178 df-pnf 8209 df-mnf 8210 df-xr 8211 df-ltxr 8212 df-le 8213 df-sub 8345 df-neg 8346 df-reap 8748 df-ap 8755 df-div 8846 df-inn 9137 df-2 9195 df-3 9196 df-4 9197 df-n0 9396 df-xnn0 9459 df-z 9473 df-uz 9749 df-q 9847 df-rp 9882 df-xneg 10000 df-xadd 10001 df-ioo 10120 df-ico 10122 df-icc 10123 df-fz 10237 df-fzo 10371 df-fl 10523 df-mod 10578 df-seqfrec 10703 df-exp 10794 df-fac 10981 df-bc 11003 df-ihash 11031 df-shft 11369 df-cj 11396 df-re 11397 df-im 11398 df-rsqrt 11552 df-abs 11553 df-clim 11833 df-sumdc 11908 df-ef 12202 df-e 12203 df-dvds 12342 df-gcd 12518 df-prm 12673 df-pc 12851 df-rest 13317 df-topgen 13336 df-psmet 14550 df-xmet 14551 df-met 14552 df-bl 14553 df-mopn 14554 df-top 14715 df-topon 14728 df-bases 14760 df-ntr 14813 df-cn 14905 df-cnp 14906 df-tx 14970 df-cncf 15288 df-limced 15373 df-dvap 15374 df-relog 15575 df-rpcxp 15576 df-sgm 15699

This theorem is referenced by: perfect1 15715

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