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Theorem sumz 15695

Description: Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)

**Assertion**
Ref	Expression
sumz	⊢ ((𝐴 ⊆ (ℤ_≥‘𝑀) ∨ 𝐴 ∈ Fin) → Σ𝑘 ∈ 𝐴 0 = 0)

Distinct variable groups: 𝐴,𝑘 𝑘,𝑀

Proof of Theorem sumz Dummy variables 𝑓 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2728	. . . . 5 ⊢ (ℤ_≥‘𝑀) = (ℤ_≥‘𝑀)
2		simpr 484	. . . . 5 ⊢ ((𝐴 ⊆ (ℤ_≥‘𝑀) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
3		simpl 482	. . . . 5 ⊢ ((𝐴 ⊆ (ℤ_≥‘𝑀) ∧ 𝑀 ∈ ℤ) → 𝐴 ⊆ (ℤ_≥‘𝑀))
4		c0ex 11233	. . . . . . . 8 ⊢ 0 ∈ V
5	4	fvconst2 7211	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (((ℤ_≥‘𝑀) × {0})‘𝑘) = 0)
6		ifid 4565	. . . . . . 7 ⊢ if(𝑘 ∈ 𝐴, 0, 0) = 0
7	5, 6	eqtr4di 2786	. . . . . 6 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (((ℤ_≥‘𝑀) × {0})‘𝑘) = if(𝑘 ∈ 𝐴, 0, 0))
8	7	adantl 481	. . . . 5 ⊢ (((𝐴 ⊆ (ℤ_≥‘𝑀) ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((ℤ_≥‘𝑀) × {0})‘𝑘) = if(𝑘 ∈ 𝐴, 0, 0))
9		0cnd 11232	. . . . 5 ⊢ (((𝐴 ⊆ (ℤ_≥‘𝑀) ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ 𝐴) → 0 ∈ ℂ)
10	1, 2, 3, 8, 9	zsum 15691	. . . 4 ⊢ ((𝐴 ⊆ (ℤ_≥‘𝑀) ∧ 𝑀 ∈ ℤ) → Σ𝑘 ∈ 𝐴 0 = ( ⇝ ‘seq𝑀( + , ((ℤ_≥‘𝑀) × {0}))))
11		fclim 15524	. . . . . 6 ⊢ ⇝ :dom ⇝ ⟶ℂ
12		ffun 6720	. . . . . 6 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ )
13	11, 12	ax-mp 5	. . . . 5 ⊢ Fun ⇝
14		serclim0 15548	. . . . . 6 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , ((ℤ_≥‘𝑀) × {0})) ⇝ 0)
15	14	adantl 481	. . . . 5 ⊢ ((𝐴 ⊆ (ℤ_≥‘𝑀) ∧ 𝑀 ∈ ℤ) → seq𝑀( + , ((ℤ_≥‘𝑀) × {0})) ⇝ 0)
16		funbrfv 6943	. . . . 5 ⊢ (Fun ⇝ → (seq𝑀( + , ((ℤ_≥‘𝑀) × {0})) ⇝ 0 → ( ⇝ ‘seq𝑀( + , ((ℤ_≥‘𝑀) × {0}))) = 0))
17	13, 15, 16	mpsyl 68	. . . 4 ⊢ ((𝐴 ⊆ (ℤ_≥‘𝑀) ∧ 𝑀 ∈ ℤ) → ( ⇝ ‘seq𝑀( + , ((ℤ_≥‘𝑀) × {0}))) = 0)
18	10, 17	eqtrd 2768	. . 3 ⊢ ((𝐴 ⊆ (ℤ_≥‘𝑀) ∧ 𝑀 ∈ ℤ) → Σ𝑘 ∈ 𝐴 0 = 0)
19		uzf 12850	. . . . . . . . 9 ⊢ ℤ_≥:ℤ⟶𝒫 ℤ
20	19	fdmi 6729	. . . . . . . 8 ⊢ dom ℤ_≥ = ℤ
21	20	eleq2i 2821	. . . . . . 7 ⊢ (𝑀 ∈ dom ℤ_≥ ↔ 𝑀 ∈ ℤ)
22		ndmfv 6927	. . . . . . 7 ⊢ (¬ 𝑀 ∈ dom ℤ_≥ → (ℤ_≥‘𝑀) = ∅)
23	21, 22	sylnbir 331	. . . . . 6 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ_≥‘𝑀) = ∅)
24	23	sseq2d 4011	. . . . 5 ⊢ (¬ 𝑀 ∈ ℤ → (𝐴 ⊆ (ℤ_≥‘𝑀) ↔ 𝐴 ⊆ ∅))
25	24	biimpac 478	. . . 4 ⊢ ((𝐴 ⊆ (ℤ_≥‘𝑀) ∧ ¬ 𝑀 ∈ ℤ) → 𝐴 ⊆ ∅)
26		ss0 4395	. . . 4 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅)
27		sumeq1 15662	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 0 = Σ𝑘 ∈ ∅ 0)
28		sum0 15694	. . . . 5 ⊢ Σ𝑘 ∈ ∅ 0 = 0
29	27, 28	eqtrdi 2784	. . . 4 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 0 = 0)
30	25, 26, 29	3syl 18	. . 3 ⊢ ((𝐴 ⊆ (ℤ_≥‘𝑀) ∧ ¬ 𝑀 ∈ ℤ) → Σ𝑘 ∈ 𝐴 0 = 0)
31	18, 30	pm2.61dan 812	. 2 ⊢ (𝐴 ⊆ (ℤ_≥‘𝑀) → Σ𝑘 ∈ 𝐴 0 = 0)
32		fz1f1o 15683	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
33		eqidd 2729	. . . . . . . . 9 ⊢ (𝑘 = (𝑓‘𝑛) → 0 = 0)
34		simpl 482	. . . . . . . . 9 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (♯‘𝐴) ∈ ℕ)
35		simpr 484	. . . . . . . . 9 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
36		0cnd 11232	. . . . . . . . 9 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑘 ∈ 𝐴) → 0 ∈ ℂ)
37		elfznn 13557	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) → 𝑛 ∈ ℕ)
38	4	fvconst2 7211	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((ℕ × {0})‘𝑛) = 0)
39	37, 38	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) → ((ℕ × {0})‘𝑛) = 0)
40	39	adantl 481	. . . . . . . . 9 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((ℕ × {0})‘𝑛) = 0)
41	33, 34, 35, 36, 40	fsum 15693	. . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 0 = (seq1( + , (ℕ × {0}))‘(♯‘𝐴)))
42		nnuz 12890	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
43	42	ser0 14046	. . . . . . . . 9 ⊢ ((♯‘𝐴) ∈ ℕ → (seq1( + , (ℕ × {0}))‘(♯‘𝐴)) = 0)
44	43	adantr 480	. . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (seq1( + , (ℕ × {0}))‘(♯‘𝐴)) = 0)
45	41, 44	eqtrd 2768	. . . . . . 7 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 0 = 0)
46	45	ex 412	. . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 0 = 0))
47	46	exlimdv 1929	. . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 0 = 0))
48	47	imp 406	. . . 4 ⊢ (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 0 = 0)
49	29, 48	jaoi 856	. . 3 ⊢ ((𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑘 ∈ 𝐴 0 = 0)
50	32, 49	syl 17	. 2 ⊢ (𝐴 ∈ Fin → Σ𝑘 ∈ 𝐴 0 = 0)
51	31, 50	jaoi 856	1 ⊢ ((𝐴 ⊆ (ℤ_≥‘𝑀) ∨ 𝐴 ∈ Fin) → Σ𝑘 ∈ 𝐴 0 = 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 846 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ⊆ wss 3945 ∅c0 4319 ifcif 4525 𝒫 cpw 4599 {csn 4625 class class class wbr 5143 × cxp 5671 dom cdm 5673 Fun wfun 6537 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7415 Fincfn 8958 ℂcc 11131 0cc0 11133 1c1 11134 + caddc 11136 ℕcn 12237 ℤcz 12583 ℤ_≥cuz 12847 ...cfz 13511 seqcseq 13993 ♯chash 14316 ⇝ cli 15455 Σcsu 15659

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-sup 9460 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-fz 13512 df-fzo 13655 df-seq 13994 df-exp 14054 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-clim 15459 df-sum 15660

This theorem is referenced by: fsum00 15771 fsumdvds 16279 pwp1fsum 16362 pcfac 16862 ovoliunnul 25430 vitalilem5 25535 itg1addlem5 25624 itg10a 25634 itg0 25703 itgz 25704 plymullem1 26142 coemullem 26178 logtayl 26588 ftalem5 27003 chp1 27093 logexprlim 27152 bposlem2 27212 rpvmasumlem 27414 axcgrid 28721 axlowdimlem16 28762 indsumin 33636 plymulx0 34174 signsplypnf 34177 fsum2dsub 34234 knoppndvlem6 35987 volsupnfl 37133 binomcxplemnn0 43777 binomcxplemnotnn0 43784 sumnnodd 45009 stoweidlem37 45416 fourierdlem103 45588 fourierdlem104 45589 etransclem24 45637 etransclem32 45645 etransclem35 45648 sge0z 45754 aacllem 48225

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