isumclim3 - Intuitionistic Logic Explorer

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Theorem isumclim3 11929

Description: The sequence of partial finite sums of a converging infinite series converges to the infinite sum of the series. Note that 𝑗 must not occur in 𝐴. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)

**Hypotheses**
Ref	Expression
isumclim3.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
isumclim3.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
isumclim3.3	⊢ (𝜑 → 𝐹 ∈ dom ⇝ )
isumclim3.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
isumclim3.5	⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = Σ𝑘 ∈ (𝑀...𝑗)𝐴)

**Assertion**
Ref	Expression
isumclim3	⊢ (𝜑 → 𝐹 ⇝ Σ𝑘 ∈ 𝑍 𝐴)

Distinct variable groups: 𝐴,𝑗 𝑗,𝑘,𝑀 𝜑,𝑗,𝑘 𝑗,𝑍,𝑘 𝑗,𝐹 Allowed substitution hints: 𝐴(𝑘) 𝐹(𝑘)

Proof of Theorem isumclim3 Dummy variables 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isumclim3.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )
2		climdm 11801	. . 3 ⊢ (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹))
3	1, 2	sylib 122	. 2 ⊢ (𝜑 → 𝐹 ⇝ ( ⇝ ‘𝐹))
4		isumclim3.1	. . . 4 ⊢ 𝑍 = (ℤ_≥‘𝑀)
5		isumclim3.2	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
6		eqidd 2230	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
7		isumclim3.4	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
8	7	fmpttd 5789	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐴):𝑍⟶ℂ)
9	8	ffvelcdmda 5769	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
10	4, 5, 6, 9	isum 11891	. . 3 ⊢ (𝜑 → Σ𝑚 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ( ⇝ ‘seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))))
11	7	ralrimiva 2603	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ)
12		sumfct 11880	. . . 4 ⊢ (∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ → Σ𝑚 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑘 ∈ 𝑍 𝐴)
13	11, 12	syl 14	. . 3 ⊢ (𝜑 → Σ𝑚 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑘 ∈ 𝑍 𝐴)
14		seqex 10666	. . . . . . 7 ⊢ seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ∈ V
15	14	a1i 9	. . . . . 6 ⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ∈ V)
16		isumclim3.5	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = Σ𝑘 ∈ (𝑀...𝑗)𝐴)
17		simpl 109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝜑)
18		fvres 5650	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (𝑀...𝑗) → (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
19		fzssuz 10257	. . . . . . . . . . . . . 14 ⊢ (𝑀...𝑗) ⊆ (ℤ_≥‘𝑀)
20	19, 4	sseqtrri 3259	. . . . . . . . . . . . 13 ⊢ (𝑀...𝑗) ⊆ 𝑍
21		resmpt 5052	. . . . . . . . . . . . 13 ⊢ ((𝑀...𝑗) ⊆ 𝑍 → ((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗)) = (𝑘 ∈ (𝑀...𝑗) ↦ 𝐴))
22	20, 21	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗)) = (𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)
23	22	fveq1i 5627	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗))‘𝑚) = ((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚)
24	18, 23	eqtr3di 2277	. . . . . . . . . 10 ⊢ (𝑚 ∈ (𝑀...𝑗) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚))
25	24	sumeq2i 11870	. . . . . . . . 9 ⊢ Σ𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚)
26		ssralv 3288	. . . . . . . . . . 11 ⊢ ((𝑀...𝑗) ⊆ 𝑍 → (∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ → ∀𝑘 ∈ (𝑀...𝑗)𝐴 ∈ ℂ))
27	20, 11, 26	mpsyl 65	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑗)𝐴 ∈ ℂ)
28		sumfct 11880	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ (𝑀...𝑗)𝐴 ∈ ℂ → Σ𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚) = Σ𝑘 ∈ (𝑀...𝑗)𝐴)
29	27, 28	syl 14	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚) = Σ𝑘 ∈ (𝑀...𝑗)𝐴)
30	25, 29	eqtrid 2274	. . . . . . . 8 ⊢ (𝜑 → Σ𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑘 ∈ (𝑀...𝑗)𝐴)
31	17, 30	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → Σ𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑘 ∈ (𝑀...𝑗)𝐴)
32		eqidd 2230	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
33		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
34	33, 4	eleqtrdi 2322	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑀))
35	4	eleq2i 2296	. . . . . . . . . 10 ⊢ (𝑚 ∈ 𝑍 ↔ 𝑚 ∈ (ℤ_≥‘𝑀))
36	35	biimpri 133	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ_≥‘𝑀) → 𝑚 ∈ 𝑍)
37	17, 36, 9	syl2an 289	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑀)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
38	32, 34, 37	fsum3ser 11903	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → Σ𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗))
39	16, 31, 38	3eqtr2rd 2269	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗) = (𝐹‘𝑗))
40	4, 15, 1, 5, 39	climeq 11805	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥 ↔ 𝐹 ⇝ 𝑥))
41	40	iotabidv 5300	. . . 4 ⊢ (𝜑 → (℩𝑥seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥) = (℩𝑥𝐹 ⇝ 𝑥))
42		df-fv 5325	. . . 4 ⊢ ( ⇝ ‘seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))) = (℩𝑥seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥)
43		df-fv 5325	. . . 4 ⊢ ( ⇝ ‘𝐹) = (℩𝑥𝐹 ⇝ 𝑥)
44	41, 42, 43	3eqtr4g 2287	. . 3 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))) = ( ⇝ ‘𝐹))
45	10, 13, 44	3eqtr3d 2270	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = ( ⇝ ‘𝐹))
46	3, 45	breqtrrd 4110	1 ⊢ (𝜑 → 𝐹 ⇝ Σ𝑘 ∈ 𝑍 𝐴)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1395 ∈ wcel 2200 ∀wral 2508 Vcvv 2799 ⊆ wss 3197 class class class wbr 4082 ↦ cmpt 4144 dom cdm 4718 ↾ cres 4720 ℩cio 5275 ‘cfv 5317 (class class class)co 6000 ℂcc 7993 + caddc 7998 ℤcz 9442 ℤ_≥cuz 9718 ...cfz 10200 seqcseq 10664 ⇝ cli 11784 Σcsu 11859

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 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113 ax-arch 8114 ax-caucvg 8115

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 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4383 df-po 4386 df-iso 4387 df-iord 4456 df-on 4458 df-ilim 4459 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-isom 5326 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-recs 6449 df-irdg 6514 df-frec 6535 df-1o 6560 df-oadd 6564 df-er 6678 df-en 6886 df-dom 6887 df-fin 6888 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 df-div 8816 df-inn 9107 df-2 9165 df-3 9166 df-4 9167 df-n0 9366 df-z 9443 df-uz 9719 df-q 9811 df-rp 9846 df-fz 10201 df-fzo 10335 df-seqfrec 10665 df-exp 10756 df-ihash 10993 df-cj 11348 df-re 11349 df-im 11350 df-rsqrt 11504 df-abs 11505 df-clim 11785 df-sumdc 11860

This theorem is referenced by: (None)

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