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Mirrors > Home > ILE Home > Th. List > fsumm1 | GIF version |
Description: Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 26-Apr-2014.) |
Ref | Expression |
---|---|
fsumm1.1 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
fsumm1.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) |
fsumm1.3 | ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
fsumm1 | ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumm1.1 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | eluzelz 9466 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
3 | 1, 2 | syl 14 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
4 | fzsn 9991 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁}) | |
5 | 3, 4 | syl 14 | . . . . 5 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁}) |
6 | 5 | ineq2d 3318 | . . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ (𝑁...𝑁)) = ((𝑀...(𝑁 − 1)) ∩ {𝑁})) |
7 | 3 | zred 9304 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
8 | 7 | ltm1d 8818 | . . . . 5 ⊢ (𝜑 → (𝑁 − 1) < 𝑁) |
9 | fzdisj 9977 | . . . . 5 ⊢ ((𝑁 − 1) < 𝑁 → ((𝑀...(𝑁 − 1)) ∩ (𝑁...𝑁)) = ∅) | |
10 | 8, 9 | syl 14 | . . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ (𝑁...𝑁)) = ∅) |
11 | 6, 10 | eqtr3d 2199 | . . 3 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅) |
12 | eluzel2 9462 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
13 | 1, 12 | syl 14 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
14 | peano2zm 9220 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ) | |
15 | 13, 14 | syl 14 | . . . . . . 7 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ) |
16 | 13 | zcnd 9305 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
17 | ax-1cn 7837 | . . . . . . . . . 10 ⊢ 1 ∈ ℂ | |
18 | npcan 8098 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 − 1) + 1) = 𝑀) | |
19 | 16, 17, 18 | sylancl 410 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀) |
20 | 19 | fveq2d 5484 | . . . . . . . 8 ⊢ (𝜑 → (ℤ≥‘((𝑀 − 1) + 1)) = (ℤ≥‘𝑀)) |
21 | 1, 20 | eleqtrrd 2244 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1))) |
22 | eluzp1m1 9480 | . . . . . . 7 ⊢ (((𝑀 − 1) ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1))) → (𝑁 − 1) ∈ (ℤ≥‘(𝑀 − 1))) | |
23 | 15, 21, 22 | syl2anc 409 | . . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘(𝑀 − 1))) |
24 | fzsuc2 10004 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ (ℤ≥‘(𝑀 − 1))) → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) | |
25 | 13, 23, 24 | syl2anc 409 | . . . . 5 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) |
26 | 3 | zcnd 9305 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
27 | npcan 8098 | . . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
28 | 26, 17, 27 | sylancl 410 | . . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
29 | 28 | oveq2d 5852 | . . . . 5 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁)) |
30 | 25, 29 | eqtr3d 2199 | . . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = (𝑀...𝑁)) |
31 | 28 | sneqd 3583 | . . . . 5 ⊢ (𝜑 → {((𝑁 − 1) + 1)} = {𝑁}) |
32 | 31 | uneq2d 3271 | . . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = ((𝑀...(𝑁 − 1)) ∪ {𝑁})) |
33 | 30, 32 | eqtr3d 2199 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁})) |
34 | 13, 3 | fzfigd 10356 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) |
35 | fsumm1.2 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) | |
36 | 11, 33, 34, 35 | fsumsplit 11334 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + Σ𝑘 ∈ {𝑁}𝐴)) |
37 | fsumm1.3 | . . . . . 6 ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵) | |
38 | 37 | eleq1d 2233 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ)) |
39 | 35 | ralrimiva 2537 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ) |
40 | eluzfz2 9957 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) | |
41 | 1, 40 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁)) |
42 | 38, 39, 41 | rspcdva 2830 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
43 | 37 | sumsn 11338 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑁}𝐴 = 𝐵) |
44 | 1, 42, 43 | syl2anc 409 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝑁}𝐴 = 𝐵) |
45 | 44 | oveq2d 5852 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + Σ𝑘 ∈ {𝑁}𝐴) = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵)) |
46 | 36, 45 | eqtrd 2197 | 1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 ∪ cun 3109 ∩ cin 3110 ∅c0 3404 {csn 3570 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 ℂcc 7742 1c1 7745 + caddc 7747 < clt 7924 − cmin 8060 ℤcz 9182 ℤ≥cuz 9457 ...cfz 9935 Σcsu 11280 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-frec 6350 df-1o 6375 df-oadd 6379 df-er 6492 df-en 6698 df-dom 6699 df-fin 6700 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-fz 9936 df-fzo 10068 df-seqfrec 10371 df-exp 10445 df-ihash 10678 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-clim 11206 df-sumdc 11281 |
This theorem is referenced by: fzosump1 11344 fsump1 11347 telfsumo 11393 fsumparts 11397 binom1dif 11414 |
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