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Mirrors > Home > ILE Home > Th. List > fzosump1 | GIF version |
Description: Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 13-Apr-2016.) |
Ref | Expression |
---|---|
fsumm1.1 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
fsumm1.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) |
fsumm1.3 | ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
fzosump1 | ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (𝑀..^𝑁)𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumm1.1 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | eluzelz 9566 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
3 | 1, 2 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
4 | fzoval 10177 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) | |
5 | 3, 4 | syl 14 | . . . 4 ⊢ (𝜑 → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
6 | 5 | sumeq1d 11405 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝐴 = Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴) |
7 | 6 | oveq1d 5910 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀..^𝑁)𝐴 + 𝐵) = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵)) |
8 | fsumm1.2 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) | |
9 | fsumm1.3 | . . 3 ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵) | |
10 | 1, 8, 9 | fsumm1 11455 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵)) |
11 | fzval3 10233 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑀...𝑁) = (𝑀..^(𝑁 + 1))) | |
12 | 3, 11 | syl 14 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) = (𝑀..^(𝑁 + 1))) |
13 | 12 | sumeq1d 11405 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ (𝑀..^(𝑁 + 1))𝐴) |
14 | 7, 10, 13 | 3eqtr2rd 2229 | 1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (𝑀..^𝑁)𝐴 + 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 ‘cfv 5235 (class class class)co 5895 ℂcc 7838 1c1 7841 + caddc 7843 − cmin 8157 ℤcz 9282 ℤ≥cuz 9557 ...cfz 10037 ..^cfzo 10171 Σcsu 11392 |
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 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulrcl 7939 ax-addcom 7940 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-1rid 7947 ax-0id 7948 ax-rnegex 7949 ax-precex 7950 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-apti 7955 ax-pre-ltadd 7956 ax-pre-mulgt0 7957 ax-pre-mulext 7958 ax-arch 7959 ax-caucvg 7960 |
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 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-isom 5244 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-recs 6329 df-irdg 6394 df-frec 6415 df-1o 6440 df-oadd 6444 df-er 6558 df-en 6766 df-dom 6767 df-fin 6768 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-reap 8561 df-ap 8568 df-div 8659 df-inn 8949 df-2 9007 df-3 9008 df-4 9009 df-n0 9206 df-z 9283 df-uz 9558 df-q 9649 df-rp 9683 df-fz 10038 df-fzo 10172 df-seqfrec 10476 df-exp 10550 df-ihash 10787 df-cj 10882 df-re 10883 df-im 10884 df-rsqrt 11038 df-abs 11039 df-clim 11318 df-sumdc 11393 |
This theorem is referenced by: (None) |
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