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Theorem fsumshftm 11591

Description: Negative index shift of a finite sum. (Contributed by NM, 28-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
fsumshftm	⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵)

Distinct variable groups: 𝐴,𝑘 𝐵,𝑗 𝑗,𝑘,𝐾 𝑗,𝑀,𝑘 𝑗,𝑁,𝑘 𝜑,𝑗,𝑘 Allowed substitution hints: 𝐴(𝑗) 𝐵(𝑘)

Proof of Theorem fsumshftm Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2336	. . 3 ⊢ Ⅎ𝑚𝐴
2		nfcsb1v 3114	. . 3 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐴
3		csbeq1a 3090	. . 3 ⊢ (𝑗 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑗⦌𝐴)
4	1, 2, 3	cbvsumi 11508	. 2 ⊢ Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑚 ∈ (𝑀...𝑁)⦋𝑚 / 𝑗⦌𝐴
5		fsumrev.1	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ)
6	5	znegcld 9444	. . . 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 2567	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ (𝑀...𝑁)𝐴 ∈ ℂ)
11	2	nfel1 2347	. . . . . 6 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ
12	3	eleq1d 2262	. . . . . 6 ⊢ (𝑗 = 𝑚 → (𝐴 ∈ ℂ ↔ ⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ))
13	11, 12	rspc 2859	. . . . 5 ⊢ (𝑚 ∈ (𝑀...𝑁) → (∀𝑗 ∈ (𝑀...𝑁)𝐴 ∈ ℂ → ⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ))
14	10, 13	mpan9 281	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀...𝑁)) → ⦋𝑚 / 𝑗⦌𝐴 ∈ ℂ)
15		csbeq1 3084	. . . 4 ⊢ (𝑚 = (𝑘 − -𝐾) → ⦋𝑚 / 𝑗⦌𝐴 = ⦋(𝑘 − -𝐾) / 𝑗⦌𝐴)
16	6, 7, 8, 14, 15	fsumshft 11590	. . 3 ⊢ (𝜑 → Σ𝑚 ∈ (𝑀...𝑁)⦋𝑚 / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 + -𝐾)...(𝑁 + -𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴)
17	7	zcnd 9443	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ)
18	5	zcnd 9443	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℂ)
19	17, 18	negsubd 8338	. . . . 5 ⊢ (𝜑 → (𝑀 + -𝐾) = (𝑀 − 𝐾))
20	8	zcnd 9443	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ)
21	20, 18	negsubd 8338	. . . . 5 ⊢ (𝜑 → (𝑁 + -𝐾) = (𝑁 − 𝐾))
22	19, 21	oveq12d 5937	. . . 4 ⊢ (𝜑 → ((𝑀 + -𝐾)...(𝑁 + -𝐾)) = ((𝑀 − 𝐾)...(𝑁 − 𝐾)))
23	22	sumeq1d 11512	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + -𝐾)...(𝑁 + -𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴)
24		elfzelz 10094	. . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾)) → 𝑘 ∈ ℤ)
25	24	zcnd 9443	. . . . . . 7 ⊢ (𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾)) → 𝑘 ∈ ℂ)
26		subneg 8270	. . . . . . 7 ⊢ ((𝑘 ∈ ℂ ∧ 𝐾 ∈ ℂ) → (𝑘 − -𝐾) = (𝑘 + 𝐾))
27	25, 18, 26	syl2anr 290	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) → (𝑘 − -𝐾) = (𝑘 + 𝐾))
28	27	csbeq1d 3088	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) → ⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = ⦋(𝑘 + 𝐾) / 𝑗⦌𝐴)
29	24	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) → 𝑘 ∈ ℤ)
30	5	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) → 𝐾 ∈ ℤ)
31	29, 30	zaddcld 9446	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) → (𝑘 + 𝐾) ∈ ℤ)
32		fsumshftm.5	. . . . . . 7 ⊢ (𝑗 = (𝑘 + 𝐾) → 𝐴 = 𝐵)
33	32	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) ∧ 𝑗 = (𝑘 + 𝐾)) → 𝐴 = 𝐵)
34	31, 33	csbied 3128	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) → ⦋(𝑘 + 𝐾) / 𝑗⦌𝐴 = 𝐵)
35	28, 34	eqtrd 2226	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))) → ⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = 𝐵)
36	35	sumeq2dv 11514	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))⦋(𝑘 − -𝐾) / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵)
37	16, 23, 36	3eqtrd 2230	. 2 ⊢ (𝜑 → Σ𝑚 ∈ (𝑀...𝑁)⦋𝑚 / 𝑗⦌𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵)
38	4, 37	eqtrid 2238	1 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 ∀wral 2472 ⦋csb 3081 (class class class)co 5919 ℂcc 7872 + caddc 7877 − cmin 8192 -cneg 8193 ℤcz 9320 ...cfz 10077 Σcsu 11499

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993 ax-caucvg 7994

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-isom 5264 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-irdg 6425 df-frec 6446 df-1o 6471 df-oadd 6475 df-er 6589 df-en 6797 df-dom 6798 df-fin 6799 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-n0 9244 df-z 9321 df-uz 9596 df-q 9688 df-rp 9723 df-fz 10078 df-fzo 10212 df-seqfrec 10522 df-exp 10613 df-ihash 10850 df-cj 10989 df-re 10990 df-im 10991 df-rsqrt 11145 df-abs 11146 df-clim 11425 df-sumdc 11500

This theorem is referenced by: telfsumo 11612 fsumparts 11616 arisum 11644 geo2sum 11660 dvply1 14943

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