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Mirrors > Home > ILE Home > Th. List > isumnn0nn | GIF version |
Description: Sum from 0 to infinity in terms of sum from 1 to infinity. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
isumnn0nn.1 | ⊢ (𝑘 = 0 → 𝐴 = 𝐵) |
isumnn0nn.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = 𝐴) |
isumnn0nn.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℂ) |
isumnn0nn.4 | ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ ) |
Ref | Expression |
---|---|
isumnn0nn | ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐴 = (𝐵 + Σ𝑘 ∈ ℕ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 9360 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 9066 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
3 | isumnn0nn.2 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = 𝐴) | |
4 | isumnn0nn.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℂ) | |
5 | isumnn0nn.4 | . . 3 ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ ) | |
6 | 1, 2, 3, 4, 5 | isum1p 11261 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐴 = ((𝐹‘0) + Σ𝑘 ∈ (ℤ≥‘(0 + 1))𝐴)) |
7 | fveq2 5421 | . . . . 5 ⊢ (𝑘 = 0 → (𝐹‘𝑘) = (𝐹‘0)) | |
8 | isumnn0nn.1 | . . . . 5 ⊢ (𝑘 = 0 → 𝐴 = 𝐵) | |
9 | 7, 8 | eqeq12d 2154 | . . . 4 ⊢ (𝑘 = 0 → ((𝐹‘𝑘) = 𝐴 ↔ (𝐹‘0) = 𝐵)) |
10 | 3 | ralrimiva 2505 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝐹‘𝑘) = 𝐴) |
11 | 0nn0 8992 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
12 | 11 | a1i 9 | . . . 4 ⊢ (𝜑 → 0 ∈ ℕ0) |
13 | 9, 10, 12 | rspcdva 2794 | . . 3 ⊢ (𝜑 → (𝐹‘0) = 𝐵) |
14 | 0p1e1 8834 | . . . . . . 7 ⊢ (0 + 1) = 1 | |
15 | 14 | fveq2i 5424 | . . . . . 6 ⊢ (ℤ≥‘(0 + 1)) = (ℤ≥‘1) |
16 | nnuz 9361 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
17 | 15, 16 | eqtr4i 2163 | . . . . 5 ⊢ (ℤ≥‘(0 + 1)) = ℕ |
18 | 17 | sumeq1i 11132 | . . . 4 ⊢ Σ𝑘 ∈ (ℤ≥‘(0 + 1))𝐴 = Σ𝑘 ∈ ℕ 𝐴 |
19 | 18 | a1i 9 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(0 + 1))𝐴 = Σ𝑘 ∈ ℕ 𝐴) |
20 | 13, 19 | oveq12d 5792 | . 2 ⊢ (𝜑 → ((𝐹‘0) + Σ𝑘 ∈ (ℤ≥‘(0 + 1))𝐴) = (𝐵 + Σ𝑘 ∈ ℕ 𝐴)) |
21 | 6, 20 | eqtrd 2172 | 1 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐴 = (𝐵 + Σ𝑘 ∈ ℕ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 dom cdm 4539 ‘cfv 5123 (class class class)co 5774 ℂcc 7618 0cc0 7620 1c1 7621 + caddc 7623 ℕcn 8720 ℕ0cn0 8977 ℤ≥cuz 9326 seqcseq 10218 ⇝ cli 11047 Σcsu 11122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-fz 9791 df-fzo 9920 df-seqfrec 10219 df-exp 10293 df-ihash 10522 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-clim 11048 df-sumdc 11123 |
This theorem is referenced by: (None) |
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