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Theorem 0sgm 26645

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 12568	. . 3 ⊢ 0 ∈ ℤ
2		sgmval2 26644	. . 3 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℕ) → (0 σ 𝐴) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0))
3	1, 2	mpan 688	. 2 ⊢ (𝐴 ∈ ℕ → (0 σ 𝐴) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0))
4		elrabi 3677	. . . . . 6 ⊢ (𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} → 𝑘 ∈ ℕ)
5	4	nncnd 12227	. . . . 5 ⊢ (𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} → 𝑘 ∈ ℂ)
6	5	exp0d 14104	. . . 4 ⊢ (𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} → (𝑘↑0) = 1)
7	6	sumeq2i 15644	. . 3 ⊢ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}1
8		fzfid 13937	. . . . 5 ⊢ (𝐴 ∈ ℕ → (1...𝐴) ∈ Fin)
9		dvdsssfz1 16260	. . . . 5 ⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ⊆ (1...𝐴))
10	8, 9	ssfid 9266	. . . 4 ⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ∈ Fin)
11		ax-1cn 11167	. . . 4 ⊢ 1 ∈ ℂ
12		fsumconst 15735	. . . 4 ⊢ (({𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}1 = ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1))
13	10, 11, 12	sylancl 586	. . 3 ⊢ (𝐴 ∈ ℕ → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}1 = ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1))
14	7, 13	eqtrid 2784	. 2 ⊢ (𝐴 ∈ ℕ → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0) = ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1))
15		hashcl 14315	. . . . 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 12533	. . 3 ⊢ (𝐴 ∈ ℕ → (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) ∈ ℂ)
18	17	mulridd 11230	. 2 ⊢ (𝐴 ∈ ℕ → ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1) = (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}))
19	3, 14, 18	3eqtrd 2776	1 ⊢ (𝐴 ∈ ℕ → (0 σ 𝐴) = (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {crab 3432 class class class wbr 5148 ‘cfv 6543 (class class class)co 7408 Fincfn 8938 ℂcc 11107 0cc0 11109 1c1 11110 · cmul 11114 ℕcn 12211 ℕ₀cn0 12471 ℤcz 12557 ...cfz 13483 ↑cexp 14026 ♯chash 14289 Σcsu 15631 ∥ cdvds 16196 σ csgm 26597

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-ioo 13327 df-ioc 13328 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-fl 13756 df-mod 13834 df-seq 13966 df-exp 14027 df-fac 14233 df-bc 14262 df-hash 14290 df-shft 15013 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-limsup 15414 df-clim 15431 df-rlim 15432 df-sum 15632 df-ef 16010 df-sin 16012 df-cos 16013 df-pi 16015 df-dvds 16197 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 18560 df-sgrp 18609 df-mnd 18625 df-submnd 18671 df-mulg 18950 df-cntz 19180 df-cmn 19649 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-fbas 20940 df-fg 20941 df-cnfld 20944 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-cld 22522 df-ntr 22523 df-cls 22524 df-nei 22601 df-lp 22639 df-perf 22640 df-cn 22730 df-cnp 22731 df-haus 22818 df-tx 23065 df-hmeo 23258 df-fil 23349 df-fm 23441 df-flim 23442 df-flf 23443 df-xms 23825 df-ms 23826 df-tms 23827 df-cncf 24393 df-limc 25382 df-dv 25383 df-log 26064 df-cxp 26065 df-sgm 26603

This theorem is referenced by: 0sgmppw 26698

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