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Mirrors > Home > MPE Home > Th. List > fsumsub | Structured version Visualization version GIF version |
Description: Split a finite sum over a subtraction. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
fsumneg.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumneg.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
fsumsub.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
Ref | Expression |
---|---|
fsumsub | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐵 − 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 − Σ𝑘 ∈ 𝐴 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumneg.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
2 | fsumneg.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
3 | fsumsub.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) | |
4 | 3 | negcld 11176 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -𝐶 ∈ ℂ) |
5 | 1, 2, 4 | fsumadd 15304 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐵 + -𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 + Σ𝑘 ∈ 𝐴 -𝐶)) |
6 | 1, 3 | fsumneg 15351 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 -𝐶 = -Σ𝑘 ∈ 𝐴 𝐶) |
7 | 6 | oveq2d 7229 | . . 3 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 + Σ𝑘 ∈ 𝐴 -𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 + -Σ𝑘 ∈ 𝐴 𝐶)) |
8 | 5, 7 | eqtrd 2777 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐵 + -𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 + -Σ𝑘 ∈ 𝐴 𝐶)) |
9 | 2, 3 | negsubd 11195 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 + -𝐶) = (𝐵 − 𝐶)) |
10 | 9 | sumeq2dv 15267 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐵 + -𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 − 𝐶)) |
11 | 1, 2 | fsumcl 15297 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
12 | 1, 3 | fsumcl 15297 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 ∈ ℂ) |
13 | 11, 12 | negsubd 11195 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 + -Σ𝑘 ∈ 𝐴 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 − Σ𝑘 ∈ 𝐴 𝐶)) |
14 | 8, 10, 13 | 3eqtr3d 2785 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐵 − 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 − Σ𝑘 ∈ 𝐴 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 (class class class)co 7213 Fincfn 8626 ℂcc 10727 + caddc 10732 − cmin 11062 -cneg 11063 Σcsu 15249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-sum 15250 |
This theorem is referenced by: fsumle 15363 fsumlt 15364 telfsumo 15366 fsumparts 15370 mertens 15450 bpolydiflem 15616 3dvds 15892 pcfac 16452 pcbc 16453 ramcl 16582 ovolicc2lem4 24417 dvfsumabs 24920 coeeulem 25118 birthdaylem2 25835 emcllem5 25882 lgamcvg2 25937 chpub 26101 logfaclbnd 26103 lgsquadlem1 26261 vmadivsum 26363 rpvmasumlem 26368 dchrmusum2 26375 dchrvmasumiflem2 26383 rpvmasum2 26393 dchrisum0lem2a 26398 dchrisum0lem2 26399 rplogsum 26408 mulogsumlem 26412 mulogsum 26413 mulog2sumlem1 26415 mulog2sumlem2 26416 mulog2sumlem3 26417 vmalogdivsum2 26419 vmalogdivsum 26420 2vmadivsumlem 26421 logsqvma 26423 selberglem1 26426 selberg3lem1 26438 selberg4lem1 26441 pntrsumo1 26446 selbergr 26449 selberg3r 26450 selberg4r 26451 selberg34r 26452 pntrlog2bndlem4 26461 pntrlog2bndlem5 26462 pntlemo 26488 ax5seglem9 27028 fwddifnp1 34204 knoppndvlem11 34439 sticksstones10 39833 sticksstones12a 39835 etransclem46 43496 |
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