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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 5209 | . . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝐻‘𝑛)) = (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ 𝐴 (𝐹‘𝑛))) |
| 3 | climfsum.1 | . . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
| 4 | uzssz 12901 | . . . . . . . 8 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
| 5 | 3, 4 | eqsstri 3986 | . . . . . . 7 ⊢ 𝑍 ⊆ ℤ |
| 6 | zssre 12616 | . . . . . . 7 ⊢ ℤ ⊆ ℝ | |
| 7 | 5, 6 | sstri 3949 | . . . . . 6 ⊢ 𝑍 ⊆ ℝ |
| 8 | 7 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝑍 ⊆ ℝ) |
| 9 | climfsum.3 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
| 10 | fvexd 6903 | . . . . 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 486 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑀 ∈ ℤ) |
| 14 | climrel 15569 | . . . . . . . . . 10 ⊢ Rel ⇝ | |
| 15 | 14 | brrelex1i 5722 | . . . . . . . . 9 ⊢ (𝐹 ⇝ 𝐵 → 𝐹 ∈ V) |
| 16 | 11, 15 | syl 18 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐹 ∈ V) |
| 17 | eqid 2766 | . . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) = (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) | |
| 18 | 3, 17 | climmpt 15648 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ V) → (𝐹 ⇝ 𝐵 ↔ (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝ 𝐵)) |
| 19 | 13, 16, 18 | syl2anc 596 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹 ⇝ 𝐵 ↔ (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝ 𝐵)) |
| 20 | 11, 19 | mpbid 235 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝ 𝐵) |
| 21 | climfsum.7 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑛 ∈ 𝑍)) → (𝐹‘𝑛) ∈ ℂ) | |
| 22 | 21 | anassrs 473 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℂ) |
| 23 | 22 | fmpttd 7117 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)):𝑍⟶ℂ) |
| 24 | 3, 13, 23 | rlimclim 15623 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝𝑟 𝐵 ↔ (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝ 𝐵)) |
| 25 | 20, 24 | mpbird 260 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝𝑟 𝐵) |
| 26 | 8, 9, 10, 25 | fsumrlim 15889 | . . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ 𝐴 (𝐹‘𝑛)) ⇝𝑟 Σ𝑘 ∈ 𝐴 𝐵) |
| 27 | 9 | adantr 486 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐴 ∈ Fin) |
| 28 | 21 | anass1rs 668 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑛) ∈ ℂ) |
| 29 | 27, 28 | fsumcl 15810 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Σ𝑘 ∈ 𝐴 (𝐹‘𝑛) ∈ ℂ) |
| 30 | 29 | fmpttd 7117 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ 𝐴 (𝐹‘𝑛)):𝑍⟶ℂ) |
| 31 | 3, 12, 30 | rlimclim 15623 | . . . 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 235 | . . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ 𝐴 (𝐹‘𝑛)) ⇝ Σ𝑘 ∈ 𝐴 𝐵) |
| 33 | 2, 32 | eqbrtrd 5138 | . 2 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝐻‘𝑛)) ⇝ Σ𝑘 ∈ 𝐴 𝐵) |
| 34 | climfsum.6 | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑊) | |
| 35 | eqid 2766 | . . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ (𝐻‘𝑛)) = (𝑛 ∈ 𝑍 ↦ (𝐻‘𝑛)) | |
| 36 | 3, 35 | climmpt 15648 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐻 ∈ 𝑊) → (𝐻 ⇝ Σ𝑘 ∈ 𝐴 𝐵 ↔ (𝑛 ∈ 𝑍 ↦ (𝐻‘𝑛)) ⇝ Σ𝑘 ∈ 𝐴 𝐵)) |
| 37 | 12, 34, 36 | syl2anc 596 | . 2 ⊢ (𝜑 → (𝐻 ⇝ Σ𝑘 ∈ 𝐴 𝐵 ↔ (𝑛 ∈ 𝑍 ↦ (𝐻‘𝑛)) ⇝ Σ𝑘 ∈ 𝐴 𝐵)) |
| 38 | 33, 37 | mpbird 260 | 1 ⊢ (𝜑 → 𝐻 ⇝ Σ𝑘 ∈ 𝐴 𝐵) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 Vcvv 3458 ⊆ wss 3908 class class class wbr 5114 ↦ cmpt 5197 ‘cfv 6543 Fincfn 8952 ℂcc 11116 ℝcr 11117 ℤcz 12609 ℤ≥cuz 12880 ⇝ cli 15561 ⇝𝑟 crli 15562 Σcsu 15763 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-inf2 9620 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-pm 8836 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-n0 12523 df-z 12610 df-uz 12881 df-rp 13035 df-fz 13554 df-fzo 13702 df-fl 13845 df-seq 14058 df-exp 14118 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-clim 15565 df-rlim 15566 df-sum 15764 |
| This theorem is used by: itg1climres 25903 plyeq0lem 26397 |
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