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Theorem sgmf 27095

Description: The divisor function is a function into the complex numbers. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 21-Jun-2015.)

**Assertion**
Ref	Expression
sgmf	⊢ σ :(ℂ × ℕ)⟶ℂ

Proof of Theorem sgmf Dummy variables 𝑘 𝑛 𝑝 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 13976	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
2		dvdsssfz1 16300	. . . . . 6 ⊢ (𝑛 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} ⊆ (1...𝑛))
3	2	adantl 480	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑛 ∈ ℕ) → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} ⊆ (1...𝑛))
4	1, 3	ssfid 9296	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑛 ∈ ℕ) → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} ∈ Fin)
5		elrabi 3676	. . . . . 6 ⊢ (𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} → 𝑘 ∈ ℕ)
6	5	nncnd 12264	. . . . 5 ⊢ (𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} → 𝑘 ∈ ℂ)
7		simpl 481	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℂ)
8		cxpcl 26626	. . . . 5 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘↑_𝑐𝑥) ∈ ℂ)
9	6, 7, 8	syl2anr 595	. . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛}) → (𝑘↑_𝑐𝑥) ∈ ℂ)
10	4, 9	fsumcl 15717	. . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑_𝑐𝑥) ∈ ℂ)
11	10	rgen2 3193	. 2 ⊢ ∀𝑥 ∈ ℂ ∀𝑛 ∈ ℕ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑_𝑐𝑥) ∈ ℂ
12		df-sgm 27052	. . 3 ⊢ σ = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑_𝑐𝑥))
13	12	fmpo 8076	. 2 ⊢ (∀𝑥 ∈ ℂ ∀𝑛 ∈ ℕ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑_𝑐𝑥) ∈ ℂ ↔ σ :(ℂ × ℕ)⟶ℂ)
14	11, 13	mpbi 229	1 ⊢ σ :(ℂ × ℕ)⟶ℂ

Colors of variables: wff setvar class

Syntax hints: ∧ wa 394 ∈ wcel 2098 ∀wral 3057 {crab 3428 ⊆ wss 3947 class class class wbr 5150 × cxp 5678 ⟶wf 6547 (class class class)co 7424 ℂcc 11142 1c1 11145 ℕcn 12248 ...cfz 13522 Σcsu 15670 ∥ cdvds 16236 ↑_𝑐ccxp 26507 σ csgm 27046

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 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 ax-addf 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-om 7875 df-1st 7997 df-2nd 7998 df-supp 8170 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-2o 8492 df-er 8729 df-map 8851 df-pm 8852 df-ixp 8921 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9539 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13130 df-xadd 13131 df-xmul 13132 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-fz 13523 df-fzo 13666 df-fl 13795 df-mod 13873 df-seq 14005 df-exp 14065 df-fac 14271 df-bc 14300 df-hash 14328 df-shft 15052 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-limsup 15453 df-clim 15470 df-rlim 15471 df-sum 15671 df-ef 16049 df-sin 16051 df-cos 16052 df-pi 16054 df-dvds 16237 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-starv 17253 df-sca 17254 df-vsca 17255 df-ip 17256 df-tset 17257 df-ple 17258 df-ds 17260 df-unif 17261 df-hom 17262 df-cco 17263 df-rest 17409 df-topn 17410 df-0g 17428 df-gsum 17429 df-topgen 17430 df-pt 17431 df-prds 17434 df-xrs 17489 df-qtop 17494 df-imas 17495 df-xps 17497 df-mre 17571 df-mrc 17572 df-acs 17574 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-submnd 18746 df-mulg 19029 df-cntz 19273 df-cmn 19742 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-lp 23058 df-perf 23059 df-cn 23149 df-cnp 23150 df-haus 23237 df-tx 23484 df-hmeo 23677 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-xms 24244 df-ms 24245 df-tms 24246 df-cncf 24816 df-limc 25813 df-dv 25814 df-log 26508 df-cxp 26509 df-sgm 27052

This theorem is referenced by: sgmcl 27096

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