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 2821 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
5 | 4 | expcn 23474 | . . . . . 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 23512 | . . . . 5 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)) |
8 | 6, 7 | eleqtrrdi 2924 | . . . 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 14850 | . . . . 5 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
13 | 1, 2, 8, 9, 10, 12 | climcncf 23502 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘𝐴)) |
14 | eqidd 2822 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) | |
15 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴) | |
16 | 15 | oveq1d 7165 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑥↑𝑁) = (𝐴↑𝑁)) |
17 | 12, 3 | expcld 13504 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
18 | 14, 16, 12, 17 | fvmptd 6769 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘𝐴) = (𝐴↑𝑁)) |
19 | 13, 18 | breqtrd 5084 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ (𝐴↑𝑁)) |
20 | climexp.9 | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑉) | |
21 | cnex 10612 | . . . . 5 ⊢ ℂ ∈ V | |
22 | 21 | mptex 6980 | . . . 4 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ V |
23 | 1 | fvexi 6678 | . . . . 5 ⊢ 𝑍 ∈ V |
24 | fex 6983 | . . . . 5 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑍 ∈ V) → 𝐹 ∈ V) | |
25 | 9, 23, 24 | sylancl 588 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ V) |
26 | coexg 7628 | . . . 4 ⊢ (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ V ∧ 𝐹 ∈ V) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ∈ V) | |
27 | 22, 25, 26 | sylancr 589 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ∈ V) |
28 | eqidd 2822 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) | |
29 | simpr 487 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 = (𝐹‘𝑗)) → 𝑥 = (𝐹‘𝑗)) | |
30 | 29 | oveq1d 7165 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 = (𝐹‘𝑗)) → (𝑥↑𝑁) = ((𝐹‘𝑗)↑𝑁)) |
31 | 9 | ffvelrnda 6845 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ) |
32 | 3 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑁 ∈ ℕ0) |
33 | 31, 32 | expcld 13504 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝐹‘𝑗)↑𝑁) ∈ ℂ) |
34 | 28, 30, 31, 33 | fvmptd 6769 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗)) = ((𝐹‘𝑗)↑𝑁)) |
35 | fvco3 6754 | . . . . 5 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑗 ∈ 𝑍) → (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗))) | |
36 | 9, 35 | sylan 582 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗))) |
37 | climexp.1 | . . . . . . 7 ⊢ Ⅎ𝑘𝜑 | |
38 | nfv 1911 | . . . . . . 7 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
39 | 37, 38 | nfan 1896 | . . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
40 | climexp.3 | . . . . . . . 8 ⊢ Ⅎ𝑘𝐻 | |
41 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑗 | |
42 | 40, 41 | nffv 6674 | . . . . . . 7 ⊢ Ⅎ𝑘(𝐻‘𝑗) |
43 | climexp.2 | . . . . . . . . 9 ⊢ Ⅎ𝑘𝐹 | |
44 | 43, 41 | nffv 6674 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
45 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑘↑ | |
46 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑁 | |
47 | 44, 45, 46 | nfov 7180 | . . . . . . 7 ⊢ Ⅎ𝑘((𝐹‘𝑗)↑𝑁) |
48 | 42, 47 | nfeq 2991 | . . . . . 6 ⊢ Ⅎ𝑘(𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁) |
49 | 39, 48 | nfim 1893 | . . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁)) |
50 | eleq1w 2895 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
51 | 50 | anbi2d 630 | . . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
52 | fveq2 6664 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗)) | |
53 | fveq2 6664 | . . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
54 | 53 | oveq1d 7165 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘)↑𝑁) = ((𝐹‘𝑗)↑𝑁)) |
55 | 52, 54 | eqeq12d 2837 | . . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = ((𝐹‘𝑘)↑𝑁) ↔ (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁))) |
56 | 51, 55 | imbi12d 347 | . . . . 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 2238 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁)) |
59 | 34, 36, 58 | 3eqtr4rd 2867 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗)) |
60 | 1, 20, 27, 2, 59 | climeq 14918 | . 2 ⊢ (𝜑 → (𝐻 ⇝ (𝐴↑𝑁) ↔ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ (𝐴↑𝑁))) |
61 | 19, 60 | mpbird 259 | 1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴↑𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 Ⅎwnfc 2961 Vcvv 3494 class class class wbr 5058 ↦ cmpt 5138 ∘ ccom 5553 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℕ0cn0 11891 ℤcz 11975 ℤ≥cuz 12237 ↑cexp 13423 ⇝ cli 14835 TopOpenctopn 16689 ℂfldccnfld 20539 Cn ccn 21826 –cn→ccncf 23478 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-icc 12739 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cn 21829 df-cnp 21830 df-tx 22164 df-hmeo 22357 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 |
This theorem is referenced by: stirlinglem8 42360 |
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