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Theorem 0sgm 25729

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 11980	. . 3 ⊢ 0 ∈ ℤ
2		sgmval2 25728	. . 3 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℕ) → (0 σ 𝐴) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0))
3	1, 2	mpan 689	. 2 ⊢ (𝐴 ∈ ℕ → (0 σ 𝐴) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0))
4		elrabi 3623	. . . . . 6 ⊢ (𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} → 𝑘 ∈ ℕ)
5	4	nncnd 11641	. . . . 5 ⊢ (𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} → 𝑘 ∈ ℂ)
6	5	exp0d 13500	. . . 4 ⊢ (𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} → (𝑘↑0) = 1)
7	6	sumeq2i 15048	. . 3 ⊢ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}1
8		fzfid 13336	. . . . 5 ⊢ (𝐴 ∈ ℕ → (1...𝐴) ∈ Fin)
9		dvdsssfz1 15660	. . . . 5 ⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ⊆ (1...𝐴))
10	8, 9	ssfid 8725	. . . 4 ⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ∈ Fin)
11		ax-1cn 10584	. . . 4 ⊢ 1 ∈ ℂ
12		fsumconst 15137	. . . 4 ⊢ (({𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}1 = ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1))
13	10, 11, 12	sylancl 589	. . 3 ⊢ (𝐴 ∈ ℕ → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}1 = ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1))
14	7, 13	syl5eq 2845	. 2 ⊢ (𝐴 ∈ ℕ → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0) = ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1))
15		hashcl 13713	. . . . 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 11945	. . 3 ⊢ (𝐴 ∈ ℕ → (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) ∈ ℂ)
18	17	mulid1d 10647	. 2 ⊢ (𝐴 ∈ ℕ → ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1) = (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}))
19	3, 14, 18	3eqtrd 2837	1 ⊢ (𝐴 ∈ ℕ → (0 σ 𝐴) = (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {crab 3110 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ℂcc 10524 0cc0 10526 1c1 10527 · cmul 10531 ℕcn 11625 ℕ₀cn0 11885 ℤcz 11969 ...cfz 12885 ↑cexp 13425 ♯chash 13686 Σcsu 15034 ∥ cdvds 15599 σ csgm 25681

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-pi 15418 df-dvds 15600 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470 df-log 25148 df-cxp 25149 df-sgm 25687

This theorem is referenced by: 0sgmppw 25782

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