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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 5256 | . . 3 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑍 ↦ -1) | |
3 | climneg.2 | . . 3 ⊢ Ⅎ𝑘𝐹 | |
4 | nfmpt1 5256 | . . 3 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) | |
5 | climneg.3 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | climneg.4 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | 5 | fvexi 6921 | . . . . . 6 ⊢ 𝑍 ∈ V |
8 | 7 | mptex 7243 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 ↦ -1) ∈ V |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -1) ∈ V) |
10 | 1cnd 11254 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
11 | 10 | negcld 11605 | . . . 4 ⊢ (𝜑 → -1 ∈ ℂ) |
12 | eqidd 2736 | . . . . . 6 ⊢ (𝑗 ∈ 𝑍 → (𝑘 ∈ 𝑍 ↦ -1) = (𝑘 ∈ 𝑍 ↦ -1)) | |
13 | eqidd 2736 | . . . . . 6 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 = 𝑗) → -1 = -1) | |
14 | id 22 | . . . . . 6 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ 𝑍) | |
15 | 1cnd 11254 | . . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → 1 ∈ ℂ) | |
16 | 15 | negcld 11605 | . . . . . 6 ⊢ (𝑗 ∈ 𝑍 → -1 ∈ ℂ) |
17 | 12, 13, 14, 16 | fvmptd 7023 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ -1)‘𝑗) = -1) |
18 | 17 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ -1)‘𝑗) = -1) |
19 | 5, 6, 9, 11, 18 | climconst 15576 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -1) ⇝ -1) |
20 | 7 | mptex 7243 | . . . 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 12378 | . . . . . 6 ⊢ -1 ∈ ℂ | |
24 | eqid 2735 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↦ -1) = (𝑘 ∈ 𝑍 ↦ -1) | |
25 | 24 | fvmpt2 7027 | . . . . . 6 ⊢ ((𝑘 ∈ 𝑍 ∧ -1 ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ -1)‘𝑘) = -1) |
26 | 23, 25 | mpan2 691 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ -1)‘𝑘) = -1) |
27 | 26, 23 | eqeltrdi 2847 | . . . 4 ⊢ (𝑘 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ -1)‘𝑘) ∈ ℂ) |
28 | 27 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ -1)‘𝑘) ∈ ℂ) |
29 | climneg.6 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
30 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍) | |
31 | 29 | negcld 11605 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -(𝐹‘𝑘) ∈ ℂ) |
32 | eqid 2735 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) = (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) | |
33 | 32 | fvmpt2 7027 | . . . . 5 ⊢ ((𝑘 ∈ 𝑍 ∧ -(𝐹‘𝑘) ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))‘𝑘) = -(𝐹‘𝑘)) |
34 | 30, 31, 33 | syl2anc 584 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))‘𝑘) = -(𝐹‘𝑘)) |
35 | 29 | mulm1d 11713 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (-1 · (𝐹‘𝑘)) = -(𝐹‘𝑘)) |
36 | 26 | eqcomd 2741 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 → -1 = ((𝑘 ∈ 𝑍 ↦ -1)‘𝑘)) |
37 | 36 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -1 = ((𝑘 ∈ 𝑍 ↦ -1)‘𝑘)) |
38 | 37 | oveq1d 7446 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (-1 · (𝐹‘𝑘)) = (((𝑘 ∈ 𝑍 ↦ -1)‘𝑘) · (𝐹‘𝑘))) |
39 | 34, 35, 38 | 3eqtr2d 2781 | . . 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 45560 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ (-1 · 𝐴)) |
41 | climcl 15532 | . . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
42 | 22, 41 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
43 | 42 | mulm1d 11713 | . 2 ⊢ (𝜑 → (-1 · 𝐴) = -𝐴) |
44 | 40, 43 | breqtrd 5174 | 1 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ -𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 Vcvv 3478 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 1c1 11154 · cmul 11158 -cneg 11491 ℤcz 12611 ℤ≥cuz 12876 ⇝ cli 15517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 |
This theorem is referenced by: climliminflimsupd 45757 |
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