isumclim3 - Intuitionistic Logic Explorer

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Theorem isumclim3 12134

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 12005	. . 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 2235	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
7		isumclim3.4	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
8	7	fmpttd 5837	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐴):𝑍⟶ℂ)
9	8	ffvelcdmda 5817	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
10	4, 5, 6, 9	isum 12096	. . 3 ⊢ (𝜑 → Σ𝑚 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ( ⇝ ‘seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))))
11	7	ralrimiva 2617	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ)
12		sumfct 12084	. . . 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 10835	. . . . . . 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 5699	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (𝑀...𝑗) → (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
19		fzssuz 10420	. . . . . . . . . . . . . 14 ⊢ (𝑀...𝑗) ⊆ (ℤ_≥‘𝑀)
20	19, 4	sseqtrri 3277	. . . . . . . . . . . . 13 ⊢ (𝑀...𝑗) ⊆ 𝑍
21		resmpt 5091	. . . . . . . . . . . . 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 5676	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗))‘𝑚) = ((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚)
24	18, 23	eqtr3di 2282	. . . . . . . . . 10 ⊢ (𝑚 ∈ (𝑀...𝑗) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚))
25	24	sumeq2i 12074	. . . . . . . . 9 ⊢ Σ𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚)
26		ssralv 3306	. . . . . . . . . . 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 12084	. . . . . . . . . 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 2279	. . . . . . . 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 2235	. . . . . . . 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 2327	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑀))
35	4	eleq2i 2301	. . . . . . . . . 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 12108	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → Σ𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗))
39	16, 31, 38	3eqtr2rd 2274	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗) = (𝐹‘𝑗))
40	4, 15, 1, 5, 39	climeq 12009	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥 ↔ 𝐹 ⇝ 𝑥))
41	40	iotabidv 5340	. . . 4 ⊢ (𝜑 → (℩𝑥seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥) = (℩𝑥𝐹 ⇝ 𝑥))
42		df-fv 5365	. . . 4 ⊢ ( ⇝ ‘seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))) = (℩𝑥seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥)
43		df-fv 5365	. . . 4 ⊢ ( ⇝ ‘𝐹) = (℩𝑥𝐹 ⇝ 𝑥)
44	41, 42, 43	3eqtr4g 2292	. . 3 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))) = ( ⇝ ‘𝐹))
45	10, 13, 44	3eqtr3d 2275	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = ( ⇝ ‘𝐹))
46	3, 45	breqtrrd 4142	1 ⊢ (𝜑 → 𝐹 ⇝ Σ𝑘 ∈ 𝑍 𝐴)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1398 ∈ wcel 2205 ∀wral 2522 Vcvv 2815 ⊆ wss 3214 class class class wbr 4114 ↦ cmpt 4176 dom cdm 4754 ↾ cres 4756 ℩cio 5315 ‘cfv 5357 (class class class)co 6058 ℂcc 8141 + caddc 8146 ℤcz 9594 ℤ_≥cuz 9871 ...cfz 10361 seqcseq 10833 ⇝ cli 11988 Σcsu 12063

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 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

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 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-oadd 6664 df-er 6780 df-en 6989 df-dom 6990 df-fin 6991 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-n0 9514 df-z 9595 df-uz 9872 df-q 9970 df-rp 10005 df-fz 10362 df-fzo 10499 df-seqfrec 10834 df-exp 10925 df-ihash 11164 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709 df-clim 11989 df-sumdc 12064

This theorem is referenced by: (None)

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