isumrpcl - Intuitionistic Logic Explorer

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Theorem isumrpcl 12048

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 2322	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
5		eluzelz 9758	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
6	4, 5	syl 14	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7		uzss 9770	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
8	4, 7	syl 14	. . . . . 6 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
9	8, 1, 3	3sstr4g 3268	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑍)
10	9	sselda 3225	. . . 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 9924	. . . 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 2306	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ⁺)
18	17	rpcnd 9926	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
19	3, 2, 18	iserex 11893	. . . 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 11983	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ)
22		fveq2 5635	. . . 4 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁))
23	22	eleq1d 2298	. . 3 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℝ⁺ ↔ (𝐹‘𝑁) ∈ ℝ⁺))
24	17	ralrimiva 2603	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℝ⁺)
25	23, 24, 2	rspcdva 2913	. 2 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ⁺)
26	8	sselda 3225	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → 𝑘 ∈ (ℤ_≥‘𝑀))
27	26, 3	eleqtrrdi 2323	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → 𝑘 ∈ 𝑍)
28	27, 17	syldan 282	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ⁺)
29		rpaddcl 9905	. . . . 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 10717	. . 3 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) = (𝐹‘𝑁))
32		uzid 9763	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ_≥‘𝑁))
33	6, 32	syl 14	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑁))
34	33, 1	eleqtrrdi 2323	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑊)
35	15	recnd 8201	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℂ)
36	1, 6, 12, 35, 20	isumclim2 11976	. . . 4 ⊢ (𝜑 → seq𝑁( + , 𝐹) ⇝ Σ𝑘 ∈ 𝑊 𝐴)
37	9	sseld 3224	. . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ 𝑊 → 𝑚 ∈ 𝑍))
38		fveq2 5635	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
39	38	eleq1d 2298	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) ∈ ℝ⁺ ↔ (𝐹‘𝑚) ∈ ℝ⁺))
40	39	rspcv 2904	. . . . . . 7 ⊢ (𝑚 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℝ⁺ → (𝐹‘𝑚) ∈ ℝ⁺))
41	37, 24, 40	syl6ci 1488	. . . . . 6 ⊢ (𝜑 → (𝑚 ∈ 𝑊 → (𝐹‘𝑚) ∈ ℝ⁺))
42	41	imp 124	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐹‘𝑚) ∈ ℝ⁺)
43	42	rpred 9924	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐹‘𝑚) ∈ ℝ)
44	42	rpge0d 9928	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → 0 ≤ (𝐹‘𝑚))
45	1, 34, 36, 43, 44	climserle 11899	. . 3 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) ≤ Σ𝑘 ∈ 𝑊 𝐴)
46	31, 45	eqbrtrrd 4110	. 2 ⊢ (𝜑 → (𝐹‘𝑁) ≤ Σ𝑘 ∈ 𝑊 𝐴)
47	21, 25, 46	rpgecld 9964	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ⁺)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1395 ∈ wcel 2200 ∀wral 2508 ⊆ wss 3198 dom cdm 4723 ‘cfv 5324 (class class class)co 6013 ℝcr 8024 + caddc 8028 ≤ cle 8208 ℤcz 9472 ℤ_≥cuz 9748 ℝ⁺crp 9881 seqcseq 10702 ⇝ cli 11832 Σcsu 11907

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 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-mulrcl 8124 ax-addcom 8125 ax-mulcom 8126 ax-addass 8127 ax-mulass 8128 ax-distr 8129 ax-i2m1 8130 ax-0lt1 8131 ax-1rid 8132 ax-0id 8133 ax-rnegex 8134 ax-precex 8135 ax-cnre 8136 ax-pre-ltirr 8137 ax-pre-ltwlin 8138 ax-pre-lttrn 8139 ax-pre-apti 8140 ax-pre-ltadd 8141 ax-pre-mulgt0 8142 ax-pre-mulext 8143 ax-arch 8144 ax-caucvg 8145

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-isom 5333 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-irdg 6531 df-frec 6552 df-1o 6577 df-oadd 6581 df-er 6697 df-en 6905 df-dom 6906 df-fin 6907 df-pnf 8209 df-mnf 8210 df-xr 8211 df-ltxr 8212 df-le 8213 df-sub 8345 df-neg 8346 df-reap 8748 df-ap 8755 df-div 8846 df-inn 9137 df-2 9195 df-3 9196 df-4 9197 df-n0 9396 df-z 9473 df-uz 9749 df-q 9847 df-rp 9882 df-fz 10237 df-fzo 10371 df-seqfrec 10703 df-exp 10794 df-ihash 11031 df-cj 11396 df-re 11397 df-im 11398 df-rsqrt 11552 df-abs 11553 df-clim 11833 df-sumdc 11908

This theorem is referenced by: effsumlt 12246 eirraplem 12331

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