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Mirrors > Home > MPE Home > Th. List > isumdivc | Structured version Visualization version GIF version |
Description: An infinite sum divided by a constant. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.) |
Ref | Expression |
---|---|
isumcl.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
isumcl.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
isumcl.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) |
isumcl.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
isumcl.5 | ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
summulc.6 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
isumdivc.7 | ⊢ (𝜑 → 𝐵 ≠ 0) |
Ref | Expression |
---|---|
isumdivc | ⊢ (𝜑 → (Σ𝑘 ∈ 𝑍 𝐴 / 𝐵) = Σ𝑘 ∈ 𝑍 (𝐴 / 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumcl.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | isumcl.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | isumcl.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | |
4 | isumcl.4 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) | |
5 | isumcl.5 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) | |
6 | summulc.6 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
7 | isumdivc.7 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 0) | |
8 | 6, 7 | reccld 11121 | . . 3 ⊢ (𝜑 → (1 / 𝐵) ∈ ℂ) |
9 | 1, 2, 3, 4, 5, 8 | isummulc1 14870 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ 𝑍 𝐴 · (1 / 𝐵)) = Σ𝑘 ∈ 𝑍 (𝐴 · (1 / 𝐵))) |
10 | 1, 2, 3, 4, 5 | isumcl 14868 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 ∈ ℂ) |
11 | 10, 6, 7 | divrecd 11131 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ 𝑍 𝐴 / 𝐵) = (Σ𝑘 ∈ 𝑍 𝐴 · (1 / 𝐵))) |
12 | 6 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) |
13 | 7 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ≠ 0) |
14 | 4, 12, 13 | divrecd 11131 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) |
15 | 14 | sumeq2dv 14811 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 (𝐴 / 𝐵) = Σ𝑘 ∈ 𝑍 (𝐴 · (1 / 𝐵))) |
16 | 9, 11, 15 | 3eqtr4d 2872 | 1 ⊢ (𝜑 → (Σ𝑘 ∈ 𝑍 𝐴 / 𝐵) = Σ𝑘 ∈ 𝑍 (𝐴 / 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 dom cdm 5343 ‘cfv 6124 (class class class)co 6906 ℂcc 10251 0cc0 10253 1c1 10254 + caddc 10256 · cmul 10258 / cdiv 11010 ℤcz 11705 ℤ≥cuz 11969 seqcseq 13096 ⇝ cli 14593 Σcsu 14794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-inf2 8816 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-se 5303 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-isom 6133 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-sup 8618 df-oi 8685 df-card 9079 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-2 11415 df-3 11416 df-n0 11620 df-z 11706 df-uz 11970 df-rp 12114 df-fz 12621 df-fzo 12762 df-seq 13097 df-exp 13156 df-hash 13412 df-cj 14217 df-re 14218 df-im 14219 df-sqrt 14353 df-abs 14354 df-clim 14597 df-sum 14795 |
This theorem is referenced by: (None) |
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