isumrpcl - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > isumrpcl		GIF version

Theorem isumrpcl 12263

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 2331	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
5		eluzelz 9933	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
6	4, 5	syl 14	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7		uzss 9945	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
8	4, 7	syl 14	. . . . . 6 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
9	8, 1, 3	3sstr4g 3291	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑍)
10	9	sselda 3248	. . . 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 10099	. . . 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 2315	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ⁺)
18	17	rpcnd 10101	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
19	3, 2, 18	iserex 12107	. . . 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 12198	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ)
22		fveq2 5695	. . . 4 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁))
23	22	eleq1d 2307	. . 3 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℝ⁺ ↔ (𝐹‘𝑁) ∈ ℝ⁺))
24	17	ralrimiva 2623	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℝ⁺)
25	23, 24, 2	rspcdva 2934	. 2 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ⁺)
26	8	sselda 3248	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → 𝑘 ∈ (ℤ_≥‘𝑀))
27	26, 3	eleqtrrdi 2332	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → 𝑘 ∈ 𝑍)
28	27, 17	syldan 282	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ⁺)
29		rpaddcl 10080	. . . . 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 10901	. . 3 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) = (𝐹‘𝑁))
32		uzid 9938	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ_≥‘𝑁))
33	6, 32	syl 14	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑁))
34	33, 1	eleqtrrdi 2332	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑊)
35	15	recnd 8354	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℂ)
36	1, 6, 12, 35, 20	isumclim2 12191	. . . 4 ⊢ (𝜑 → seq𝑁( + , 𝐹) ⇝ Σ𝑘 ∈ 𝑊 𝐴)
37	9	sseld 3247	. . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ 𝑊 → 𝑚 ∈ 𝑍))
38		fveq2 5695	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
39	38	eleq1d 2307	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) ∈ ℝ⁺ ↔ (𝐹‘𝑚) ∈ ℝ⁺))
40	39	rspcv 2925	. . . . . . 7 ⊢ (𝑚 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℝ⁺ → (𝐹‘𝑚) ∈ ℝ⁺))
41	37, 24, 40	syl6ci 1495	. . . . . 6 ⊢ (𝜑 → (𝑚 ∈ 𝑊 → (𝐹‘𝑚) ∈ ℝ⁺))
42	41	imp 124	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐹‘𝑚) ∈ ℝ⁺)
43	42	rpred 10099	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐹‘𝑚) ∈ ℝ)
44	42	rpge0d 10103	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → 0 ≤ (𝐹‘𝑚))
45	1, 34, 36, 43, 44	climserle 12113	. . 3 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) ≤ Σ𝑘 ∈ 𝑊 𝐴)
46	31, 45	eqbrtrrd 4154	. 2 ⊢ (𝜑 → (𝐹‘𝑁) ≤ Σ𝑘 ∈ 𝑊 𝐴)
47	21, 25, 46	rpgecld 10139	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ⁺)

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 = wceq 1402 ∈ wcel 2209 ∀wral 2528 ⊆ wss 3220 dom cdm 4774 ‘cfv 5377 (class class class)co 6085 ℝcr 8178 + caddc 8182 ≤ cle 8361 ℤcz 9646 ℤ_≥cuz 9923 ℝ⁺crp 10056 seqcseq 10886 ⇝ cli 12046 Σcsu 12121

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-oadd 6691 df-er 6807 df-en 7023 df-dom 7024 df-fin 7025 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8904 df-ap 8911 df-div 9004 df-inn 9306 df-2 9364 df-3 9365 df-4 9366 df-n0 9566 df-z 9647 df-uz 9924 df-q 10022 df-rp 10057 df-fz 10414 df-fzo 10552 df-seqfrec 10887 df-exp 10978 df-ihash 11217 df-cj 11609 df-re 11610 df-im 11611 df-rsqrt 11766 df-abs 11767 df-clim 12047 df-sumdc 12122

This theorem is used by: effsumlt 12461 eirraplem 12546

W3C validator