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 2737 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
5 | 4 | expcn 23769 | . . . . . 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 23808 | . . . . 5 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)) |
8 | 6, 7 | eleqtrrdi 2849 | . . . 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 15060 | . . . . 5 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
13 | 1, 2, 8, 9, 10, 12 | climcncf 23797 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘𝐴)) |
14 | eqidd 2738 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) | |
15 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴) | |
16 | 15 | oveq1d 7228 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑥↑𝑁) = (𝐴↑𝑁)) |
17 | 12, 3 | expcld 13716 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
18 | 14, 16, 12, 17 | fvmptd 6825 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘𝐴) = (𝐴↑𝑁)) |
19 | 13, 18 | breqtrd 5079 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ (𝐴↑𝑁)) |
20 | climexp.9 | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑉) | |
21 | cnex 10810 | . . . . 5 ⊢ ℂ ∈ V | |
22 | 21 | mptex 7039 | . . . 4 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ V |
23 | 1 | fvexi 6731 | . . . . 5 ⊢ 𝑍 ∈ V |
24 | fex 7042 | . . . . 5 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑍 ∈ V) → 𝐹 ∈ V) | |
25 | 9, 23, 24 | sylancl 589 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ V) |
26 | coexg 7707 | . . . 4 ⊢ (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ V ∧ 𝐹 ∈ V) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ∈ V) | |
27 | 22, 25, 26 | sylancr 590 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ∈ V) |
28 | eqidd 2738 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) | |
29 | simpr 488 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 = (𝐹‘𝑗)) → 𝑥 = (𝐹‘𝑗)) | |
30 | 29 | oveq1d 7228 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 = (𝐹‘𝑗)) → (𝑥↑𝑁) = ((𝐹‘𝑗)↑𝑁)) |
31 | 9 | ffvelrnda 6904 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ) |
32 | 3 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑁 ∈ ℕ0) |
33 | 31, 32 | expcld 13716 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝐹‘𝑗)↑𝑁) ∈ ℂ) |
34 | 28, 30, 31, 33 | fvmptd 6825 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗)) = ((𝐹‘𝑗)↑𝑁)) |
35 | fvco3 6810 | . . . . 5 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑗 ∈ 𝑍) → (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗))) | |
36 | 9, 35 | sylan 583 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑗))) |
37 | climexp.1 | . . . . . . 7 ⊢ Ⅎ𝑘𝜑 | |
38 | nfv 1922 | . . . . . . 7 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
39 | 37, 38 | nfan 1907 | . . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
40 | climexp.3 | . . . . . . . 8 ⊢ Ⅎ𝑘𝐻 | |
41 | nfcv 2904 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑗 | |
42 | 40, 41 | nffv 6727 | . . . . . . 7 ⊢ Ⅎ𝑘(𝐻‘𝑗) |
43 | climexp.2 | . . . . . . . . 9 ⊢ Ⅎ𝑘𝐹 | |
44 | 43, 41 | nffv 6727 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
45 | nfcv 2904 | . . . . . . . 8 ⊢ Ⅎ𝑘↑ | |
46 | nfcv 2904 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑁 | |
47 | 44, 45, 46 | nfov 7243 | . . . . . . 7 ⊢ Ⅎ𝑘((𝐹‘𝑗)↑𝑁) |
48 | 42, 47 | nfeq 2917 | . . . . . 6 ⊢ Ⅎ𝑘(𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁) |
49 | 39, 48 | nfim 1904 | . . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁)) |
50 | eleq1w 2820 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
51 | 50 | anbi2d 632 | . . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
52 | fveq2 6717 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗)) | |
53 | fveq2 6717 | . . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
54 | 53 | oveq1d 7228 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘)↑𝑁) = ((𝐹‘𝑗)↑𝑁)) |
55 | 52, 54 | eqeq12d 2753 | . . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = ((𝐹‘𝑘)↑𝑁) ↔ (𝐻‘𝑗) = ((𝐹‘𝑗)↑𝑁))) |
56 | 51, 55 | imbi12d 348 | . . . . 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 2788 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)‘𝑗)) |
60 | 1, 20, 27, 2, 59 | climeq 15128 | . 2 ⊢ (𝜑 → (𝐻 ⇝ (𝐴↑𝑁) ↔ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ⇝ (𝐴↑𝑁))) |
61 | 19, 60 | mpbird 260 | 1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴↑𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 Ⅎwnfc 2884 Vcvv 3408 class class class wbr 5053 ↦ cmpt 5135 ∘ ccom 5555 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 ℕ0cn0 12090 ℤcz 12176 ℤ≥cuz 12438 ↑cexp 13635 ⇝ cli 15045 TopOpenctopn 16926 ℂfldccnfld 20363 Cn ccn 22121 –cn→ccncf 23773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-mulf 10809 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-icc 12942 df-fz 13096 df-fzo 13239 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cn 22124 df-cnp 22125 df-tx 22459 df-hmeo 22652 df-xms 23218 df-ms 23219 df-tms 23220 df-cncf 23775 |
This theorem is referenced by: stirlinglem8 43297 |
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