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Mirrors > Home > MPE Home > Th. List > serclim0 | Structured version Visualization version GIF version |
Description: The zero series converges to zero. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 31-Jan-2014.) |
Ref | Expression |
---|---|
serclim0 | ⊢ (𝑀 ∈ ℤ → seq𝑀( + , ((ℤ≥‘𝑀) × {0})) ⇝ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . 3 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
2 | 1 | ser0f 13419 | . 2 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , ((ℤ≥‘𝑀) × {0})) = ((ℤ≥‘𝑀) × {0})) |
3 | 0cn 10622 | . . 3 ⊢ 0 ∈ ℂ | |
4 | ssid 3937 | . . . 4 ⊢ (ℤ≥‘𝑀) ⊆ (ℤ≥‘𝑀) | |
5 | fvex 6658 | . . . 4 ⊢ (ℤ≥‘𝑀) ∈ V | |
6 | 4, 5 | climconst2 14897 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑀 ∈ ℤ) → ((ℤ≥‘𝑀) × {0}) ⇝ 0) |
7 | 3, 6 | mpan 689 | . 2 ⊢ (𝑀 ∈ ℤ → ((ℤ≥‘𝑀) × {0}) ⇝ 0) |
8 | 2, 7 | eqbrtrd 5052 | 1 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , ((ℤ≥‘𝑀) × {0})) ⇝ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 {csn 4525 class class class wbr 5030 × cxp 5517 ‘cfv 6324 ℂcc 10524 0cc0 10526 + caddc 10529 ℤcz 11969 ℤ≥cuz 12231 seqcseq 13364 ⇝ cli 14833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 |
This theorem is referenced by: iserge0 15009 sum0 15070 sumz 15071 geomulcvg 15224 ovoliunnul 24111 vitalilem5 24216 lgam1 25649 binomcxplemnn0 41053 |
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