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Mirrors > Home > MPE Home > Th. List > climfsum | Structured version Visualization version GIF version |
Description: Limit of a finite sum of converging sequences. Note that 𝐹(𝑘) is a collection of functions with implicit parameter 𝑘, each of which converges to 𝐵(𝑘) as 𝑛 ⇝ +∞. (Contributed by Mario Carneiro, 22-Jul-2014.) (Proof shortened by Mario Carneiro, 22-May-2016.) |
Ref | Expression |
---|---|
climfsum.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climfsum.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climfsum.3 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
climfsum.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐹 ⇝ 𝐵) |
climfsum.6 | ⊢ (𝜑 → 𝐻 ∈ 𝑊) |
climfsum.7 | ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑛 ∈ 𝑍)) → (𝐹‘𝑛) ∈ ℂ) |
climfsum.8 | ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = Σ𝑘 ∈ 𝐴 (𝐹‘𝑛)) |
Ref | Expression |
---|---|
climfsum | ⊢ (𝜑 → 𝐻 ⇝ Σ𝑘 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climfsum.8 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = Σ𝑘 ∈ 𝐴 (𝐹‘𝑛)) | |
2 | 1 | mpteq2dva 5152 | . . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝐻‘𝑛)) = (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ 𝐴 (𝐹‘𝑛))) |
3 | climfsum.1 | . . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | uzssz 12252 | . . . . . . . 8 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
5 | 3, 4 | eqsstri 3998 | . . . . . . 7 ⊢ 𝑍 ⊆ ℤ |
6 | zssre 11976 | . . . . . . 7 ⊢ ℤ ⊆ ℝ | |
7 | 5, 6 | sstri 3973 | . . . . . 6 ⊢ 𝑍 ⊆ ℝ |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝑍 ⊆ ℝ) |
9 | climfsum.3 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
10 | fvexd 6678 | . . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑘 ∈ 𝐴)) → (𝐹‘𝑛) ∈ V) | |
11 | climfsum.5 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐹 ⇝ 𝐵) | |
12 | climfsum.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
13 | 12 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑀 ∈ ℤ) |
14 | climrel 14837 | . . . . . . . . . 10 ⊢ Rel ⇝ | |
15 | 14 | brrelex1i 5601 | . . . . . . . . 9 ⊢ (𝐹 ⇝ 𝐵 → 𝐹 ∈ V) |
16 | 11, 15 | syl 17 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐹 ∈ V) |
17 | eqid 2818 | . . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) = (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) | |
18 | 3, 17 | climmpt 14916 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ V) → (𝐹 ⇝ 𝐵 ↔ (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝ 𝐵)) |
19 | 13, 16, 18 | syl2anc 584 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹 ⇝ 𝐵 ↔ (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝ 𝐵)) |
20 | 11, 19 | mpbid 233 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝ 𝐵) |
21 | climfsum.7 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑛 ∈ 𝑍)) → (𝐹‘𝑛) ∈ ℂ) | |
22 | 21 | anassrs 468 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℂ) |
23 | 22 | fmpttd 6871 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)):𝑍⟶ℂ) |
24 | 3, 13, 23 | rlimclim 14891 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝𝑟 𝐵 ↔ (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝ 𝐵)) |
25 | 20, 24 | mpbird 258 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝𝑟 𝐵) |
26 | 8, 9, 10, 25 | fsumrlim 15154 | . . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ 𝐴 (𝐹‘𝑛)) ⇝𝑟 Σ𝑘 ∈ 𝐴 𝐵) |
27 | 9 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐴 ∈ Fin) |
28 | 21 | anass1rs 651 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑛) ∈ ℂ) |
29 | 27, 28 | fsumcl 15078 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Σ𝑘 ∈ 𝐴 (𝐹‘𝑛) ∈ ℂ) |
30 | 29 | fmpttd 6871 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ 𝐴 (𝐹‘𝑛)):𝑍⟶ℂ) |
31 | 3, 12, 30 | rlimclim 14891 | . . . 4 ⊢ (𝜑 → ((𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ 𝐴 (𝐹‘𝑛)) ⇝𝑟 Σ𝑘 ∈ 𝐴 𝐵 ↔ (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ 𝐴 (𝐹‘𝑛)) ⇝ Σ𝑘 ∈ 𝐴 𝐵)) |
32 | 26, 31 | mpbid 233 | . . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ 𝐴 (𝐹‘𝑛)) ⇝ Σ𝑘 ∈ 𝐴 𝐵) |
33 | 2, 32 | eqbrtrd 5079 | . 2 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝐻‘𝑛)) ⇝ Σ𝑘 ∈ 𝐴 𝐵) |
34 | climfsum.6 | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑊) | |
35 | eqid 2818 | . . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ (𝐻‘𝑛)) = (𝑛 ∈ 𝑍 ↦ (𝐻‘𝑛)) | |
36 | 3, 35 | climmpt 14916 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐻 ∈ 𝑊) → (𝐻 ⇝ Σ𝑘 ∈ 𝐴 𝐵 ↔ (𝑛 ∈ 𝑍 ↦ (𝐻‘𝑛)) ⇝ Σ𝑘 ∈ 𝐴 𝐵)) |
37 | 12, 34, 36 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐻 ⇝ Σ𝑘 ∈ 𝐴 𝐵 ↔ (𝑛 ∈ 𝑍 ↦ (𝐻‘𝑛)) ⇝ Σ𝑘 ∈ 𝐴 𝐵)) |
38 | 33, 37 | mpbird 258 | 1 ⊢ (𝜑 → 𝐻 ⇝ Σ𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 class class class wbr 5057 ↦ cmpt 5137 ‘cfv 6348 Fincfn 8497 ℂcc 10523 ℝcr 10524 ℤcz 11969 ℤ≥cuz 12231 ⇝ cli 14829 ⇝𝑟 crli 14830 Σcsu 15030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-rlim 14834 df-sum 15031 |
This theorem is referenced by: itg1climres 24242 plyeq0lem 24727 |
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