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Mirrors > Home > MPE Home > Th. List > fsumdivc | Structured version Visualization version GIF version |
Description: A finite sum divided by a constant. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
fsummulc2.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsummulc2.2 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
fsummulc2.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
fsumdivc.4 | ⊢ (𝜑 → 𝐶 ≠ 0) |
Ref | Expression |
---|---|
fsumdivc | ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 / 𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 / 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsummulc2.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
2 | fsummulc2.2 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
3 | fsumdivc.4 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ 0) | |
4 | 2, 3 | reccld 11412 | . . 3 ⊢ (𝜑 → (1 / 𝐶) ∈ ℂ) |
5 | fsummulc2.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
6 | 1, 4, 5 | fsummulc1 15143 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 · (1 / 𝐶)) = Σ𝑘 ∈ 𝐴 (𝐵 · (1 / 𝐶))) |
7 | 1, 5 | fsumcl 15093 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
8 | 7, 2, 3 | divrecd 11422 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 / 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 · (1 / 𝐶))) |
9 | 2 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
10 | 3 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ≠ 0) |
11 | 5, 9, 10 | divrecd 11422 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 / 𝐶) = (𝐵 · (1 / 𝐶))) |
12 | 11 | sumeq2dv 15063 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐵 / 𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 · (1 / 𝐶))) |
13 | 6, 8, 12 | 3eqtr4d 2869 | 1 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 / 𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 / 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 (class class class)co 7159 Fincfn 8512 ℂcc 10538 0cc0 10540 1c1 10541 · cmul 10545 / cdiv 11300 Σcsu 15045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-sum 15046 |
This theorem is referenced by: efaddlem 15449 fsumdvds 15661 ovolscalem1 24117 plyeq0lem 24803 aareccl 24918 birthdaylem3 25534 logexprlim 25804 logfacrlim2 25805 dchrvmasumlem1 26074 dchrisum0lem1 26095 dchrisum0 26099 vmalogdivsum2 26117 selberglem2 26125 selberg4lem1 26139 selberg4r 26149 pntrlog2bndlem5 26160 pntrlog2bndlem6 26162 pntlemo 26186 axsegconlem9 26714 signsplypnf 31824 dirkertrigeqlem2 42391 fourierdlem83 42481 elaa2lem 42525 etransclem38 42564 etransclem44 42570 etransclem45 42571 |
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