fsumcl - Metamath Proof Explorer

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Theorem fsumcl 15709

Description: Closure of a finite sum of complex numbers 𝐴(𝑘). (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)

**Hypotheses**
Ref	Expression
fsumcl.1	⊢ (𝜑 → 𝐴 ∈ Fin)
fsumcl.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)

**Assertion**
Ref	Expression
fsumcl	⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℂ)

Distinct variable groups: 𝐴,𝑘 𝜑,𝑘 Allowed substitution hint: 𝐵(𝑘)

Proof of Theorem fsumcl Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssidd 3996	. 2 ⊢ (𝜑 → ℂ ⊆ ℂ)
2		addcl 11218	. . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
3	2	adantl 480	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
4		fsumcl.1	. 2 ⊢ (𝜑 → 𝐴 ∈ Fin)
5		fsumcl.2	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
6		0cnd 11235	. 2 ⊢ (𝜑 → 0 ∈ ℂ)
7	1, 3, 4, 5, 6	fsumcllem 15708	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℂ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 (class class class)co 7415 Fincfn 8960 ℂcc 11134 + caddc 11139 Σcsu 15662

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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214

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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-n0 12501 df-z 12587 df-uz 12851 df-rp 13005 df-fz 13515 df-fzo 13658 df-seq 13997 df-exp 14057 df-hash 14320 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-clim 15462 df-sum 15663

This theorem is referenced by: fsumclf 15714 fsum2dlem 15746 fsum0diag2 15759 fsummulc1 15761 fsumdivc 15762 fsumneg 15763 fsumsub 15764 fsum2mul 15765 fsumabs 15777 telfsumo 15778 fsumparts 15782 o1fsum 15789 cvgcmpce 15794 climfsum 15796 fsumiun 15797 binom1dif 15809 incexclem 15812 incexc 15813 isumsplit 15816 arisum2 15837 geoserg 15842 pwdif 15844 mertenslem1 15860 mertens 15862 binomfallfaclem2 16014 bpolycl 16026 bpolysum 16027 bpolydiflem 16028 fsumkthpow 16030 fprodefsum 16069 eirrlem 16178 pwp1fsum 16365 pcfac 16865 sylow2a 19576 itg1addlem5 25646 itgcl 25729 dvmptfsum 25923 dvfsumabs 25973 dvfsumlem1 25976 plyf 26148 plymullem1 26164 coeeulem 26174 coemullem 26200 plycjlem 26227 taylpf 26316 mtest 26356 mtestbdd 26357 pserdvlem2 26381 abelthlem6 26389 abelthlem7 26391 advlogexp 26605 log2tlbnd 26893 birthdaylem2 26900 fsumharmonic 26960 lgamcvg2 27003 ftalem1 27021 ftalem5 27025 sgmf 27093 chtdif 27106 fsumdvdscom 27133 fsumdvdsmul 27143 fsumdvdsmulOLD 27145 logexprlim 27174 dchrsum2 27217 sumdchr2 27219 rpvmasumlem 27436 dchrisumlem1 27438 dchrisumlem2 27439 dchrisum 27441 dchrmusum2 27443 dchrvmasum2if 27446 dchrvmasumlem3 27448 dchrvmasumiflem1 27450 dchrvmasumiflem2 27451 rpvmasum2 27461 dchrisum0lem1b 27464 dchrisum0lem1 27465 dchrisum0lem2a 27466 dchrisum0lem2 27467 dchrisum0lem3 27468 dchrmusumlem 27471 dchrvmasumlem 27472 mudivsum 27479 mulogsumlem 27480 mulogsum 27481 mulog2sumlem1 27483 mulog2sumlem2 27484 mulog2sumlem3 27485 vmalogdivsum 27488 logsqvma 27491 selberglem1 27494 selberglem2 27495 selberg2lem 27499 selberg2 27500 selberg3lem1 27506 pntrsumo1 27514 pntrsumbnd 27515 selbergr 27517 selberg4r 27519 pntrlog2bndlem2 27527 pntrlog2bndlem4 27529 pntrlog2bndlem5 27530 pntlemo 27556 ax5seglem6 28787 axlowdimlem16 28810 finsumvtxdg2ssteplem4 29404 dipcl 30564 indsumin 33697 esumcvg 33761 fsum2dsub 34295 reprsuc 34303 breprexplemc 34320 breprexp 34321 breprexpnat 34322 vtscl 34326 circlemeth 34328 hgt750lemd 34336 tgoldbachgtde 34348 subfacval2 34853 subfaclim 34854 fwddifnp1 35817 knoppndvlem11 36053 aks4d1p1p1 41589 sticksstones12a 41684 sumcubes 41937 fltnltalem 42150 jm2.23 42481 fsumsermpt 45029 sumnnodd 45080 dvnmul 45393 dvnprodlem1 45396 dvnprodlem2 45397 stoweidlem26 45476 dirkertrigeqlem2 45549 dirkeritg 45552 fourierdlem73 45629 fourierdlem83 45639 elaa2lem 45683 etransclem23 45707 etransclem27 45711 etransclem31 45715 etransclem33 45717 etransclem39 45723 etransclem46 45730 etransclem47 45731 etransclem48 45732 altgsumbcALT 47528 nn0sumshdiglemA 47803 amgmlemALT 48347

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