Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climexp | Structured version Visualization version GIF version |
Description: The limit of natural powers, is the natural power of the limit. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
climexp.1 | ⊢ Ⅎ𝑘𝜑 |
climexp.2 | ⊢ Ⅎ𝑘𝐹 |
climexp.3 | ⊢ Ⅎ𝑘𝐻 |
climexp.4 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climexp.5 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climexp.6 | ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) |
climexp.7 | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
climexp.8 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
climexp.9 | ⊢ (𝜑 → 𝐻 ∈ 𝑉) |
climexp.10 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘)↑𝑁)) |
Ref | Expression |
---|---|
climexp | ⊢ (𝜑 → 𝐻 ⇝ (𝐴↑𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climexp.4 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climexp.5 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | climexp.8 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
4 | eqid 2818 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
5 | 4 | expcn 23407 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
6 | 3, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
7 | 4 | cncfcn1 23445 | . . . . 5 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)) |
8 | 6, 7 | eleqtrrdi 2921 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ)) |
9 | climexp.6 | . . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) | |
10 | climexp.7 | . . . 4 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
11 | climcl 14844 | . . . . 5 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
13 | 1, 2, 8, 9, 10, 12 | climcncf 23435 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘𝐴)) |
14 | eqidd 2819 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) | |
15 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴) | |
16 | 15 | oveq1d 7160 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑥↑𝑁) = (𝐴↑𝑁)) |
17 | 12, 3 | expcld 13498 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
18 | 14, 16, 12, 17 | fvmptd 6767 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘𝐴) = (𝐴↑𝑁)) |
19 | 13, 18 | breqtrd 5083 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ (𝐴↑𝑁)) |
20 | climexp.9 | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑉) | |
21 | cnex 10606 | . . . . 5 ⊢ ℂ ∈ V | |
22 | 21 | mptex 6977 | . . . 4 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ V |
23 | 1 | fvexi 6677 | . . . . 5 ⊢ 𝑍 ∈ V |
24 | fex 6980 | . . . . 5 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑍 ∈ V) → 𝐹 ∈ V) | |
25 | 9, 23, 24 | sylancl 586 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ V) |
26 | coexg 7623 | . . . 4 ⊢ (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ V ∧ 𝐹 ∈ V) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ∈ V) | |
27 | 22, 25, 26 | sylancr 587 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ∈ V) |
28 | eqidd 2819 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) | |
29 | simpr 485 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 = (𝐹‘𝑗)) → 𝑥 = (𝐹‘𝑗)) | |
30 | 29 | oveq1d 7160 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 = (𝐹‘𝑗)) → (𝑥↑𝑁) = ((𝐹‘𝑗)↑𝑁)) |
31 | 9 | ffvelrnda 6843 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ) |
32 | 3 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑁 ∈ ℕ0) |
33 | 31, 32 | expcld 13498 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝐹‘𝑗)↑𝑁) ∈ ℂ) |
34 | 28, 30, 31, 33 | fvmptd 6767 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗)) = ((𝐹‘𝑗)↑𝑁)) |
35 | fvco3 6753 | . . . . 5 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑗 ∈ 𝑍) → (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗))) | |
36 | 9, 35 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗))) |
37 | climexp.1 | . . . . . . 7 ⊢ Ⅎ𝑘𝜑 | |
38 | nfv 1906 | . . . . . . 7 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
39 | 37, 38 | nfan 1891 | . . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
40 | climexp.3 | . . . . . . . 8 ⊢ Ⅎ𝑘𝐻 | |
41 | nfcv 2974 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑗 | |
42 | 40, 41 | nffv 6673 | . . . . . . 7 ⊢ Ⅎ𝑘(𝐻‘𝑗) |
43 | climexp.2 | . . . . . . . . 9 ⊢ Ⅎ𝑘𝐹 | |
44 | 43, 41 | nffv 6673 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
45 | nfcv 2974 | . . . . . . . 8 ⊢ Ⅎ𝑘↑ | |
46 | nfcv 2974 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑁 | |
47 | 44, 45, 46 | nfov 7175 | . . . . . . 7 ⊢ Ⅎ𝑘((𝐹‘𝑗)↑𝑁) |
48 | 42, 47 | nfeq 2988 | . . . . . 6 ⊢ Ⅎ𝑘(𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁) |
49 | 39, 48 | nfim 1888 | . . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁)) |
50 | eleq1w 2892 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
51 | 50 | anbi2d 628 | . . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
52 | fveq2 6663 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗)) | |
53 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
54 | 53 | oveq1d 7160 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘)↑𝑁) = ((𝐹‘𝑗)↑𝑁)) |
55 | 52, 54 | eqeq12d 2834 | . . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = ((𝐹‘𝑘)↑𝑁) ↔ (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁))) |
56 | 51, 55 | imbi12d 346 | . . . . 5 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘)↑𝑁)) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁)))) |
57 | climexp.10 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘)↑𝑁)) | |
58 | 49, 56, 57 | chvarfv 2232 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁)) |
59 | 34, 36, 58 | 3eqtr4rd 2864 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗)) |
60 | 1, 20, 27, 2, 59 | climeq 14912 | . 2 ⊢ (𝜑 → (𝐻 ⇝ (𝐴↑𝑁) ↔ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ (𝐴↑𝑁))) |
61 | 19, 60 | mpbird 258 | 1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴↑𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 Ⅎwnfc 2958 Vcvv 3492 class class class wbr 5057 ↦ cmpt 5137 ∘ ccom 5552 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℕ0cn0 11885 ℤcz 11969 ℤ≥cuz 12231 ↑cexp 13417 ⇝ cli 14829 TopOpenctopn 16683 ℂfldccnfld 20473 Cn ccn 21760 –cn→ccncf 23411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-icc 12733 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cn 21763 df-cnp 21764 df-tx 22098 df-hmeo 22291 df-xms 22857 df-ms 22858 df-tms 22859 df-cncf 23413 |
This theorem is referenced by: stirlinglem8 42243 |
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