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Mirrors > Home > ILE Home > Th. List > fsumsplitsn | GIF version |
Description: Separate out a term in a finite sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
fsumsplitsn.ph | ⊢ Ⅎ𝑘𝜑 |
fsumsplitsn.kd | ⊢ Ⅎ𝑘𝐷 |
fsumsplitsn.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumsplitsn.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
fsumsplitsn.ba | ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴) |
fsumsplitsn.c | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
fsumsplitsn.d | ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷) |
fsumsplitsn.dcn | ⊢ (𝜑 → 𝐷 ∈ ℂ) |
Ref | Expression |
---|---|
fsumsplitsn | ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumsplitsn.ph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | fsumsplitsn.ba | . . . 4 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴) | |
3 | disjsn 3585 | . . . 4 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) | |
4 | 2, 3 | sylibr 133 | . . 3 ⊢ (𝜑 → (𝐴 ∩ {𝐵}) = ∅) |
5 | eqidd 2140 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) = (𝐴 ∪ {𝐵})) | |
6 | fsumsplitsn.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
7 | fsumsplitsn.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
8 | unsnfi 6807 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ ¬ 𝐵 ∈ 𝐴) → (𝐴 ∪ {𝐵}) ∈ Fin) | |
9 | 6, 7, 2, 8 | syl3anc 1216 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ∈ Fin) |
10 | fsumsplitsn.c | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) | |
11 | 10 | adantlr 468 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
12 | fsumsplitsn.d | . . . . . . 7 ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷) | |
13 | 12 | adantl 275 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐷) |
14 | fsumsplitsn.dcn | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
15 | 14 | adantr 274 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐷 ∈ ℂ) |
16 | 13, 15 | eqeltrd 2216 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 ∈ ℂ) |
17 | 16 | adantlr 468 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 = 𝐵) → 𝐶 ∈ ℂ) |
18 | elun 3217 | . . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝐵})) | |
19 | elsni 3545 | . . . . . . 7 ⊢ (𝑘 ∈ {𝐵} → 𝑘 = 𝐵) | |
20 | 19 | orim2i 750 | . . . . . 6 ⊢ ((𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝐵}) → (𝑘 ∈ 𝐴 ∨ 𝑘 = 𝐵)) |
21 | 18, 20 | sylbi 120 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∪ {𝐵}) → (𝑘 ∈ 𝐴 ∨ 𝑘 = 𝐵)) |
22 | 21 | adantl 275 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) → (𝑘 ∈ 𝐴 ∨ 𝑘 = 𝐵)) |
23 | 11, 17, 22 | mpjaodan 787 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) → 𝐶 ∈ ℂ) |
24 | 1, 4, 5, 9, 23 | fsumsplitf 11177 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ {𝐵}𝐶)) |
25 | fsumsplitsn.kd | . . . . 5 ⊢ Ⅎ𝑘𝐷 | |
26 | 25, 12 | sumsnf 11178 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ ℂ) → Σ𝑘 ∈ {𝐵}𝐶 = 𝐷) |
27 | 7, 14, 26 | syl2anc 408 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐵}𝐶 = 𝐷) |
28 | 27 | oveq2d 5790 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ {𝐵}𝐶) = (Σ𝑘 ∈ 𝐴 𝐶 + 𝐷)) |
29 | 24, 28 | eqtrd 2172 | 1 ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 697 = wceq 1331 Ⅎwnf 1436 ∈ wcel 1480 Ⅎwnfc 2268 ∪ cun 3069 ∩ cin 3070 ∅c0 3363 {csn 3527 (class class class)co 5774 Fincfn 6634 ℂcc 7618 + caddc 7623 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: fsumrelem 11240 trilpolemeq1 13233 |
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