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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 10821 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -𝐶 ∈ ℂ) |
5 | 1, 2, 4 | fsumadd 14917 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐵 + -𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 + Σ𝑘 ∈ 𝐴 -𝐶)) |
6 | 1, 3 | fsumneg 14963 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 -𝐶 = -Σ𝑘 ∈ 𝐴 𝐶) |
7 | 6 | oveq2d 7023 | . . 3 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 + Σ𝑘 ∈ 𝐴 -𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 + -Σ𝑘 ∈ 𝐴 𝐶)) |
8 | 5, 7 | eqtrd 2829 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐵 + -𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 + -Σ𝑘 ∈ 𝐴 𝐶)) |
9 | 2, 3 | negsubd 10840 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 + -𝐶) = (𝐵 − 𝐶)) |
10 | 9 | sumeq2dv 14881 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐵 + -𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 − 𝐶)) |
11 | 1, 2 | fsumcl 14911 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
12 | 1, 3 | fsumcl 14911 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 ∈ ℂ) |
13 | 11, 12 | negsubd 10840 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 + -Σ𝑘 ∈ 𝐴 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 − Σ𝑘 ∈ 𝐴 𝐶)) |
14 | 8, 10, 13 | 3eqtr3d 2837 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐵 − 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 − Σ𝑘 ∈ 𝐴 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1520 ∈ wcel 2079 (class class class)co 7007 Fincfn 8347 ℂcc 10370 + caddc 10375 − cmin 10706 -cneg 10707 Σcsu 14864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-inf2 8939 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-fal 1533 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-se 5395 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-isom 6226 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-oadd 7948 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-sup 8742 df-oi 8810 df-card 9203 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-n0 11735 df-z 11819 df-uz 12083 df-rp 12229 df-fz 12732 df-fzo 12873 df-seq 13208 df-exp 13268 df-hash 13529 df-cj 14280 df-re 14281 df-im 14282 df-sqrt 14416 df-abs 14417 df-clim 14667 df-sum 14865 |
This theorem is referenced by: fsumle 14975 fsumlt 14976 telfsumo 14978 fsumparts 14982 mertens 15063 bpolydiflem 15229 3dvds 15501 pcfac 16052 pcbc 16053 ramcl 16182 ovolicc2lem4 23792 dvfsumabs 24291 coeeulem 24485 birthdaylem2 25200 emcllem5 25247 lgamcvg2 25302 chpub 25466 logfaclbnd 25468 lgsquadlem1 25626 vmadivsum 25728 rpvmasumlem 25733 dchrmusum2 25740 dchrvmasumiflem2 25748 rpvmasum2 25758 dchrisum0lem2a 25763 dchrisum0lem2 25764 rplogsum 25773 mulogsumlem 25777 mulogsum 25778 mulog2sumlem1 25780 mulog2sumlem2 25781 mulog2sumlem3 25782 vmalogdivsum2 25784 vmalogdivsum 25785 2vmadivsumlem 25786 logsqvma 25788 selberglem1 25791 selberg3lem1 25803 selberg4lem1 25806 pntrsumo1 25811 selbergr 25814 selberg3r 25815 selberg4r 25816 selberg34r 25817 pntrlog2bndlem4 25826 pntrlog2bndlem5 25827 pntlemo 25853 ax5seglem9 26394 fwddifnp1 33180 knoppndvlem11 33414 etransclem46 42061 |
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