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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 9521 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 9224 | . . 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 11455 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐴 = ((𝐹‘0) + Σ𝑘 ∈ (ℤ≥‘(0 + 1))𝐴)) |
7 | fveq2 5496 | . . . . 5 ⊢ (𝑘 = 0 → (𝐹‘𝑘) = (𝐹‘0)) | |
8 | isumnn0nn.1 | . . . . 5 ⊢ (𝑘 = 0 → 𝐴 = 𝐵) | |
9 | 7, 8 | eqeq12d 2185 | . . . 4 ⊢ (𝑘 = 0 → ((𝐹‘𝑘) = 𝐴 ↔ (𝐹‘0) = 𝐵)) |
10 | 3 | ralrimiva 2543 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝐹‘𝑘) = 𝐴) |
11 | 0nn0 9150 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
12 | 11 | a1i 9 | . . . 4 ⊢ (𝜑 → 0 ∈ ℕ0) |
13 | 9, 10, 12 | rspcdva 2839 | . . 3 ⊢ (𝜑 → (𝐹‘0) = 𝐵) |
14 | 0p1e1 8992 | . . . . . . 7 ⊢ (0 + 1) = 1 | |
15 | 14 | fveq2i 5499 | . . . . . 6 ⊢ (ℤ≥‘(0 + 1)) = (ℤ≥‘1) |
16 | nnuz 9522 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
17 | 15, 16 | eqtr4i 2194 | . . . . 5 ⊢ (ℤ≥‘(0 + 1)) = ℕ |
18 | 17 | sumeq1i 11326 | . . . 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 5871 | . 2 ⊢ (𝜑 → ((𝐹‘0) + Σ𝑘 ∈ (ℤ≥‘(0 + 1))𝐴) = (𝐵 + Σ𝑘 ∈ ℕ 𝐴)) |
21 | 6, 20 | eqtrd 2203 | 1 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐴 = (𝐵 + Σ𝑘 ∈ ℕ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 dom cdm 4611 ‘cfv 5198 (class class class)co 5853 ℂcc 7772 0cc0 7774 1c1 7775 + caddc 7777 ℕcn 8878 ℕ0cn0 9135 ℤ≥cuz 9487 seqcseq 10401 ⇝ cli 11241 Σcsu 11316 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-exp 10476 df-ihash 10710 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-sumdc 11317 |
This theorem is referenced by: (None) |
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