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Theorem 0sgm 26992

Description: The value of the sum-of-divisors function, usually denoted σ<SUB>0</SUB>(<i>n</i>). (Contributed by Mario Carneiro, 21-Jun-2015.)

**Assertion**
Ref	Expression
0sgm	⊢ (𝐴 ∈ ℕ → (0 σ 𝐴) = (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}))

Distinct variable group: 𝐴,𝑝

Proof of Theorem 0sgm Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0z 12566	. . 3 ⊢ 0 ∈ ℤ
2		sgmval2 26991	. . 3 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℕ) → (0 σ 𝐴) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0))
3	1, 2	mpan 687	. 2 ⊢ (𝐴 ∈ ℕ → (0 σ 𝐴) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0))
4		elrabi 3669	. . . . . 6 ⊢ (𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} → 𝑘 ∈ ℕ)
5	4	nncnd 12225	. . . . 5 ⊢ (𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} → 𝑘 ∈ ℂ)
6	5	exp0d 14102	. . . 4 ⊢ (𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} → (𝑘↑0) = 1)
7	6	sumeq2i 15642	. . 3 ⊢ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}1
8		fzfid 13935	. . . . 5 ⊢ (𝐴 ∈ ℕ → (1...𝐴) ∈ Fin)
9		dvdsssfz1 16258	. . . . 5 ⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ⊆ (1...𝐴))
10	8, 9	ssfid 9263	. . . 4 ⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ∈ Fin)
11		ax-1cn 11164	. . . 4 ⊢ 1 ∈ ℂ
12		fsumconst 15733	. . . 4 ⊢ (({𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}1 = ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1))
13	10, 11, 12	sylancl 585	. . 3 ⊢ (𝐴 ∈ ℕ → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}1 = ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1))
14	7, 13	eqtrid 2776	. 2 ⊢ (𝐴 ∈ ℕ → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0) = ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1))
15		hashcl 14313	. . . . 5 ⊢ ({𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ∈ Fin → (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) ∈ ℕ₀)
16	10, 15	syl 17	. . . 4 ⊢ (𝐴 ∈ ℕ → (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) ∈ ℕ₀)
17	16	nn0cnd 12531	. . 3 ⊢ (𝐴 ∈ ℕ → (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) ∈ ℂ)
18	17	mulridd 11228	. 2 ⊢ (𝐴 ∈ ℕ → ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1) = (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}))
19	3, 14, 18	3eqtrd 2768	1 ⊢ (𝐴 ∈ ℕ → (0 σ 𝐴) = (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3424 class class class wbr 5138 ‘cfv 6533 (class class class)co 7401 Fincfn 8935 ℂcc 11104 0cc0 11106 1c1 11107 · cmul 11111 ℕcn 12209 ℕ₀cn0 12469 ℤcz 12555 ...cfz 13481 ↑cexp 14024 ♯chash 14287 Σcsu 15629 ∥ cdvds 16194 σ csgm 26944

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-ef 16008 df-sin 16010 df-cos 16011 df-pi 16013 df-dvds 16195 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18704 df-mulg 18986 df-cntz 19223 df-cmn 19692 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-fbas 21225 df-fg 21226 df-cnfld 21229 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-cld 22845 df-ntr 22846 df-cls 22847 df-nei 22924 df-lp 22962 df-perf 22963 df-cn 23053 df-cnp 23054 df-haus 23141 df-tx 23388 df-hmeo 23581 df-fil 23672 df-fm 23764 df-flim 23765 df-flf 23766 df-xms 24148 df-ms 24149 df-tms 24150 df-cncf 24720 df-limc 25717 df-dv 25718 df-log 26407 df-cxp 26408 df-sgm 26950

This theorem is referenced by: 0sgmppw 27047

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