Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climneg | Structured version Visualization version GIF version |
Description: Complex limit of the negative of a sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
climneg.1 | ⊢ Ⅎ𝑘𝜑 |
climneg.2 | ⊢ Ⅎ𝑘𝐹 |
climneg.3 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climneg.4 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climneg.5 | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
climneg.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
Ref | Expression |
---|---|
climneg | ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ -𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climneg.1 | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | nfmpt1 5166 | . . 3 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑍 ↦ -1) | |
3 | climneg.2 | . . 3 ⊢ Ⅎ𝑘𝐹 | |
4 | nfmpt1 5166 | . . 3 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) | |
5 | climneg.3 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | climneg.4 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | 5 | fvexi 6686 | . . . . . 6 ⊢ 𝑍 ∈ V |
8 | 7 | mptex 6988 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 ↦ -1) ∈ V |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -1) ∈ V) |
10 | 1cnd 10638 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
11 | 10 | negcld 10986 | . . . 4 ⊢ (𝜑 → -1 ∈ ℂ) |
12 | eqidd 2824 | . . . . . 6 ⊢ (𝑗 ∈ 𝑍 → (𝑘 ∈ 𝑍 ↦ -1) = (𝑘 ∈ 𝑍 ↦ -1)) | |
13 | eqidd 2824 | . . . . . 6 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 = 𝑗) → -1 = -1) | |
14 | id 22 | . . . . . 6 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ 𝑍) | |
15 | 1cnd 10638 | . . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → 1 ∈ ℂ) | |
16 | 15 | negcld 10986 | . . . . . 6 ⊢ (𝑗 ∈ 𝑍 → -1 ∈ ℂ) |
17 | 12, 13, 14, 16 | fvmptd 6777 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ -1)‘𝑗) = -1) |
18 | 17 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ -1)‘𝑗) = -1) |
19 | 5, 6, 9, 11, 18 | climconst 14902 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -1) ⇝ -1) |
20 | 7 | mptex 6988 | . . . 4 ⊢ (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ∈ V |
21 | 20 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ∈ V) |
22 | climneg.5 | . . 3 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
23 | neg1cn 11754 | . . . . . 6 ⊢ -1 ∈ ℂ | |
24 | eqid 2823 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↦ -1) = (𝑘 ∈ 𝑍 ↦ -1) | |
25 | 24 | fvmpt2 6781 | . . . . . 6 ⊢ ((𝑘 ∈ 𝑍 ∧ -1 ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ -1)‘𝑘) = -1) |
26 | 23, 25 | mpan2 689 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ -1)‘𝑘) = -1) |
27 | 26, 23 | eqeltrdi 2923 | . . . 4 ⊢ (𝑘 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ -1)‘𝑘) ∈ ℂ) |
28 | 27 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ -1)‘𝑘) ∈ ℂ) |
29 | climneg.6 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
30 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍) | |
31 | 29 | negcld 10986 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -(𝐹‘𝑘) ∈ ℂ) |
32 | eqid 2823 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) = (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) | |
33 | 32 | fvmpt2 6781 | . . . . 5 ⊢ ((𝑘 ∈ 𝑍 ∧ -(𝐹‘𝑘) ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))‘𝑘) = -(𝐹‘𝑘)) |
34 | 30, 31, 33 | syl2anc 586 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))‘𝑘) = -(𝐹‘𝑘)) |
35 | 29 | mulm1d 11094 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (-1 · (𝐹‘𝑘)) = -(𝐹‘𝑘)) |
36 | 26 | eqcomd 2829 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 → -1 = ((𝑘 ∈ 𝑍 ↦ -1)‘𝑘)) |
37 | 36 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -1 = ((𝑘 ∈ 𝑍 ↦ -1)‘𝑘)) |
38 | 37 | oveq1d 7173 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (-1 · (𝐹‘𝑘)) = (((𝑘 ∈ 𝑍 ↦ -1)‘𝑘) · (𝐹‘𝑘))) |
39 | 34, 35, 38 | 3eqtr2d 2864 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))‘𝑘) = (((𝑘 ∈ 𝑍 ↦ -1)‘𝑘) · (𝐹‘𝑘))) |
40 | 1, 2, 3, 4, 5, 6, 19, 21, 22, 28, 29, 39 | climmulf 41892 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ (-1 · 𝐴)) |
41 | climcl 14858 | . . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
42 | 22, 41 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
43 | 42 | mulm1d 11094 | . 2 ⊢ (𝜑 → (-1 · 𝐴) = -𝐴) |
44 | 40, 43 | breqtrd 5094 | 1 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ -𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2963 Vcvv 3496 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 1c1 10540 · cmul 10544 -cneg 10873 ℤcz 11984 ℤ≥cuz 12246 ⇝ cli 14843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 |
This theorem is referenced by: climliminflimsupd 42089 |
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