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 5164 | . . 3 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑍 ↦ -1) | |
3 | climneg.2 | . . 3 ⊢ Ⅎ𝑘𝐹 | |
4 | nfmpt1 5164 | . . 3 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) | |
5 | climneg.3 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | climneg.4 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | 5 | fvexi 6684 | . . . . . 6 ⊢ 𝑍 ∈ V |
8 | 7 | mptex 6986 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 ↦ -1) ∈ V |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -1) ∈ V) |
10 | 1cnd 10636 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
11 | 10 | negcld 10984 | . . . 4 ⊢ (𝜑 → -1 ∈ ℂ) |
12 | eqidd 2822 | . . . . . 6 ⊢ (𝑗 ∈ 𝑍 → (𝑘 ∈ 𝑍 ↦ -1) = (𝑘 ∈ 𝑍 ↦ -1)) | |
13 | eqidd 2822 | . . . . . 6 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 = 𝑗) → -1 = -1) | |
14 | id 22 | . . . . . 6 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ 𝑍) | |
15 | 1cnd 10636 | . . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → 1 ∈ ℂ) | |
16 | 15 | negcld 10984 | . . . . . 6 ⊢ (𝑗 ∈ 𝑍 → -1 ∈ ℂ) |
17 | 12, 13, 14, 16 | fvmptd 6775 | . . . . 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 14900 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -1) ⇝ -1) |
20 | 7 | mptex 6986 | . . . 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 11752 | . . . . . 6 ⊢ -1 ∈ ℂ | |
24 | eqid 2821 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↦ -1) = (𝑘 ∈ 𝑍 ↦ -1) | |
25 | 24 | fvmpt2 6779 | . . . . . 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 2921 | . . . 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 10984 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -(𝐹‘𝑘) ∈ ℂ) |
32 | eqid 2821 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) = (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) | |
33 | 32 | fvmpt2 6779 | . . . . 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 11092 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (-1 · (𝐹‘𝑘)) = -(𝐹‘𝑘)) |
36 | 26 | eqcomd 2827 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 → -1 = ((𝑘 ∈ 𝑍 ↦ -1)‘𝑘)) |
37 | 36 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -1 = ((𝑘 ∈ 𝑍 ↦ -1)‘𝑘)) |
38 | 37 | oveq1d 7171 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (-1 · (𝐹‘𝑘)) = (((𝑘 ∈ 𝑍 ↦ -1)‘𝑘) · (𝐹‘𝑘))) |
39 | 34, 35, 38 | 3eqtr2d 2862 | . . 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 41905 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ (-1 · 𝐴)) |
41 | climcl 14856 | . . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
42 | 22, 41 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
43 | 42 | mulm1d 11092 | . 2 ⊢ (𝜑 → (-1 · 𝐴) = -𝐴) |
44 | 40, 43 | breqtrd 5092 | 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 2961 Vcvv 3494 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 1c1 10538 · cmul 10542 -cneg 10871 ℤcz 11982 ℤ≥cuz 12244 ⇝ cli 14841 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 |
This theorem is referenced by: climliminflimsupd 42102 |
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