![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 9114 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 8823 | . . 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 10947 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐴 = ((𝐹‘0) + Σ𝑘 ∈ (ℤ≥‘(0 + 1))𝐴)) |
7 | fveq2 5318 | . . . . 5 ⊢ (𝑘 = 0 → (𝐹‘𝑘) = (𝐹‘0)) | |
8 | isumnn0nn.1 | . . . . 5 ⊢ (𝑘 = 0 → 𝐴 = 𝐵) | |
9 | 7, 8 | eqeq12d 2103 | . . . 4 ⊢ (𝑘 = 0 → ((𝐹‘𝑘) = 𝐴 ↔ (𝐹‘0) = 𝐵)) |
10 | 3 | ralrimiva 2447 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝐹‘𝑘) = 𝐴) |
11 | 0nn0 8749 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
12 | 11 | a1i 9 | . . . 4 ⊢ (𝜑 → 0 ∈ ℕ0) |
13 | 9, 10, 12 | rspcdva 2728 | . . 3 ⊢ (𝜑 → (𝐹‘0) = 𝐵) |
14 | 0p1e1 8597 | . . . . . . 7 ⊢ (0 + 1) = 1 | |
15 | 14 | fveq2i 5321 | . . . . . 6 ⊢ (ℤ≥‘(0 + 1)) = (ℤ≥‘1) |
16 | nnuz 9115 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
17 | 15, 16 | eqtr4i 2112 | . . . . 5 ⊢ (ℤ≥‘(0 + 1)) = ℕ |
18 | 17 | sumeq1i 10813 | . . . 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 5684 | . 2 ⊢ (𝜑 → ((𝐹‘0) + Σ𝑘 ∈ (ℤ≥‘(0 + 1))𝐴) = (𝐵 + Σ𝑘 ∈ ℕ 𝐴)) |
21 | 6, 20 | eqtrd 2121 | 1 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐴 = (𝐵 + Σ𝑘 ∈ ℕ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1290 ∈ wcel 1439 dom cdm 4452 ‘cfv 5028 (class class class)co 5666 ℂcc 7409 0cc0 7411 1c1 7412 + caddc 7414 ℕcn 8483 ℕ0cn0 8734 ℤ≥cuz 9080 seqcseq 9913 ⇝ cli 10727 Σcsu 10803 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-mulrcl 7505 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-mulass 7509 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-1rid 7513 ax-0id 7514 ax-rnegex 7515 ax-precex 7516 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-apti 7521 ax-pre-ltadd 7522 ax-pre-mulgt0 7523 ax-pre-mulext 7524 ax-arch 7525 ax-caucvg 7526 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-if 3398 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-po 4132 df-iso 4133 df-iord 4202 df-on 4204 df-ilim 4205 df-suc 4207 df-iom 4419 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-isom 5037 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-irdg 6149 df-frec 6170 df-1o 6195 df-oadd 6199 df-er 6306 df-en 6512 df-dom 6513 df-fin 6514 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 df-sub 7716 df-neg 7717 df-reap 8113 df-ap 8120 df-div 8201 df-inn 8484 df-2 8542 df-3 8543 df-4 8544 df-n0 8735 df-z 8812 df-uz 9081 df-q 9166 df-rp 9196 df-fz 9486 df-fzo 9615 df-iseq 9914 df-seq3 9915 df-exp 10016 df-ihash 10245 df-cj 10337 df-re 10338 df-im 10339 df-rsqrt 10492 df-abs 10493 df-clim 10728 df-isum 10804 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |