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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 2908 | . . 3 ⊢ Ⅎ𝑚𝐴 | |
2 | nfcsb1v 3885 | . . 3 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐴 | |
3 | csbeq1a 3874 | . . 3 ⊢ (𝑗 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑗⦌𝐴) | |
4 | 1, 2, 3 | cbvsumi 15589 | . 2 ⊢ Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑚 ∈ (𝑀...𝑁)⦋𝑚 / 𝑗⦌𝐴 |
5 | fsumrev.1 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ) | |
6 | 5 | znegcld 12616 | . . . 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 3144 | . . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ (𝑀...𝑁)𝐴 ∈ ℂ) |
11 | 2 | nfel1 2924 | . . . . . 6 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ |
12 | 3 | eleq1d 2823 | . . . . . 6 ⊢ (𝑗 = 𝑚 → (𝐴 ∈ ℂ ↔ ⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ)) |
13 | 11, 12 | rspc 3572 | . . . . 5 ⊢ (𝑚 ∈ (𝑀...𝑁) → (∀𝑗 ∈ (𝑀...𝑁)𝐴 ∈ ℂ → ⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ)) |
14 | 10, 13 | mpan9 508 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀...𝑁)) → ⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ) |
15 | csbeq1 3863 | . . . 4 ⊢ (𝑚 = (𝑘 − -𝐾) → ⦋𝑚 / 𝑗⦌𝐴 = ⦋(𝑘 − -𝐾) / 𝑗⦌𝐴) | |
16 | 6, 7, 8, 14, 15 | fsumshft 15672 | . . 3 ⊢ (𝜑 → Σ𝑚 ∈ (𝑀...𝑁)⦋𝑚 / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 + -𝐾)...(𝑁 + -𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴) |
17 | 7 | zcnd 12615 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
18 | 5 | zcnd 12615 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
19 | 17, 18 | negsubd 11525 | . . . . 5 ⊢ (𝜑 → (𝑀 + -𝐾) = (𝑀 − 𝐾)) |
20 | 8 | zcnd 12615 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
21 | 20, 18 | negsubd 11525 | . . . . 5 ⊢ (𝜑 → (𝑁 + -𝐾) = (𝑁 − 𝐾)) |
22 | 19, 21 | oveq12d 7380 | . . . 4 ⊢ (𝜑 → ((𝑀 + -𝐾)...(𝑁 + -𝐾)) = ((𝑀 − 𝐾)...(𝑁 − 𝐾))) |
23 | 22 | sumeq1d 15593 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + -𝐾)...(𝑁 + -𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴) |
24 | elfzelz 13448 | . . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾)) → 𝑘 ∈ ℤ) | |
25 | 24 | zcnd 12615 | . . . . . . 7 ⊢ (𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾)) → 𝑘 ∈ ℂ) |
26 | subneg 11457 | . . . . . . 7 ⊢ ((𝑘 ∈ ℂ ∧ 𝐾 ∈ ℂ) → (𝑘 − -𝐾) = (𝑘 + 𝐾)) | |
27 | 25, 18, 26 | syl2anr 598 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) → (𝑘 − -𝐾) = (𝑘 + 𝐾)) |
28 | 27 | csbeq1d 3864 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) → ⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = ⦋(𝑘 + 𝐾) / 𝑗⦌𝐴) |
29 | ovex 7395 | . . . . . 6 ⊢ (𝑘 + 𝐾) ∈ V | |
30 | fsumshftm.5 | . . . . . 6 ⊢ (𝑗 = (𝑘 + 𝐾) → 𝐴 = 𝐵) | |
31 | 29, 30 | csbie 3896 | . . . . 5 ⊢ ⦋(𝑘 + 𝐾) / 𝑗⦌𝐴 = 𝐵 |
32 | 28, 31 | eqtrdi 2793 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) → ⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = 𝐵) |
33 | 32 | sumeq2dv 15595 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵) |
34 | 16, 23, 33 | 3eqtrd 2781 | . 2 ⊢ (𝜑 → Σ𝑚 ∈ (𝑀...𝑁)⦋𝑚 / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵) |
35 | 4, 34 | eqtrid 2789 | 1 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ⦋csb 3860 (class class class)co 7362 ℂcc 11056 + caddc 11061 − cmin 11392 -cneg 11393 ℤcz 12506 ...cfz 13431 Σcsu 15577 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-fz 13432 df-fzo 13575 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-sum 15578 |
This theorem is referenced by: telfsumo 15694 fsumparts 15698 arisum 15752 pwdif 15760 geo2sum 15765 ovolicc2lem4 24900 uniioombllem3 24965 dvply1 25660 pserdvlem2 25803 advlogexp 26026 dchrisumlem1 26853 pntpbnd2 26951 nn0sumshdiglemA 46779 nn0sumshdiglemB 46780 |
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