isumrpcl - Intuitionistic Logic Explorer

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Theorem isumrpcl 12208

Description: The infinite sum of positive reals is positive. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)

**Hypotheses**
Ref	Expression
isumrpcl.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
isumrpcl.2	⊢ 𝑊 = (ℤ_≥‘𝑁)
isumrpcl.3	⊢ (𝜑 → 𝑁 ∈ 𝑍)
isumrpcl.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
isumrpcl.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ⁺)
isumrpcl.6	⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

**Assertion**
Ref	Expression
isumrpcl	⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ⁺)

Distinct variable groups: 𝑘,𝐹 𝑘,𝑀 𝑘,𝑁 𝜑,𝑘 𝑘,𝑊 𝑘,𝑍 Allowed substitution hint: 𝐴(𝑘)

Proof of Theorem isumrpcl Dummy variables 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isumrpcl.2	. . 3 ⊢ 𝑊 = (ℤ_≥‘𝑁)
2		isumrpcl.3	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
3		isumrpcl.1	. . . . 5 ⊢ 𝑍 = (ℤ_≥‘𝑀)
4	2, 3	eleqtrdi 2327	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
5		eluzelz 9884	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
6	4, 5	syl 14	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7		uzss 9896	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
8	4, 7	syl 14	. . . . . 6 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
9	8, 1, 3	3sstr4g 3285	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑍)
10	9	sselda 3242	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑍)
11		isumrpcl.4	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
12	10, 11	syldan 282	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘𝑘) = 𝐴)
13		isumrpcl.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ⁺)
14	13	rpred 10050	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ)
15	10, 14	syldan 282	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℝ)
16		isumrpcl.6	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
17	11, 13	eqeltrd 2311	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ⁺)
18	17	rpcnd 10052	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
19	3, 2, 18	iserex 12052	. . . 4 ⊢ (𝜑 → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑁( + , 𝐹) ∈ dom ⇝ ))
20	16, 19	mpbid 147	. . 3 ⊢ (𝜑 → seq𝑁( + , 𝐹) ∈ dom ⇝ )
21	1, 6, 12, 15, 20	isumrecl 12143	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ)
22		fveq2 5675	. . . 4 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁))
23	22	eleq1d 2303	. . 3 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℝ⁺ ↔ (𝐹‘𝑁) ∈ ℝ⁺))
24	17	ralrimiva 2617	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℝ⁺)
25	23, 24, 2	rspcdva 2928	. 2 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ⁺)
26	8	sselda 3242	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → 𝑘 ∈ (ℤ_≥‘𝑀))
27	26, 3	eleqtrrdi 2328	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → 𝑘 ∈ 𝑍)
28	27, 17	syldan 282	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ⁺)
29		rpaddcl 10031	. . . . 5 ⊢ ((𝑘 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → (𝑘 + 𝑦) ∈ ℝ⁺)
30	29	adantl 277	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → (𝑘 + 𝑦) ∈ ℝ⁺)
31	6, 28, 30	seq3-1 10851	. . 3 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) = (𝐹‘𝑁))
32		uzid 9889	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ_≥‘𝑁))
33	6, 32	syl 14	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑁))
34	33, 1	eleqtrrdi 2328	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑊)
35	15	recnd 8318	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℂ)
36	1, 6, 12, 35, 20	isumclim2 12136	. . . 4 ⊢ (𝜑 → seq𝑁( + , 𝐹) ⇝ Σ𝑘 ∈ 𝑊 𝐴)
37	9	sseld 3241	. . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ 𝑊 → 𝑚 ∈ 𝑍))
38		fveq2 5675	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
39	38	eleq1d 2303	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) ∈ ℝ⁺ ↔ (𝐹‘𝑚) ∈ ℝ⁺))
40	39	rspcv 2919	. . . . . . 7 ⊢ (𝑚 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℝ⁺ → (𝐹‘𝑚) ∈ ℝ⁺))
41	37, 24, 40	syl6ci 1491	. . . . . 6 ⊢ (𝜑 → (𝑚 ∈ 𝑊 → (𝐹‘𝑚) ∈ ℝ⁺))
42	41	imp 124	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐹‘𝑚) ∈ ℝ⁺)
43	42	rpred 10050	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐹‘𝑚) ∈ ℝ)
44	42	rpge0d 10054	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → 0 ≤ (𝐹‘𝑚))
45	1, 34, 36, 43, 44	climserle 12058	. . 3 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) ≤ Σ𝑘 ∈ 𝑊 𝐴)
46	31, 45	eqbrtrrd 4138	. 2 ⊢ (𝜑 → (𝐹‘𝑁) ≤ Σ𝑘 ∈ 𝑊 𝐴)
47	21, 25, 46	rpgecld 10090	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ⁺)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1398 ∈ wcel 2205 ∀wral 2522 ⊆ wss 3214 dom cdm 4754 ‘cfv 5357 (class class class)co 6058 ℝcr 8142 + caddc 8146 ≤ cle 8325 ℤcz 9597 ℤ_≥cuz 9874 ℝ⁺crp 10007 seqcseq 10836 ⇝ cli 11991 Σcsu 12066

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-oadd 6664 df-er 6780 df-en 6989 df-dom 6990 df-fin 6991 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8463 df-neg 8464 df-reap 8867 df-ap 8874 df-div 8967 df-inn 9258 df-2 9316 df-3 9317 df-4 9318 df-n0 9517 df-z 9598 df-uz 9875 df-q 9973 df-rp 10008 df-fz 10365 df-fzo 10502 df-seqfrec 10837 df-exp 10928 df-ihash 11167 df-cj 11555 df-re 11556 df-im 11557 df-rsqrt 11711 df-abs 11712 df-clim 11992 df-sumdc 12067

This theorem is referenced by: effsumlt 12406 eirraplem 12491

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