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Theorem 0sgm 25333

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 11744	. . 3 ⊢ 0 ∈ ℤ
2		sgmval2 25332	. . 3 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℕ) → (0 σ 𝐴) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0))
3	1, 2	mpan 680	. 2 ⊢ (𝐴 ∈ ℕ → (0 σ 𝐴) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0))
4		elrabi 3567	. . . . . 6 ⊢ (𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} → 𝑘 ∈ ℕ)
5	4	nncnd 11397	. . . . 5 ⊢ (𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} → 𝑘 ∈ ℂ)
6	5	exp0d 13326	. . . 4 ⊢ (𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} → (𝑘↑0) = 1)
7	6	sumeq2i 14846	. . 3 ⊢ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}1
8		fzfid 13096	. . . . 5 ⊢ (𝐴 ∈ ℕ → (1...𝐴) ∈ Fin)
9		dvdsssfz1 15457	. . . . 5 ⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ⊆ (1...𝐴))
10		ssfi 8470	. . . . 5 ⊢ (((1...𝐴) ∈ Fin ∧ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ⊆ (1...𝐴)) → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ∈ Fin)
11	8, 9, 10	syl2anc 579	. . . 4 ⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ∈ Fin)
12		ax-1cn 10332	. . . 4 ⊢ 1 ∈ ℂ
13		fsumconst 14935	. . . 4 ⊢ (({𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}1 = ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1))
14	11, 12, 13	sylancl 580	. . 3 ⊢ (𝐴 ∈ ℕ → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}1 = ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1))
15	7, 14	syl5eq 2826	. 2 ⊢ (𝐴 ∈ ℕ → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0) = ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1))
16		hashcl 13468	. . . . 5 ⊢ ({𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ∈ Fin → (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) ∈ ℕ₀)
17	11, 16	syl 17	. . . 4 ⊢ (𝐴 ∈ ℕ → (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) ∈ ℕ₀)
18	17	nn0cnd 11709	. . 3 ⊢ (𝐴 ∈ ℕ → (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) ∈ ℂ)
19	18	mulid1d 10396	. 2 ⊢ (𝐴 ∈ ℕ → ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1) = (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}))
20	3, 15, 19	3eqtrd 2818	1 ⊢ (𝐴 ∈ ℕ → (0 σ 𝐴) = (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 {crab 3094 ⊆ wss 3792 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 Fincfn 8243 ℂcc 10272 0cc0 10274 1c1 10275 · cmul 10279 ℕcn 11379 ℕ₀cn0 11647 ℤcz 11733 ...cfz 12648 ↑cexp 13183 ♯chash 13441 Σcsu 14833 ∥ cdvds 15396 σ csgm 25285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-fi 8607 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-dec 11851 df-uz 11998 df-q 12101 df-rp 12143 df-xneg 12262 df-xadd 12263 df-xmul 12264 df-ioo 12496 df-ioc 12497 df-ico 12498 df-icc 12499 df-fz 12649 df-fzo 12790 df-fl 12917 df-mod 12993 df-seq 13125 df-exp 13184 df-fac 13385 df-bc 13414 df-hash 13442 df-shft 14220 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 df-limsup 14619 df-clim 14636 df-rlim 14637 df-sum 14834 df-ef 15209 df-sin 15211 df-cos 15212 df-pi 15214 df-dvds 15397 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-starv 16364 df-sca 16365 df-vsca 16366 df-ip 16367 df-tset 16368 df-ple 16369 df-ds 16371 df-unif 16372 df-hom 16373 df-cco 16374 df-rest 16480 df-topn 16481 df-0g 16499 df-gsum 16500 df-topgen 16501 df-pt 16502 df-prds 16505 df-xrs 16559 df-qtop 16564 df-imas 16565 df-xps 16567 df-mre 16643 df-mrc 16644 df-acs 16646 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-submnd 17733 df-mulg 17939 df-cntz 18144 df-cmn 18592 df-psmet 20145 df-xmet 20146 df-met 20147 df-bl 20148 df-mopn 20149 df-fbas 20150 df-fg 20151 df-cnfld 20154 df-top 21117 df-topon 21134 df-topsp 21156 df-bases 21169 df-cld 21242 df-ntr 21243 df-cls 21244 df-nei 21321 df-lp 21359 df-perf 21360 df-cn 21450 df-cnp 21451 df-haus 21538 df-tx 21785 df-hmeo 21978 df-fil 22069 df-fm 22161 df-flim 22162 df-flf 22163 df-xms 22544 df-ms 22545 df-tms 22546 df-cncf 23100 df-limc 24078 df-dv 24079 df-log 24751 df-cxp 24752 df-sgm 25291

This theorem is referenced by: 0sgmppw 25386

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