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Theorem 0sgm 15715

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 9490	. . 3 ⊢ 0 ∈ ℤ
2		sgmval2 15714	. . 3 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℕ) → (0 σ 𝐴) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0))
3	1, 2	mpan 424	. 2 ⊢ (𝐴 ∈ ℕ → (0 σ 𝐴) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0))
4		elrabi 2959	. . . . . 6 ⊢ (𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} → 𝑘 ∈ ℕ)
5	4	nncnd 9157	. . . . 5 ⊢ (𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} → 𝑘 ∈ ℂ)
6	5	exp0d 10930	. . . 4 ⊢ (𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} → (𝑘↑0) = 1)
7	6	sumeq2i 11929	. . 3 ⊢ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}1
8		dvdsfi 12816	. . . 4 ⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ∈ Fin)
9		ax-1cn 8125	. . . 4 ⊢ 1 ∈ ℂ
10		fsumconst 12020	. . . 4 ⊢ (({𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}1 = ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1))
11	8, 9, 10	sylancl 413	. . 3 ⊢ (𝐴 ∈ ℕ → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}1 = ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1))
12	7, 11	eqtrid 2276	. 2 ⊢ (𝐴 ∈ ℕ → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} (𝑘↑0) = ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1))
13		hashcl 11044	. . . . 5 ⊢ ({𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ∈ Fin → (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) ∈ ℕ₀)
14	8, 13	syl 14	. . . 4 ⊢ (𝐴 ∈ ℕ → (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) ∈ ℕ₀)
15	14	nn0cnd 9457	. . 3 ⊢ (𝐴 ∈ ℕ → (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) ∈ ℂ)
16	15	mulridd 8196	. 2 ⊢ (𝐴 ∈ ℕ → ((♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) · 1) = (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}))
17	3, 12, 16	3eqtrd 2268	1 ⊢ (𝐴 ∈ ℕ → (0 σ 𝐴) = (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1397 ∈ wcel 2202 {crab 2514 class class class wbr 4088 ‘cfv 5326 (class class class)co 6018 Fincfn 6909 ℂcc 8030 0cc0 8032 1c1 8033 · cmul 8037 ℕcn 9143 ℕ₀cn0 9402 ℤcz 9479 ↑cexp 10801 ♯chash 11038 Σcsu 11918 ∥ cdvds 12353 σ csgm 15711

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152 ax-pre-suploc 8153 ax-addf 8154 ax-mulf 8155

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-disj 4065 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-of 6235 df-1st 6303 df-2nd 6304 df-recs 6471 df-irdg 6536 df-frec 6557 df-1o 6582 df-oadd 6586 df-er 6702 df-map 6819 df-pm 6820 df-en 6910 df-dom 6911 df-fin 6912 df-sup 7183 df-inf 7184 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-xneg 10007 df-xadd 10008 df-ioo 10127 df-ico 10129 df-icc 10130 df-fz 10244 df-fzo 10378 df-fl 10531 df-mod 10586 df-seqfrec 10711 df-exp 10802 df-fac 10989 df-bc 11011 df-ihash 11039 df-shft 11380 df-cj 11407 df-re 11408 df-im 11409 df-rsqrt 11563 df-abs 11564 df-clim 11844 df-sumdc 11919 df-ef 12214 df-e 12215 df-dvds 12354 df-rest 13329 df-topgen 13348 df-psmet 14563 df-xmet 14564 df-met 14565 df-bl 14566 df-mopn 14567 df-top 14728 df-topon 14741 df-bases 14773 df-ntr 14826 df-cn 14918 df-cnp 14919 df-tx 14983 df-cncf 15301 df-limced 15386 df-dvap 15387 df-relog 15588 df-rpcxp 15589 df-sgm 15712

This theorem is referenced by: 0sgmppw 15723

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