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Mirrors > Home > ILE Home > Th. List > ser0 | GIF version |
Description: The value of the partial sums in a zero-valued infinite series. (Contributed by Mario Carneiro, 31-Aug-2013.) (Revised by Mario Carneiro, 15-Dec-2014.) |
Ref | Expression |
---|---|
ser0.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
ser0 | ⊢ (𝑁 ∈ 𝑍 → (seq𝑀( + , (𝑍 × {0}))‘𝑁) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 00id 7927 | . . 3 ⊢ (0 + 0) = 0 | |
2 | 1 | a1i 9 | . 2 ⊢ (𝑁 ∈ 𝑍 → (0 + 0) = 0) |
3 | ser0.1 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | 3 | eleq2i 2207 | . . 3 ⊢ (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (ℤ≥‘𝑀)) |
5 | 4 | biimpi 119 | . 2 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ≥‘𝑀)) |
6 | 0cn 7782 | . . 3 ⊢ 0 ∈ ℂ | |
7 | elfzuz 9833 | . . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
8 | 7, 3 | eleqtrrdi 2234 | . . . 4 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ 𝑍) |
9 | 8 | adantl 275 | . . 3 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ 𝑍) |
10 | fvconst2g 5642 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑘 ∈ 𝑍) → ((𝑍 × {0})‘𝑘) = 0) | |
11 | 6, 9, 10 | sylancr 411 | . 2 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑍 × {0})‘𝑘) = 0) |
12 | 0cnd 7783 | . 2 ⊢ (𝑁 ∈ 𝑍 → 0 ∈ ℂ) | |
13 | 3 | eleq2i 2207 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (ℤ≥‘𝑀)) |
14 | 13 | biimpri 132 | . . . . 5 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ 𝑍) |
15 | 14 | adantl 275 | . . . 4 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ 𝑍) |
16 | 6, 15, 10 | sylancr 411 | . . 3 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝑍 × {0})‘𝑘) = 0) |
17 | 16, 6 | eqeltrdi 2231 | . 2 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝑍 × {0})‘𝑘) ∈ ℂ) |
18 | addcl 7769 | . . 3 ⊢ ((𝑘 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑘 + 𝑣) ∈ ℂ) | |
19 | 18 | adantl 275 | . 2 ⊢ ((𝑁 ∈ 𝑍 ∧ (𝑘 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑘 + 𝑣) ∈ ℂ) |
20 | 2, 5, 11, 12, 17, 19 | seq3id3 10311 | 1 ⊢ (𝑁 ∈ 𝑍 → (seq𝑀( + , (𝑍 × {0}))‘𝑁) = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 {csn 3532 × cxp 4545 ‘cfv 5131 (class class class)co 5782 ℂcc 7642 0cc0 7644 + caddc 7647 ℤ≥cuz 9350 ...cfz 9821 seqcseq 10249 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 df-uz 9351 df-fz 9822 df-fzo 9951 df-seqfrec 10250 |
This theorem is referenced by: ser0f 10319 isumz 11190 |
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