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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 2899 | . . 3 ⊢ Ⅎ𝑚𝐴 | |
2 | nfcsb1v 3917 | . . 3 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐴 | |
3 | csbeq1a 3906 | . . 3 ⊢ (𝑗 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑗⦌𝐴) | |
4 | 1, 2, 3 | cbvsumi 15676 | . 2 ⊢ Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑚 ∈ (𝑀...𝑁)⦋𝑚 / 𝑗⦌𝐴 |
5 | fsumrev.1 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ) | |
6 | 5 | znegcld 12699 | . . . 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 3143 | . . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ (𝑀...𝑁)𝐴 ∈ ℂ) |
11 | 2 | nfel1 2916 | . . . . . 6 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ |
12 | 3 | eleq1d 2814 | . . . . . 6 ⊢ (𝑗 = 𝑚 → (𝐴 ∈ ℂ ↔ ⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ)) |
13 | 11, 12 | rspc 3597 | . . . . 5 ⊢ (𝑚 ∈ (𝑀...𝑁) → (∀𝑗 ∈ (𝑀...𝑁)𝐴 ∈ ℂ → ⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ)) |
14 | 10, 13 | mpan9 506 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀...𝑁)) → ⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ) |
15 | csbeq1 3895 | . . . 4 ⊢ (𝑚 = (𝑘 − -𝐾) → ⦋𝑚 / 𝑗⦌𝐴 = ⦋(𝑘 − -𝐾) / 𝑗⦌𝐴) | |
16 | 6, 7, 8, 14, 15 | fsumshft 15759 | . . 3 ⊢ (𝜑 → Σ𝑚 ∈ (𝑀...𝑁)⦋𝑚 / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 + -𝐾)...(𝑁 + -𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴) |
17 | 7 | zcnd 12698 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
18 | 5 | zcnd 12698 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
19 | 17, 18 | negsubd 11608 | . . . . 5 ⊢ (𝜑 → (𝑀 + -𝐾) = (𝑀 − 𝐾)) |
20 | 8 | zcnd 12698 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
21 | 20, 18 | negsubd 11608 | . . . . 5 ⊢ (𝜑 → (𝑁 + -𝐾) = (𝑁 − 𝐾)) |
22 | 19, 21 | oveq12d 7438 | . . . 4 ⊢ (𝜑 → ((𝑀 + -𝐾)...(𝑁 + -𝐾)) = ((𝑀 − 𝐾)...(𝑁 − 𝐾))) |
23 | 22 | sumeq1d 15680 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + -𝐾)...(𝑁 + -𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴) |
24 | elfzelz 13534 | . . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾)) → 𝑘 ∈ ℤ) | |
25 | 24 | zcnd 12698 | . . . . . . 7 ⊢ (𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾)) → 𝑘 ∈ ℂ) |
26 | subneg 11540 | . . . . . . 7 ⊢ ((𝑘 ∈ ℂ ∧ 𝐾 ∈ ℂ) → (𝑘 − -𝐾) = (𝑘 + 𝐾)) | |
27 | 25, 18, 26 | syl2anr 596 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) → (𝑘 − -𝐾) = (𝑘 + 𝐾)) |
28 | 27 | csbeq1d 3896 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) → ⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = ⦋(𝑘 + 𝐾) / 𝑗⦌𝐴) |
29 | ovex 7453 | . . . . . 6 ⊢ (𝑘 + 𝐾) ∈ V | |
30 | fsumshftm.5 | . . . . . 6 ⊢ (𝑗 = (𝑘 + 𝐾) → 𝐴 = 𝐵) | |
31 | 29, 30 | csbie 3928 | . . . . 5 ⊢ ⦋(𝑘 + 𝐾) / 𝑗⦌𝐴 = 𝐵 |
32 | 28, 31 | eqtrdi 2784 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) → ⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = 𝐵) |
33 | 32 | sumeq2dv 15682 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵) |
34 | 16, 23, 33 | 3eqtrd 2772 | . 2 ⊢ (𝜑 → Σ𝑚 ∈ (𝑀...𝑁)⦋𝑚 / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵) |
35 | 4, 34 | eqtrid 2780 | 1 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3058 ⦋csb 3892 (class class class)co 7420 ℂcc 11137 + caddc 11142 − cmin 11475 -cneg 11476 ℤcz 12589 ...cfz 13517 Σcsu 15665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-fz 13518 df-fzo 13661 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-sum 15666 |
This theorem is referenced by: telfsumo 15781 fsumparts 15785 arisum 15839 pwdif 15847 geo2sum 15852 ovolicc2lem4 25462 uniioombllem3 25527 dvply1 26231 pserdvlem2 26378 advlogexp 26602 dchrisumlem1 27435 pntpbnd2 27533 sumcubes 41873 nn0sumshdiglemA 47692 nn0sumshdiglemB 47693 |
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