isumrpcl - Intuitionistic Logic Explorer

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Theorem isumrpcl 11971

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 2302	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
5		eluzelz 9699	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
6	4, 5	syl 14	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7		uzss 9711	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
8	4, 7	syl 14	. . . . . 6 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
9	8, 1, 3	3sstr4g 3247	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑍)
10	9	sselda 3204	. . . 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 9860	. . . 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 2286	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ⁺)
18	17	rpcnd 9862	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
19	3, 2, 18	iserex 11816	. . . 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 11906	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ)
22		fveq2 5603	. . . 4 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁))
23	22	eleq1d 2278	. . 3 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℝ⁺ ↔ (𝐹‘𝑁) ∈ ℝ⁺))
24	17	ralrimiva 2583	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℝ⁺)
25	23, 24, 2	rspcdva 2892	. 2 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ⁺)
26	8	sselda 3204	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → 𝑘 ∈ (ℤ_≥‘𝑀))
27	26, 3	eleqtrrdi 2303	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → 𝑘 ∈ 𝑍)
28	27, 17	syldan 282	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ⁺)
29		rpaddcl 9841	. . . . 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 10651	. . 3 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) = (𝐹‘𝑁))
32		uzid 9704	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ_≥‘𝑁))
33	6, 32	syl 14	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑁))
34	33, 1	eleqtrrdi 2303	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑊)
35	15	recnd 8143	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℂ)
36	1, 6, 12, 35, 20	isumclim2 11899	. . . 4 ⊢ (𝜑 → seq𝑁( + , 𝐹) ⇝ Σ𝑘 ∈ 𝑊 𝐴)
37	9	sseld 3203	. . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ 𝑊 → 𝑚 ∈ 𝑍))
38		fveq2 5603	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
39	38	eleq1d 2278	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) ∈ ℝ⁺ ↔ (𝐹‘𝑚) ∈ ℝ⁺))
40	39	rspcv 2883	. . . . . . 7 ⊢ (𝑚 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℝ⁺ → (𝐹‘𝑚) ∈ ℝ⁺))
41	37, 24, 40	syl6ci 1468	. . . . . 6 ⊢ (𝜑 → (𝑚 ∈ 𝑊 → (𝐹‘𝑚) ∈ ℝ⁺))
42	41	imp 124	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐹‘𝑚) ∈ ℝ⁺)
43	42	rpred 9860	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐹‘𝑚) ∈ ℝ)
44	42	rpge0d 9864	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → 0 ≤ (𝐹‘𝑚))
45	1, 34, 36, 43, 44	climserle 11822	. . 3 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) ≤ Σ𝑘 ∈ 𝑊 𝐴)
46	31, 45	eqbrtrrd 4086	. 2 ⊢ (𝜑 → (𝐹‘𝑁) ≤ Σ𝑘 ∈ 𝑊 𝐴)
47	21, 25, 46	rpgecld 9900	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ⁺)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1375 ∈ wcel 2180 ∀wral 2488 ⊆ wss 3177 dom cdm 4696 ‘cfv 5294 (class class class)co 5974 ℝcr 7966 + caddc 7970 ≤ cle 8150 ℤcz 9414 ℤ_≥cuz 9690 ℝ⁺crp 9817 seqcseq 10636 ⇝ cli 11755 Σcsu 11830

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 617 ax-in2 618 ax-io 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-coll 4178 ax-sep 4181 ax-nul 4189 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-setind 4606 ax-iinf 4657 ax-cnex 8058 ax-resscn 8059 ax-1cn 8060 ax-1re 8061 ax-icn 8062 ax-addcl 8063 ax-addrcl 8064 ax-mulcl 8065 ax-mulrcl 8066 ax-addcom 8067 ax-mulcom 8068 ax-addass 8069 ax-mulass 8070 ax-distr 8071 ax-i2m1 8072 ax-0lt1 8073 ax-1rid 8074 ax-0id 8075 ax-rnegex 8076 ax-precex 8077 ax-cnre 8078 ax-pre-ltirr 8079 ax-pre-ltwlin 8080 ax-pre-lttrn 8081 ax-pre-apti 8082 ax-pre-ltadd 8083 ax-pre-mulgt0 8084 ax-pre-mulext 8085 ax-arch 8086 ax-caucvg 8087

This theorem depends on definitions: df-bi 117 df-dc 839 df-3or 984 df-3an 985 df-tru 1378 df-fal 1381 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-nel 2476 df-ral 2493 df-rex 2494 df-reu 2495 df-rmo 2496 df-rab 2497 df-v 2781 df-sbc 3009 df-csb 3105 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-nul 3472 df-if 3583 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-uni 3868 df-int 3903 df-iun 3946 df-br 4063 df-opab 4125 df-mpt 4126 df-tr 4162 df-id 4361 df-po 4364 df-iso 4365 df-iord 4434 df-on 4436 df-ilim 4437 df-suc 4439 df-iom 4660 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708 df-ima 4709 df-iota 5254 df-fun 5296 df-fn 5297 df-f 5298 df-f1 5299 df-fo 5300 df-f1o 5301 df-fv 5302 df-isom 5303 df-riota 5927 df-ov 5977 df-oprab 5978 df-mpo 5979 df-1st 6256 df-2nd 6257 df-recs 6421 df-irdg 6486 df-frec 6507 df-1o 6532 df-oadd 6536 df-er 6650 df-en 6858 df-dom 6859 df-fin 6860 df-pnf 8151 df-mnf 8152 df-xr 8153 df-ltxr 8154 df-le 8155 df-sub 8287 df-neg 8288 df-reap 8690 df-ap 8697 df-div 8788 df-inn 9079 df-2 9137 df-3 9138 df-4 9139 df-n0 9338 df-z 9415 df-uz 9691 df-q 9783 df-rp 9818 df-fz 10173 df-fzo 10307 df-seqfrec 10637 df-exp 10728 df-ihash 10965 df-cj 11319 df-re 11320 df-im 11321 df-rsqrt 11475 df-abs 11476 df-clim 11756 df-sumdc 11831

This theorem is referenced by: effsumlt 12169 eirraplem 12254

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