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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 2977 | . . 3 ⊢ Ⅎ𝑚𝐴 | |
2 | nfcsb1v 3907 | . . 3 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐴 | |
3 | csbeq1a 3897 | . . 3 ⊢ (𝑗 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑗⦌𝐴) | |
4 | 1, 2, 3 | cbvsumi 15054 | . 2 ⊢ Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑚 ∈ (𝑀...𝑁)⦋𝑚 / 𝑗⦌𝐴 |
5 | fsumrev.1 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ) | |
6 | 5 | znegcld 12090 | . . . 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 3182 | . . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ (𝑀...𝑁)𝐴 ∈ ℂ) |
11 | 2 | nfel1 2994 | . . . . . 6 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ |
12 | 3 | eleq1d 2897 | . . . . . 6 ⊢ (𝑗 = 𝑚 → (𝐴 ∈ ℂ ↔ ⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ)) |
13 | 11, 12 | rspc 3611 | . . . . 5 ⊢ (𝑚 ∈ (𝑀...𝑁) → (∀𝑗 ∈ (𝑀...𝑁)𝐴 ∈ ℂ → ⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ)) |
14 | 10, 13 | mpan9 509 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀...𝑁)) → ⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ) |
15 | csbeq1 3886 | . . . 4 ⊢ (𝑚 = (𝑘 − -𝐾) → ⦋𝑚 / 𝑗⦌𝐴 = ⦋(𝑘 − -𝐾) / 𝑗⦌𝐴) | |
16 | 6, 7, 8, 14, 15 | fsumshft 15135 | . . 3 ⊢ (𝜑 → Σ𝑚 ∈ (𝑀...𝑁)⦋𝑚 / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 + -𝐾)...(𝑁 + -𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴) |
17 | 7 | zcnd 12089 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
18 | 5 | zcnd 12089 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
19 | 17, 18 | negsubd 11003 | . . . . 5 ⊢ (𝜑 → (𝑀 + -𝐾) = (𝑀 − 𝐾)) |
20 | 8 | zcnd 12089 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
21 | 20, 18 | negsubd 11003 | . . . . 5 ⊢ (𝜑 → (𝑁 + -𝐾) = (𝑁 − 𝐾)) |
22 | 19, 21 | oveq12d 7174 | . . . 4 ⊢ (𝜑 → ((𝑀 + -𝐾)...(𝑁 + -𝐾)) = ((𝑀 − 𝐾)...(𝑁 − 𝐾))) |
23 | 22 | sumeq1d 15058 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + -𝐾)...(𝑁 + -𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴) |
24 | elfzelz 12909 | . . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾)) → 𝑘 ∈ ℤ) | |
25 | 24 | zcnd 12089 | . . . . . . 7 ⊢ (𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾)) → 𝑘 ∈ ℂ) |
26 | subneg 10935 | . . . . . . 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 3887 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) → ⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = ⦋(𝑘 + 𝐾) / 𝑗⦌𝐴) |
29 | ovex 7189 | . . . . . 6 ⊢ (𝑘 + 𝐾) ∈ V | |
30 | fsumshftm.5 | . . . . . 6 ⊢ (𝑗 = (𝑘 + 𝐾) → 𝐴 = 𝐵) | |
31 | 29, 30 | csbie 3918 | . . . . 5 ⊢ ⦋(𝑘 + 𝐾) / 𝑗⦌𝐴 = 𝐵 |
32 | 28, 31 | syl6eq 2872 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) → ⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = 𝐵) |
33 | 32 | sumeq2dv 15060 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵) |
34 | 16, 23, 33 | 3eqtrd 2860 | . 2 ⊢ (𝜑 → Σ𝑚 ∈ (𝑀...𝑁)⦋𝑚 / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵) |
35 | 4, 34 | syl5eq 2868 | 1 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⦋csb 3883 (class class class)co 7156 ℂcc 10535 + caddc 10540 − cmin 10870 -cneg 10871 ℤcz 11982 ...cfz 12893 Σcsu 15042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 |
This theorem is referenced by: telfsumo 15157 fsumparts 15161 arisum 15215 pwdif 15223 geo2sum 15229 ovolicc2lem4 24121 uniioombllem3 24186 dvply1 24873 pserdvlem2 25016 advlogexp 25238 dchrisumlem1 26065 pntpbnd2 26163 nn0sumshdiglemA 44699 nn0sumshdiglemB 44700 |
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