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Mirrors > Home > MPE Home > Th. List > fsumshftm | Structured version Visualization version GIF version |
Description: Negative index shift of a finite sum. (Contributed by NM, 28-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
fsumrev.1 | ⊢ (𝜑 → 𝐾 ∈ ℤ) |
fsumrev.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
fsumrev.3 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
fsumrev.4 | ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) |
fsumshftm.5 | ⊢ (𝑗 = (𝑘 + 𝐾) → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
fsumshftm | ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2902 | . . 3 ⊢ Ⅎ𝑚𝐴 | |
2 | nfcsb1v 3690 | . . 3 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐴 | |
3 | csbeq1a 3683 | . . 3 ⊢ (𝑗 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑗⦌𝐴) | |
4 | 1, 2, 3 | cbvsumi 14626 | . 2 ⊢ Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑚 ∈ (𝑀...𝑁)⦋𝑚 / 𝑗⦌𝐴 |
5 | fsumrev.1 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ) | |
6 | 5 | znegcld 11676 | . . . 4 ⊢ (𝜑 → -𝐾 ∈ ℤ) |
7 | fsumrev.2 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
8 | fsumrev.3 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
9 | fsumrev.4 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) | |
10 | 9 | ralrimiva 3104 | . . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ (𝑀...𝑁)𝐴 ∈ ℂ) |
11 | 2 | nfel1 2917 | . . . . . 6 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ |
12 | 3 | eleq1d 2824 | . . . . . 6 ⊢ (𝑗 = 𝑚 → (𝐴 ∈ ℂ ↔ ⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ)) |
13 | 11, 12 | rspc 3443 | . . . . 5 ⊢ (𝑚 ∈ (𝑀...𝑁) → (∀𝑗 ∈ (𝑀...𝑁)𝐴 ∈ ℂ → ⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ)) |
14 | 10, 13 | mpan9 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀...𝑁)) → ⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ) |
15 | csbeq1 3677 | . . . 4 ⊢ (𝑚 = (𝑘 − -𝐾) → ⦋𝑚 / 𝑗⦌𝐴 = ⦋(𝑘 − -𝐾) / 𝑗⦌𝐴) | |
16 | 6, 7, 8, 14, 15 | fsumshft 14711 | . . 3 ⊢ (𝜑 → Σ𝑚 ∈ (𝑀...𝑁)⦋𝑚 / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 + -𝐾)...(𝑁 + -𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴) |
17 | 7 | zcnd 11675 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
18 | 5 | zcnd 11675 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
19 | 17, 18 | negsubd 10590 | . . . . 5 ⊢ (𝜑 → (𝑀 + -𝐾) = (𝑀 − 𝐾)) |
20 | 8 | zcnd 11675 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
21 | 20, 18 | negsubd 10590 | . . . . 5 ⊢ (𝜑 → (𝑁 + -𝐾) = (𝑁 − 𝐾)) |
22 | 19, 21 | oveq12d 6831 | . . . 4 ⊢ (𝜑 → ((𝑀 + -𝐾)...(𝑁 + -𝐾)) = ((𝑀 − 𝐾)...(𝑁 − 𝐾))) |
23 | 22 | sumeq1d 14630 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + -𝐾)...(𝑁 + -𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴) |
24 | elfzelz 12535 | . . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾)) → 𝑘 ∈ ℤ) | |
25 | 24 | zcnd 11675 | . . . . . . 7 ⊢ (𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾)) → 𝑘 ∈ ℂ) |
26 | subneg 10522 | . . . . . . 7 ⊢ ((𝑘 ∈ ℂ ∧ 𝐾 ∈ ℂ) → (𝑘 − -𝐾) = (𝑘 + 𝐾)) | |
27 | 25, 18, 26 | syl2anr 496 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) → (𝑘 − -𝐾) = (𝑘 + 𝐾)) |
28 | 27 | csbeq1d 3681 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) → ⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = ⦋(𝑘 + 𝐾) / 𝑗⦌𝐴) |
29 | ovex 6841 | . . . . . 6 ⊢ (𝑘 + 𝐾) ∈ V | |
30 | fsumshftm.5 | . . . . . 6 ⊢ (𝑗 = (𝑘 + 𝐾) → 𝐴 = 𝐵) | |
31 | 29, 30 | csbie 3700 | . . . . 5 ⊢ ⦋(𝑘 + 𝐾) / 𝑗⦌𝐴 = 𝐵 |
32 | 28, 31 | syl6eq 2810 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) → ⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = 𝐵) |
33 | 32 | sumeq2dv 14632 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵) |
34 | 16, 23, 33 | 3eqtrd 2798 | . 2 ⊢ (𝜑 → Σ𝑚 ∈ (𝑀...𝑁)⦋𝑚 / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵) |
35 | 4, 34 | syl5eq 2806 | 1 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ∀wral 3050 ⦋csb 3674 (class class class)co 6813 ℂcc 10126 + caddc 10131 − cmin 10458 -cneg 10459 ℤcz 11569 ...cfz 12519 Σcsu 14615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-sup 8513 df-oi 8580 df-card 8955 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-n0 11485 df-z 11570 df-uz 11880 df-rp 12026 df-fz 12520 df-fzo 12660 df-seq 12996 df-exp 13055 df-hash 13312 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-clim 14418 df-sum 14616 |
This theorem is referenced by: telfsumo 14733 fsumparts 14737 arisum 14791 geo2sum 14803 ovolicc2lem4 23488 uniioombllem3 23553 dvply1 24238 pserdvlem2 24381 advlogexp 24600 dchrisumlem1 25377 pntpbnd2 25475 pwdif 42011 nn0sumshdiglemA 42923 nn0sumshdiglemB 42924 |
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