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Mirrors > Home > ILE Home > Th. List > fsum1p | GIF version |
Description: Separate out the first term in a finite sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.) |
Ref | Expression |
---|---|
fsumm1.1 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
fsumm1.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) |
fsum1p.3 | ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
fsum1p | ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (𝐵 + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumm1.1 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | eluzel2 9014 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
3 | 1, 2 | syl 14 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
4 | fzsn 9469 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) | |
5 | 3, 4 | syl 14 | . . . . 5 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀}) |
6 | 5 | ineq1d 3200 | . . . 4 ⊢ (𝜑 → ((𝑀...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ({𝑀} ∩ ((𝑀 + 1)...𝑁))) |
7 | 3 | zred 8858 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
8 | 7 | ltp1d 8381 | . . . . 5 ⊢ (𝜑 → 𝑀 < (𝑀 + 1)) |
9 | fzdisj 9456 | . . . . 5 ⊢ (𝑀 < (𝑀 + 1) → ((𝑀...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) | |
10 | 8, 9 | syl 14 | . . . 4 ⊢ (𝜑 → ((𝑀...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
11 | 6, 10 | eqtr3d 2122 | . . 3 ⊢ (𝜑 → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅) |
12 | eluzfz1 9435 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
13 | 1, 12 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
14 | fzsplit 9455 | . . . . 5 ⊢ (𝑀 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...𝑀) ∪ ((𝑀 + 1)...𝑁))) | |
15 | 13, 14 | syl 14 | . . . 4 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
16 | 5 | uneq1d 3153 | . . . 4 ⊢ (𝜑 → ((𝑀...𝑀) ∪ ((𝑀 + 1)...𝑁)) = ({𝑀} ∪ ((𝑀 + 1)...𝑁))) |
17 | 15, 16 | eqtrd 2120 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁))) |
18 | eluzelz 9018 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
19 | 1, 18 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
20 | 3, 19 | fzfigd 9826 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) |
21 | fsumm1.2 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) | |
22 | 11, 17, 20, 21 | fsumsplit 10788 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ {𝑀}𝐴 + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) |
23 | fsum1p.3 | . . . . . 6 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) | |
24 | 23 | eleq1d 2156 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ)) |
25 | 21 | ralrimiva 2446 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ) |
26 | 24, 25, 13 | rspcdva 2727 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
27 | 23 | sumsn 10792 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
28 | 3, 26, 27 | syl2anc 403 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
29 | 28 | oveq1d 5659 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ {𝑀}𝐴 + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴) = (𝐵 + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) |
30 | 22, 29 | eqtrd 2120 | 1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (𝐵 + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1289 ∈ wcel 1438 ∪ cun 2997 ∩ cin 2998 ∅c0 3286 {csn 3444 class class class wbr 3843 ‘cfv 5010 (class class class)co 5644 ℂcc 7338 1c1 7341 + caddc 7343 < clt 7512 ℤcz 8740 ℤ≥cuz 9009 ...cfz 9414 Σcsu 10729 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3952 ax-sep 3955 ax-nul 3963 ax-pow 4007 ax-pr 4034 ax-un 4258 ax-setind 4351 ax-iinf 4401 ax-cnex 7426 ax-resscn 7427 ax-1cn 7428 ax-1re 7429 ax-icn 7430 ax-addcl 7431 ax-addrcl 7432 ax-mulcl 7433 ax-mulrcl 7434 ax-addcom 7435 ax-mulcom 7436 ax-addass 7437 ax-mulass 7438 ax-distr 7439 ax-i2m1 7440 ax-0lt1 7441 ax-1rid 7442 ax-0id 7443 ax-rnegex 7444 ax-precex 7445 ax-cnre 7446 ax-pre-ltirr 7447 ax-pre-ltwlin 7448 ax-pre-lttrn 7449 ax-pre-apti 7450 ax-pre-ltadd 7451 ax-pre-mulgt0 7452 ax-pre-mulext 7453 ax-arch 7454 ax-caucvg 7455 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rmo 2367 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-if 3392 df-pw 3429 df-sn 3450 df-pr 3451 df-op 3453 df-uni 3652 df-int 3687 df-iun 3730 df-br 3844 df-opab 3898 df-mpt 3899 df-tr 3935 df-id 4118 df-po 4121 df-iso 4122 df-iord 4191 df-on 4193 df-ilim 4194 df-suc 4196 df-iom 4404 df-xp 4442 df-rel 4443 df-cnv 4444 df-co 4445 df-dm 4446 df-rn 4447 df-res 4448 df-ima 4449 df-iota 4975 df-fun 5012 df-fn 5013 df-f 5014 df-f1 5015 df-fo 5016 df-f1o 5017 df-fv 5018 df-isom 5019 df-riota 5600 df-ov 5647 df-oprab 5648 df-mpt2 5649 df-1st 5903 df-2nd 5904 df-recs 6062 df-irdg 6127 df-frec 6148 df-1o 6173 df-oadd 6177 df-er 6282 df-en 6448 df-dom 6449 df-fin 6450 df-pnf 7514 df-mnf 7515 df-xr 7516 df-ltxr 7517 df-le 7518 df-sub 7645 df-neg 7646 df-reap 8042 df-ap 8049 df-div 8130 df-inn 8413 df-2 8471 df-3 8472 df-4 8473 df-n0 8664 df-z 8741 df-uz 9010 df-q 9095 df-rp 9125 df-fz 9415 df-fzo 9542 df-iseq 9841 df-seq3 9842 df-exp 9943 df-ihash 10172 df-cj 10264 df-re 10265 df-im 10266 df-rsqrt 10419 df-abs 10420 df-clim 10654 df-isum 10730 |
This theorem is referenced by: telfsumo 10847 fsumparts 10851 arisum2 10880 |
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