isumrpcl - Intuitionistic Logic Explorer

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Theorem isumrpcl 12173

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 2325	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
5		eluzelz 9859	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
6	4, 5	syl 14	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7		uzss 9871	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
8	4, 7	syl 14	. . . . . 6 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
9	8, 1, 3	3sstr4g 3280	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑍)
10	9	sselda 3237	. . . 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 10025	. . . 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 2309	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ⁺)
18	17	rpcnd 10027	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
19	3, 2, 18	iserex 12017	. . . 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 12108	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ)
22		fveq2 5669	. . . 4 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁))
23	22	eleq1d 2301	. . 3 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℝ⁺ ↔ (𝐹‘𝑁) ∈ ℝ⁺))
24	17	ralrimiva 2615	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℝ⁺)
25	23, 24, 2	rspcdva 2925	. 2 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ⁺)
26	8	sselda 3237	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → 𝑘 ∈ (ℤ_≥‘𝑀))
27	26, 3	eleqtrrdi 2326	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → 𝑘 ∈ 𝑍)
28	27, 17	syldan 282	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ⁺)
29		rpaddcl 10006	. . . . 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 10820	. . 3 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) = (𝐹‘𝑁))
32		uzid 9864	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ_≥‘𝑁))
33	6, 32	syl 14	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑁))
34	33, 1	eleqtrrdi 2326	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑊)
35	15	recnd 8298	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℂ)
36	1, 6, 12, 35, 20	isumclim2 12101	. . . 4 ⊢ (𝜑 → seq𝑁( + , 𝐹) ⇝ Σ𝑘 ∈ 𝑊 𝐴)
37	9	sseld 3236	. . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ 𝑊 → 𝑚 ∈ 𝑍))
38		fveq2 5669	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
39	38	eleq1d 2301	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) ∈ ℝ⁺ ↔ (𝐹‘𝑚) ∈ ℝ⁺))
40	39	rspcv 2916	. . . . . . 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 10025	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐹‘𝑚) ∈ ℝ)
44	42	rpge0d 10029	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → 0 ≤ (𝐹‘𝑚))
45	1, 34, 36, 43, 44	climserle 12023	. . 3 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) ≤ Σ𝑘 ∈ 𝑊 𝐴)
46	31, 45	eqbrtrrd 4132	. 2 ⊢ (𝜑 → (𝐹‘𝑁) ≤ Σ𝑘 ∈ 𝑊 𝐴)
47	21, 25, 46	rpgecld 10065	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ⁺)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1398 ∈ wcel 2203 ∀wral 2520 ⊆ wss 3210 dom cdm 4748 ‘cfv 5351 (class class class)co 6049 ℝcr 8122 + caddc 8126 ≤ cle 8305 ℤcz 9573 ℤ_≥cuz 9849 ℝ⁺crp 9982 seqcseq 10805 ⇝ cli 11956 Σcsu 12031

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 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-nul 4235 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709 ax-cnex 8214 ax-resscn 8215 ax-1cn 8216 ax-1re 8217 ax-icn 8218 ax-addcl 8219 ax-addrcl 8220 ax-mulcl 8221 ax-mulrcl 8222 ax-addcom 8223 ax-mulcom 8224 ax-addass 8225 ax-mulass 8226 ax-distr 8227 ax-i2m1 8228 ax-0lt1 8229 ax-1rid 8230 ax-0id 8231 ax-rnegex 8232 ax-precex 8233 ax-cnre 8234 ax-pre-ltirr 8235 ax-pre-ltwlin 8236 ax-pre-lttrn 8237 ax-pre-apti 8238 ax-pre-ltadd 8239 ax-pre-mulgt0 8240 ax-pre-mulext 8241 ax-arch 8242 ax-caucvg 8243

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 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-nul 3508 df-if 3620 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-br 4109 df-opab 4171 df-mpt 4172 df-tr 4208 df-id 4413 df-po 4416 df-iso 4417 df-iord 4486 df-on 4488 df-ilim 4489 df-suc 4491 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-isom 5360 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-1st 6333 df-2nd 6334 df-recs 6535 df-irdg 6600 df-frec 6621 df-1o 6646 df-oadd 6650 df-er 6766 df-en 6975 df-dom 6976 df-fin 6977 df-pnf 8306 df-mnf 8307 df-xr 8308 df-ltxr 8309 df-le 8310 df-sub 8442 df-neg 8443 df-reap 8845 df-ap 8852 df-div 8943 df-inn 9234 df-2 9292 df-3 9293 df-4 9294 df-n0 9493 df-z 9574 df-uz 9850 df-q 9948 df-rp 9983 df-fz 10339 df-fzo 10473 df-seqfrec 10806 df-exp 10897 df-ihash 11134 df-cj 11520 df-re 11521 df-im 11522 df-rsqrt 11676 df-abs 11677 df-clim 11957 df-sumdc 12032

This theorem is referenced by: effsumlt 12371 eirraplem 12456

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