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Mirrors > Home > ILE Home > Th. List > fsumcllem | GIF version |
Description: - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 3-Jun-2014.) |
Ref | Expression |
---|---|
fsumcllem.1 | ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
fsumcllem.2 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
fsumcllem.3 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumcllem.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) |
fsumcllem.5 | ⊢ (𝜑 → 0 ∈ 𝑆) |
Ref | Expression |
---|---|
fsumcllem | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐴 = ∅) | |
2 | 1 | sumeq1d 11135 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = ∅) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵) |
3 | sum0 11157 | . . . 4 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0 | |
4 | 2, 3 | syl6eq 2188 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → Σ𝑘 ∈ 𝐴 𝐵 = 0) |
5 | fsumcllem.5 | . . . 4 ⊢ (𝜑 → 0 ∈ 𝑆) | |
6 | 5 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 0 ∈ 𝑆) |
7 | 4, 6 | eqeltrd 2216 | . 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
8 | fsumcllem.1 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ ℂ) | |
9 | 8 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ ∃𝑧 𝑧 ∈ 𝐴) → 𝑆 ⊆ ℂ) |
10 | fsumcllem.2 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
11 | 10 | adantlr 468 | . . 3 ⊢ (((𝜑 ∧ ∃𝑧 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
12 | fsumcllem.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
13 | 12 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ ∃𝑧 𝑧 ∈ 𝐴) → 𝐴 ∈ Fin) |
14 | fsumcllem.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) | |
15 | 14 | adantlr 468 | . . 3 ⊢ (((𝜑 ∧ ∃𝑧 𝑧 ∈ 𝐴) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) |
16 | n0r 3376 | . . . 4 ⊢ (∃𝑧 𝑧 ∈ 𝐴 → 𝐴 ≠ ∅) | |
17 | 16 | adantl 275 | . . 3 ⊢ ((𝜑 ∧ ∃𝑧 𝑧 ∈ 𝐴) → 𝐴 ≠ ∅) |
18 | 9, 11, 13, 15, 17 | fsumcl2lem 11167 | . 2 ⊢ ((𝜑 ∧ ∃𝑧 𝑧 ∈ 𝐴) → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
19 | fin0or 6780 | . . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ∃𝑧 𝑧 ∈ 𝐴)) | |
20 | 12, 19 | syl 14 | . 2 ⊢ (𝜑 → (𝐴 = ∅ ∨ ∃𝑧 𝑧 ∈ 𝐴)) |
21 | 7, 18, 20 | mpjaodan 787 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 697 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ≠ wne 2308 ⊆ wss 3071 ∅c0 3363 (class class class)co 5774 Fincfn 6634 ℂcc 7618 0cc0 7620 + 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: fsumcl 11169 fsumrecl 11170 fsumzcl 11171 fsumnn0cl 11172 fsumge0 11228 |
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